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In [[algebraic topology]], a '''Steenrod algebra''' was defined by {{harvtxt|Cartan|1955}} to be the algebra of stable [[cohomology operation]]s for mod ''p'' cohomology. | |||
For a given [[prime number]] ''p'', the Steenrod algebra ''A''<sub>''p''</sub> is the graded [[Hopf algebra]] over the field '''F'''<sub>''p''</sub> of order ''p'', consisting of all stable [[cohomology operation]]s for mod ''p'' [[cohomology]]. It is generated by the '''Steenrod squares''' introduced by {{harvtxt|Steenrod|1947}} for ''p''=2, and by the '''Steenrod reduced ''p''th powers''' introduced in {{harvtxt|Steenrod|1953}} and the [[Bockstein homomorphism]] for ''p''>2. | |||
The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a [[generalized cohomology theory]]. | |||
==Cohomology operations== | |||
A cohomology operation is a [[natural transformation]] between cohomology functors. For example, if we take cohomology with coefficients in a [[ring (mathematics)|ring]], the [[cup product]] squaring operation yields a family of cohomology operations: | |||
:<math>H^n(X;R) \to H^{2n}(X;R)</math> | |||
:<math>x \mapsto x \smile x.</math> | |||
Cohomology operations need not be homomorphisms of graded rings, see the Cartan formula below. | |||
These operations do not commute with [[suspension (topology)|suspension]], that is they are unstable. (This is because if ''Y'' is a suspension of a space ''X'', the cup product on the cohomology of ''Y'' is trivial.) [[Norman Steenrod]] constructed stable operations | |||
:<math>Sq^i \colon H^n(X;\mathbf{Z}/2) \to H^{n+i}(X;\mathbf{Z}/2)</math> | |||
:<math>P^i \colon H^n(X;\mathbf{Z}/p) \to H^{n+2i(p-1)}(X;\mathbf{Z}/p)</math> | |||
for all ''i'' greater than zero. The notation ''Sq'' and their name, the Steenrod squares, comes from the fact that ''Sq''<sup>''n''</sup> restricted to classes of degree ''n'' is the cup square. There are analogous operations for odd primary coefficients, usually denoted ''P''<sup>''i''</sup> and called the reduced ''p''-th power operations. The ''Sq''<sup>''i''</sup> generate a connected graded algebra over ''Z/2'', where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case ''p'' > 2, the mod ''p'' Steenrod algebra is generated by the ''P''<sup>''i''</sup> and the [[Bockstein operation]] β associated to the [[short exact sequence]] | |||
:<math>0 \to \mathbf{Z}/p \to \mathbf{Z}/p^2 \to \mathbf{Z}/p \to 0.</math> | |||
In the case ''p''=2, the Bockstein element is ''Sq''<sup>1</sup> and the reduced ''p''-th power ''P''<sup>''i''</sup> is ''Sq''<sup>2''i''</sup>. | |||
==Axiomatic characterization== | |||
{{harvtxt|Steenrod|Epstein|1962}} showed that the Steenrod squares ''Sq''<sup>''n''</sup>:H<sup>''m''</sup>→H<sup>''m''+''n''</sup> are characterized by the following 5 axioms: | |||
#Naturality: ''Sq''<sup>''n''</sup> is an additive homomorphism from H<sup>''m''</sup>(''X'','''Z'''/2'''Z''') to H<sup>''m''+''n''</sup>(''X'','''Z'''/2'''Z'''), and is natural meaning that for any map ''f'' : ''X'' → ''Y'', ''f''*(''Sq''<sup>''n''</sup>''x'') = ''Sq''<sup>''n''</sup>''f''*(x). | |||
#''Sq''<sup>0</sup> is the identity homomorphism. | |||
#''Sq''<sup>''n''</sup> is the cup square on classes of degree ''n''. | |||
#If ''n''>dim(''X'') then ''Sq''<sup>''n''</sup>(''x'') = 0 | |||
#Cartan Formula:<math>Sq^n(x \smile y) = \sum_{i+j=n} (Sq^i x) \smile (Sq^j y)</math> | |||
In addition the Steenrod squares have the following properties: | |||
*''Sq''<sup>1</sup> is the Bockstein homomorphism of the exact sequence <math>0 \to \mathbf{Z}/2 \to \mathbf{Z}/4 \to \mathbf{Z}/2 \to 0.</math> | |||
