|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Adams spectral sequence''' is a [[spectral sequence]] introduced by {{harvs|txt|authorlink=Frank Adams|last=Adams|year=1958}}. Like all spectral sequences, it is a computational tool; it relates [[homology (mathematics)|homology]] theory to what is now called [[stable homotopy theory]]. It is a reformulation using [[homological algebra]], and an extension, of a technique called 'killing homotopy groups' applied by the French school of [[Henri Cartan]] and [[Jean-Pierre Serre]].
| | Nice to meet you, my name is Ling and I completely dig that name. I currently live in Alabama. Interviewing is what she does but quickly she'll be on her own. What she enjoys performing is bottle tops collecting and she is attempting to make it a profession.<br><br>My webpage: [http://product-dev.younetco.com/xuannth/fox_ios1/index.php?do=/profile-700/info/ extended auto warranty] |
| | |
| ==Motivation==
| |
| | |
| For everything below, we need to once and for all fix a prime ''p''. All spaces are assumed to be [[CW complex]]es. The [[singular cohomology|ordinary]] [[cohomology group]]s ''H''<sup>*</sup>(''X'') are understood to mean ''H''<sup>*</sup>(''X''; '''Z'''/''p'''''Z''').
| |
| | |
| The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is ''S<sup>n</sup>'', these maps form the ''n''th [[homotopy group]] of ''Y''. A more reasonable (but still very difficult!) goal is to understand [''X'', ''Y''], the maps (up to homotopy) that remain after we apply the [[Suspension (topology)|suspension functor]] a large number of times. We call this the collection of stable maps from ''X'' to ''Y''. (This is the starting point of [[stable homotopy theory]]; more modern treatments of this topic begin with the concept of a [[Spectrum (homotopy theory)|spectrum]]. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.)
| |
| | |
| [''X'', ''Y''] turns out to be an abelian group, and if ''X'' and ''Y'' are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime ''p''. In an attempt to compute the ''p''-torsion of [''X'', ''Y''], we look at cohomology: send [''X'', ''Y''] to Hom(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')). This is a good idea because cohomology groups are usually tractable to compute.
| |
| | |
| The key idea is that ''H''<sup>*</sup>(''X'') is more than just a graded [[abelian group]], and more still than a graded [[ring (mathematics)|ring]] (via the [[cup product]]). The representability of the cohomology functor makes ''H''<sup>*</sup>(''X'') a [[module (mathematics)|module]] over the algebra of its stable [[cohomology operation]]s, the [[Steenrod algebra]] ''A''. Thinking about ''H''<sup>*</sup>(''X'') as an ''A''-module forgets some cup product structure, but the gain is enormous: Hom(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')) can now be taken to be ''A''-linear! A priori, the ''A''-module sees no more of [''X'', ''Y''] than it did when we considered it to be a map of vector spaces over F<sub>''p''</sub>. But we can now consider the derived functors of Hom in the category of ''A''-modules, [[Ext functor|Ext]]<sub>''A''</sub><sup>''r''</sup>(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')). These acquire a second grading from the grading on ''H''<sup>*</sup>(''Y''), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.
| |
| | |
| The point of all this is that ''A'' is so large that the above sheet of cohomological data contains all the information we need to recover the ''p''-primary part of [''X'', ''Y''], which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.
| |
| | |
| == Classical formulation ==
| |
| | |
| For ''X'' and ''Y'' spaces of finite type, with ''Y'' a finite dimensional CW-complex, there is a spectral sequence, called the '''classical Adams spectral sequence''', converging to the ''p''-torsion in [''X'', ''Y''], with ''E''<sub>2</sub>-term given by
| |
| | |
| :''E''<sub>2</sub><sup>''t'',''s''</sup> = Ext<sub>''A''</sub><sup>''t'',''s''</sup>(''H''<sup>*</sup>(''Y''), ''H''<sup>*</sup>(''X'')), | |
| | |
| and differentials of bidegree (''r'', ''r'' − 1).
| |
| | |
| == Calculations ==
| |
| | |
| The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.
| |
| | |
| Adams' original use for his spectral sequence was the first proof of the [[Hopf invariant]] 1 problem: <math>\mathbb{R}^n</math> admits a division algebra structure only for ''n'' = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in [[K-theory]].
| |
| | |
| The [[Thom space|Thom isomorphism theorem]] relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, Milnor and Novikov used the Adams spectral sequence to compute the coefficient ring of [[complex cobordism]]. Further, Milnor and Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented [[cobordism]] ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel–Whitney numbers agree.
| |
| | |
| == Generalizations ==
| |
| | |
| The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by {{harvtxt|Novikov|1967}} where ordinary cohomology is replaced by a [[generalized cohomology theory]], often [[complex bordism]] or [[Brown–Peterson cohomology]]. This requires knowledge of the algebra of stable cohomology operations for the cohomology theory in question, but enables calculations which are completely intractable with the classical Adams spectral sequence.
| |
| | |
| ==References==
| |
| | |
| *{{Citation | last1=Adams | first1=J. Frank | title=On the structure and applications of the Steenrod algebra | doi=10.1007/BF02564578 | mr=0096219 | year=1958 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=32 | issue=1 | pages=180–214}}
| |
| *{{citation|mr=0185597|last=Adams|first= J. Frank |title=Stable homotopy theory|publisher=Springer-Verlag,|series=Lecture notes in mathematics|volume=3|publication-place= Berlin–Göttingen–Heidelberg–New York|year= 1964}}
| |
| *{{citation|last=Botvinnik|first=Boris|title=Manifolds with Singularities and the Adams–Novikov Spectral Sequence |series=London Mathematical Society Lecture Note Series|year=1992|isbn= 0-521-42608-1|publisher=Cambridge Univ. Press|location=Cambridge}}
| |
| * {{Citation | last1=McCleary | first1=John | title=A User's Guide to Spectral Sequences | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-56759-6 | doi=10.2277/0521567599 | mr=1793722 |date=February 2001 | volume=58 | edition = 2nd}}
| |
| *{{citation|first=S.|last=Novikov|title=Methods of algebraic topology from the point of view of cobordism theory|journal= Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya|volume=31|year=1967|language=Russian|pages=855–951}}
| |
| *{{Citation | last1=Ravenel | first1=Douglas C. | editor1-last=Barratt | editor1-first=M. G. | editor2-last=Mahowald | editor2-first=Mark E. | title=Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-08859-2 | doi= 10.1007/BFb0068728 | mr=513586 | year=1978 | volume=658 | chapter=A novice's guide to the Adams–Novikov spectral sequence | pages=404–475}}
| |
| * {{citation
| |
| |last= Ravenel
| |
| |first= Douglas C.
| |
| |title= Complex cobordism and stable homotopy groups of spheres
| |
| |edition= 2nd
| |
| |url