*They satisfy the Adem relations, described below. | |||
*They commute with the suspension homomorphism and the boundary operator. | |||
Similarly the following axioms characterize the reduced ''p''-th powers for ''p'' > 2. | |||
#Naturality: ''P''<sup>''n''</sup> is an additive homomorphism from H<sup>''m''</sup>(''X'','''Z'''/''p'''''Z''') to H<sup>''m''+2''n''(''p''−1)</sup>(''X'','''Z'''/''p'''''Z'''), and is natural. | |||
#''P''<sup>0</sup> is the identity homomorphism. | |||
#''P''<sup>''n''</sup> is the cup ''p''-th power on classes of degree 2''n''. | |||
#If 2''n''>dim(''X'') then ''P''<sup>''n''</sup>(''x'') = 0 | |||
#Cartan Formula:<math>P^n(x \smile y) = \sum_{i+j=n} (P^i x) \smile (P^j y)</math> | |||
As before, the reduced ''p''-th powers also satisfy Adem relations and commute with the suspension and boundary operators. | |||
==Adem relations== | |||
The Adem relations for ''p''=2 were conjectured by {{harvtxt|Wu|1952}} and proved by {{harvs|txt|authorlink=José Adem|first=José|last=Adem|year=1952}} and are given by | |||
:<math>Sq^i Sq^j = \sum_{k=0}^{[i/2]} {j-k-1 \choose i-2k} Sq^{i+j-k} Sq^k</math> | |||
for all ''i'', ''j'' > 0 such that ''i'' < 2''j''. (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre-Cartan basis elements. | |||
For odd ''p'' the Adem relations are | |||
:<math>P^{a}P^{b} = \sum_i (-1)^{a+i}{(p-1)(b-i)-1 \choose a-pi} P^{a+b-i}P^i</math> | |||
for ''a''<''pb'' and | |||
:<math>P^{a}\beta P^{b} = \sum_i (-1)^{a+i}{(p-1)(b-i) \choose a-pi} \beta P^{a+b-i}P^i+ | |||
\sum_i (-1)^{a+i+1}{(p-1)(b-i)-1 \choose a-pi-1} P^{a+b-i}\beta P^i</math> | |||
for ''a''≤''pb'' | |||
===Bullett–Macdonald identities=== | |||
{{harvtxt|Bullett|Macdonald|1982}} reformulated the Adem relations as the following Bullett–Macdonald identities. | |||
For ''p''=2 put | |||
:<math>P(t)=\sum_{i\geq 0}t^i\text{Sq}^i</math> | |||
then the Adem relations are equivalent to | |||
:<math>P(s^2+st)\cdot P(t^2)=P(t^2+st)\cdot P(s^2)</math> | |||
For ''p''>2 put | |||
:<math>P(t)=\sum_{i\geq 0}t^i\text{P}^i</math> | |||
then the Adem relations are equivalent to the statement that | |||
:<math> (1+s\text{Ad} \beta)P(t^p+t^{p-1}s+\cdots+ts^{p-1})P(s^p)</math> | |||
is symmetric in ''s'' and ''t''. Here β is the Bockstein operation and (Ad β)''P'' = β''P''−''P''β. | |||
==Construction== | |||
Suppose that π is any degree ''n'' subgroup of the symmetric group on ''n'' points, ''u'' a cohomology class in H<sup>''q''</sup>(''X'',''B''), ''A'' an abelian group acted on by π, and ''c'' a cohomology class in H<sub>''i''</sub>(π,''A''). | |||
{{harvtxt|Steenrod|1953}} showed how to construct a reduced power ''u''<sup>''n''</sup>/''c'' in H<sup>''kq−''i''</sup>(''X'',(''A''⊗''B''⊗''B''⊗...⊗''B'')/π) as follows. | |||
# Taking the external product of ''u'' with itself ''n'' times gives an equivariant cocycle on ''X''<sup>''n''</sup> with coefficients in ''B''⊗''B''⊗...⊗''B''. | |||
#Choose ''E'' to be a contractible space on which π acts freely and an equivariant map from ''E''× ''X'' to ''X''<sup>''n''</sup>. Pulling back ''u''<sup>''n''</sup> by this map gives an equivariant cocyle on ''E''× ''X'' and therefore a cocycle of ''E''/π×''X'' with coefficients in ''B''⊗''B''⊗...⊗''B''. | |||
#Taking a slant product with ''c'' in H<sub>''i''</sub>(''E''/π,''A'')gives a cocycle of ''X'' with coefficients in H<sub>0</sub>(π,''A''⊗''B''⊗''B''⊗...⊗''B'') | |||
The Steenrod squares and reduced powers are special cases of this construction where π is a cyclic group of prime order ''p''=''n'' acting as a cyclic permutation of ''n'' elements, and the groups ''A'' and ''B'' are cyclic of order ''p'', so that H<sub>0</sub>(π,''A''⊗''B''⊗''B''⊗...⊗''B'') is also cyclic of order ''p''. | |||
==The structure of the Steenrod algebra== | |||
{{harvs|txt|last=Serre|year=1953}} (for ''p''=2) and {{harvs|txt|authorlink=Henri Cartan|last=Cartan|year=1954|year2=1955}} (for ''p''>2) described the structure of the Steenrod algebra of stable mod ''p'' cohomology operations, showing that it is generated by the Bockstein homomorphism together with the Steenrod reduced powers, and the Adem relations generate the ideal of relations between these generators. In particular they found an explicit basis for the Steenrod algebra. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence | |||
:<math>i_1, i_2, \ldots, i_n</math> | |||
is admissible if for each ''j'', ''i''<sub>''j''</sub> ≥ 2''i''<sub>''j''+1</sub>. Then the elements | |||
:<math>Sq^I = Sq^{i_1} \ldots Sq^{i_n},</math> | |||
where ''I'' is an admissible sequence, form a basis (the Serre-Cartan basis) for the mod 2 Steenrod algebra. There is a similar basis for the case ''p'' > 2 consisting of the elements | |||
:<math>Sq_p^I = Sq_p^{i_1} \ldots Sq_p^{i_n},</math> | |||
such that | |||
:<math>i_j\ge pi_{j+1}</math> | |||
:<math>i_j\equiv 0,1\bmod 2(p-1)</math> | |||
:<math>Sq_p^{2k(p-1)} = P^k</math> | |||
:<math>Sq_p^{2k(p-1)+1} = \beta P^k</math> | |||
==Hopf algebra structure and the Milnor basis== | |||
The Steenrod algebra has more structure than a graded '''F'''<sub>p</sub>-algebra. It is also a [[Hopf algebra]], so that in particular there is a diagonal or [[comultiplication]] map | |||
:<math>\psi \colon A \to A \otimes A.</math> | |||
induced by the Cartan formula for the action of the Steenrod algebra on the cup product. | |||
It is easier to describe than the product map, and is given by | |||
:<math>\psi(Sq^k) = \sum_{i+j=k} Sq^i \otimes Sq^j</math> | |||
:<math>\psi(P^k) = \sum_{i+j=k} P^i \otimes P^j</math> | |||
:<math>\psi(\beta) = \beta\otimes1+1\otimes\beta.</math> | |||
The linear dual of ψ makes the (graded) [[dual space|linear dual]] ''A''<sub>*</sub> of ''A'' into an algebra. {{harvtxt|Milnor|1958}} proved, for ''p'' = 2, that ''A''<sub>*</sub> is a [[polynomial algebra]], with one generator ξ<sub>k</sub> of degree 2<sup>''k''</sup> - 1, for every ''k'', and for ''p''>2 the dual Steenrod algebra A<sub>*</sub> is the tensor product of the polynomial algebra in generators | |||
ξ<sub>k</sub> of degree 2''p''<sup>''k''</sup> - 2 (''k''≥1) and the exterior algebra in generators τ<sub>k</sub> of degree 2''p''<sup>''k''</sup> - 1 (''k''≥0). The monomial basis for ''A''<sub>*</sub> then gives another choice of basis for ''A'', called the Milnor basis. The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative. The comultiplication for ''A''<sub>*</sub> is the dual of the product on ''A''; it is given by | |||
:<math>\psi(\xi_n) = \sum_{i=0}^n \xi_{n-i}^{p^i} \otimes \xi_i.</math> where ξ<sub>0</sub>=1, and | |||
:<math>\psi(\tau_n) = \tau_n\otimes 1 + \sum_{i=0}^n \xi_{n-i}^{p^i} \otimes \tau_i</math> if ''p''>2 | |||
The only [[primitive element]]s of ''A''<sub>*</sub> for ''p''=2 are the <math>\xi_1^{2^i}</math>, and these are dual to the <math>Sq^{2^i}</math> (the only indecomposables of ''A''). | |||
==Relation to formal groups== | |||
The dual Steenrod algebras are supercommutative Hopf algebras, so their spectra are algebra supergroup schemes. These group schemes are closely related to the automorphisms of 1-dimensional additive formal groups. For example, if ''p''=2 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme ''x''+''y'' that are the identity to first order. These automorphisms are of the form | |||
:<math>x\rightarrow x + \xi_1x^2+\xi_2x^4+\xi_3x^8+\cdots</math> | |||
==Algebraic construction== | |||
{{harvtxt|Smith|2007}} gave the following algebraic construction of the Steenrod algebra over a [[finite field]] '''F'''<sub>''q''</sub> of order ''q''. If ''V'' is a [[vector space]] over '''F'''<sub>''q''</sub> then write ''SV'' for the [[symmetric algebra]] of ''V''. There is an [[algebra homomorphism]] ''P''(''x'') | |||
:<math>P(x):SV[[x]]\rightarrow SV[[x]]</math> | |||
such that | |||
:<math>\displaystyle P(x)(v) = v+F(v)x=v+v^qx</math> | |||
for ''v''∈''V'', | |||
where ''F'' is the [[Frobenius endomorphism]] of ''SV''. | |||
If we put | |||
:<math>P(x)(f)=\sum P^i(f)x^i </math> (for ''p''>2) | |||
or | |||
:<math>P(x)(f)=\sum Sq^{2i}(f)x^i</math> (for ''p''=2) | |||
for ''f''∈''SV'' then if ''V'' is infinite dimensional the elements ''P''<sup>''i''</sup> generate an algebra isomorphism to the subalgebra of the Steenrod algebra generated by the reduced ''p''′th powers for ''p'' odd, or the even Steenrod squares Sq<sup>2''i''</sup> for ''p''=2. | |||
==Applications== | |||
The most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by [[J. Frank Adams]] of the [[Hopf invariant one]] problem and the [[vector fields on spheres]] problem. Independently Milnor and Bott, as well as Kervaire, gave a second solution of the Hopf invariant one problem, using operations in [[K-theory]]; these are the [[Adams operation]]s. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem. | |||
'''Theorem'''. If there is a map S<sup>2''n'' - 1</sup> → S<sup>''n''</sup> of [[Hopf invariant one]], then ''n'' is a power of 2. | |||
The proof uses the fact that each ''Sq''<sup>''k''</sup> is decomposable for ''k'' which is not a power of 2; | |||
that is, such an element is a product of squares of strictly smaller degree. | |||
==Connection to the Adams spectral sequence and the homotopy groups of spheres== | |||
The cohomology of the Steenrod algebra is the ''E''<sub>2</sub> term for the (''p''-local) [[Adams spectral sequence]], whose abutment is the ''p''-component of the stable homotopy groups of spheres. More specifically, the ''E''<sub>2</sub> term of this spectral sequence may be identified as | |||
:<math>\mathrm{Ext}^{s,t}_{A}(\mathbf{F}_p, \mathbf{F}_p).</math> | |||
This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres." | |||
==See also== | |||
*[[Pontryagin cohomology operation]] | |||
==References== | |||
*{{Citation | last1=Adem | first1=José | title=The iteration of the Steenrod squares in algebraic topology | jstor=88494 | mr=0050278 | year=1952 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=38 | pages=720–726}} | |||
*{{Citation | last1=Bullett | first1=S. R. | last2=Macdonald | first2=I. G. | author2-link=Ian G. Macdonald | title=On the Adem relations | doi=10.1016/0040-9383(82)90015-5 | mr=649764 | year=1982 | journal=[[Topology (journal)|Topology. An International Journal of Mathematics]] | issn=0040-9383 | volume=21 | issue=3 | pages=329–332}} | |||
*{{Citation | last1=Cartan | first1=Henri | title=Sur les groupes d'Eilenberg-Mac Lane. II | jstor=88981 | mr=0065161 | year=1954 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=40 | pages=704–707}} | |||
*{{Citation | last1=Cartan | first1=Henri | title=Sur l'itération des opérations de Steenrod | doi=10.1007/BF02564270 | mr=0068219 | year=1955 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=29 | issue=1 | pages=40–58}} | |||
* Allen Hatcher, ''Algebraic Topology''. Cambridge University Press, 2002. Available free online from the [http://www.math.cornell.edu/~hatcher/AT/ATpage.html author's home page]. | |||
*{{eom|id=S/s087550|title=Steenrod reduced power |first=S.N.|last= Malygin|first2=M.M.|last2= Postnikov}} | |||
*{{eom|id=S/s087560|title=Steenrod square |first=S.N.|last= Malygin|first2=M.M.|last2= Postnikov}} | |||
*{{Citation | last1=May | first1=J. Peter | author1-link=J. Peter May | title=The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod's Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) | url=http://www.math.uchicago.edu/~may/PAPERS/10.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0058524 | mr=0281196 | year=1970 | volume=168 | chapter=A general algebraic approach to Steenrod operations | pages=153–231}} | |||
*{{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=The Steenrod algebra and its dual | jstor=1969932 | mr=0099653 | year=1958 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=67 | pages=150–171}} | |||
*{{Citation | last1=Mosher | first1=Robert E. | last2=Tangora | first2=Martin C. | title=Cohomology operations and applications in homotopy theory | origyear=1968 | url=http://books.google.com/books?id=FFCaPwAACAAJ | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-46664-4 | mr=0226634 | year=2008}} | |||
*{{eom|id=S/s087500|first=Yu.B. |last=Rudyak}} | |||
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Cohomologie modulo 2 des complexes d'Eilenberg-MacLane | doi=10.1007/BF02564562 | mr=0060234 | year=1953 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=27 | issue=1 | pages=198–232}} | |||
*{{Citation | last1=Smith | first1=Larry | editor1-last=Hubbuck | editor1-first=John | editor2-last=Hu'ng | editor2-first=Nguyễn H. V. | editor3-last=Schwartz | editor3-first=Lionel | title=Proceedings of the School and Conference in Algebraic Topology | arxiv=0903.4997 | publisher=Geometry & Topology Publications, Coventry | series=Geometry & Topology Monographs | mr=2402812 | year=2007 | volume=11 | chapter=An algebraic introduction to the Steenrod algebra | pages=327–348}} | |||
*{{Citation | last1=Steenrod | first1=N. E. | title=Products of cocycles and extensions of mappings | jstor=1969172 | mr=0022071 | year=1947 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=48 | pages=290–320}} | |||
*{{Citation | last1=Steenrod | first1=N. E. | title=Homology groups of symmetric groups and reduced power operations | jstor=88780 | mr=0054964 | year=1953 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=39 | pages=213–217}} | |||
*{{Citation | last1=Steenrod | first1=N. E. | title=Cyclic reduced powers of cohomology classes | jstor=88781 | mr=0054965 | year=1953 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=39 | pages=217–223}} | |||
*{{Citation | last1=Steenrod | first1=N. E. | editor1-last=Epstein | editor1-first=D. B. A. | editor1-link=David B. A. Epstein | title=Cohomology operations | url=http://books.google.com/books?id=CF3bt4oYZ2oC | publisher=[[Princeton University Press]] | series= Annals of Mathematics Studies | isbn=978-0-691-07924-0 | mr=0145525 | year=1962 | volume=50}} | |||
*{{Citation | last1=Wu | first1=Wen-tsün | title=Sur les puissances de Steenrod | publisher=La Bibliothèque Nationale et Universitaire de Strasbourg | series=Colloque de Topologie de Strasbourg | mr=0051510 | year=1952 | volume=IX}} | |||
[[Category:Algebraic topology]] | |||
[[Category:Hopf algebras]] |
Revision as of 08:47, 28 October 2013
In algebraic topology, a Steenrod algebra was defined by Template:Harvtxt to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by Template:Harvtxt for p=2, and by the Steenrod reduced pth powers introduced in Template:Harvtxt and the Bockstein homomorphism for p>2.
The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory.
Cohomology operations
A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring, the cup product squaring operation yields a family of cohomology operations:
Cohomology operations need not be homomorphisms of graded rings, see the Cartan formula below.
These operations do not commute with suspension, that is they are unstable. (This is because if Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod constructed stable operations
for all i greater than zero. The notation Sq and their name, the Steenrod squares, comes from the fact that Sqn restricted to classes of degree n is the cup square. There are analogous operations for odd primary coefficients, usually denoted Pi and called the reduced p-th power operations. The Sqi generate a connected graded algebra over Z/2, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case p > 2, the mod p Steenrod algebra is generated by the Pi and the Bockstein operation β associated to the short exact sequence
In the case p=2, the Bockstein element is Sq1 and the reduced p-th power Pi is Sq2i.
Axiomatic characterization
Template:Harvtxt showed that the Steenrod squares Sqn:Hm→Hm+n are characterized by the following 5 axioms:
- Naturality: Sqn is an additive homomorphism from Hm(X,Z/2Z) to Hm+n(X,Z/2Z), and is natural meaning that for any map f : X → Y, f*(Sqnx) = Sqnf*(x).
- Sq0 is the identity homomorphism.
- Sqn is the cup square on classes of degree n.
- If n>dim(X) then Sqn(x) = 0
- Cartan Formula:
In addition the Steenrod squares have the following properties:
- Sq1 is the Bockstein homomorphism of the exact sequence
- They satisfy the Adem relations, described below.
- They commute with the suspension homomorphism and the boundary operator.
Similarly the following axioms characterize the reduced p-th powers for p > 2.
- Naturality: Pn is an additive homomorphism from Hm(X,Z/pZ) to Hm+2n(p−1)(X,Z/pZ), and is natural.
- P0 is the identity homomorphism.
- Pn is the cup p-th power on classes of degree 2n.
- If 2n>dim(X) then Pn(x) = 0
- Cartan Formula:
As before, the reduced p-th powers also satisfy Adem relations and commute with the suspension and boundary operators.
Adem relations
The Adem relations for p=2 were conjectured by Template:Harvtxt and proved by Template:Harvs and are given by
for all i, j > 0 such that i < 2j. (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one to write an arbitrary composition of Steenrod squares as a sum of Serre-Cartan basis elements.
For odd p the Adem relations are
for a<pb and
for a≤pb
Bullett–Macdonald identities
Template:Harvtxt reformulated the Adem relations as the following Bullett–Macdonald identities.
For p=2 put
then the Adem relations are equivalent to
For p>2 put
then the Adem relations are equivalent to the statement that
is symmetric in s and t. Here β is the Bockstein operation and (Ad β)P = βP−Pβ.
Construction
Suppose that π is any degree n subgroup of the symmetric group on n points, u a cohomology class in Hq(X,B), A an abelian group acted on by π, and c a cohomology class in Hi(π,A). Template:Harvtxt showed how to construct a reduced power un/c in Hkq−i(X,(A⊗B⊗B⊗...⊗B)/π) as follows.
- Taking the external product of u with itself n times gives an equivariant cocycle on Xn with coefficients in B⊗B⊗...⊗B.
- Choose E to be a contractible space on which π acts freely and an equivariant map from E× X to Xn. Pulling back un by this map gives an equivariant cocyle on E× X and therefore a cocycle of E/π×X with coefficients in B⊗B⊗...⊗B.
- Taking a slant product with c in Hi(E/π,A)gives a cocycle of X with coefficients in H0(π,A⊗B⊗B⊗...⊗B)
The Steenrod squares and reduced powers are special cases of this construction where π is a cyclic group of prime order p=n acting as a cyclic permutation of n elements, and the groups A and B are cyclic of order p, so that H0(π,A⊗B⊗B⊗...⊗B) is also cyclic of order p.
The structure of the Steenrod algebra
Template:Harvs (for p=2) and Template:Harvs (for p>2) described the structure of the Steenrod algebra of stable mod p cohomology operations, showing that it is generated by the Bockstein homomorphism together with the Steenrod reduced powers, and the Adem relations generate the ideal of relations between these generators. In particular they found an explicit basis for the Steenrod algebra. This basis relies on a certain notion of admissibility for integer sequences. We say a sequence
is admissible if for each j, ij ≥ 2ij+1. Then the elements
where I is an admissible sequence, form a basis (the Serre-Cartan basis) for the mod 2 Steenrod algebra. There is a similar basis for the case p > 2 consisting of the elements
such that
Hopf algebra structure and the Milnor basis
The Steenrod algebra has more structure than a graded Fp-algebra. It is also a Hopf algebra, so that in particular there is a diagonal or comultiplication map
induced by the Cartan formula for the action of the Steenrod algebra on the cup product. It is easier to describe than the product map, and is given by
The linear dual of ψ makes the (graded) linear dual A* of A into an algebra. Template:Harvtxt proved, for p = 2, that A* is a polynomial algebra, with one generator ξk of degree 2k - 1, for every k, and for p>2 the dual Steenrod algebra A* is the tensor product of the polynomial algebra in generators ξk of degree 2pk - 2 (k≥1) and the exterior algebra in generators τk of degree 2pk - 1 (k≥0). The monomial basis for A* then gives another choice of basis for A, called the Milnor basis. The dual to the Steenrod algebra is often more convenient to work with, because the multiplication is (super) commutative. The comultiplication for A* is the dual of the product on A; it is given by
The only primitive elements of A* for p=2 are the , and these are dual to the (the only indecomposables of A).
Relation to formal groups
The dual Steenrod algebras are supercommutative Hopf algebras, so their spectra are algebra supergroup schemes. These group schemes are closely related to the automorphisms of 1-dimensional additive formal groups. For example, if p=2 then the dual Steenrod algebra is the group scheme of automorphisms of the 1-dimensional additive formal group scheme x+y that are the identity to first order. These automorphisms are of the form
Algebraic construction
Template:Harvtxt gave the following algebraic construction of the Steenrod algebra over a finite field Fq of order q. If V is a vector space over Fq then write SV for the symmetric algebra of V. There is an algebra homomorphism P(x)
such that
for v∈V, where F is the Frobenius endomorphism of SV. If we put
or
for f∈SV then if V is infinite dimensional the elements Pi generate an algebra isomorphism to the subalgebra of the Steenrod algebra generated by the reduced p′th powers for p odd, or the even Steenrod squares Sq2i for p=2.
Applications
The most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by J. Frank Adams of the Hopf invariant one problem and the vector fields on spheres problem. Independently Milnor and Bott, as well as Kervaire, gave a second solution of the Hopf invariant one problem, using operations in K-theory; these are the Adams operations. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.
Theorem. If there is a map S2n - 1 → Sn of Hopf invariant one, then n is a power of 2.
The proof uses the fact that each Sqk is decomposable for k which is not a power of 2; that is, such an element is a product of squares of strictly smaller degree.
Connection to the Adams spectral sequence and the homotopy groups of spheres
The cohomology of the Steenrod algebra is the E2 term for the (p-local) Adams spectral sequence, whose abutment is the p-component of the stable homotopy groups of spheres. More specifically, the E2 term of this spectral sequence may be identified as
This is what is meant by the aphorism "the cohomology of the Steenrod algebra is an approximation to the stable homotopy groups of spheres."
See also
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Allen Hatcher, Algebraic Topology. Cambridge University Press, 2002. Available free online from the author's home page.
- Template:Eom
- Template:Eom
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Template:Eom
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010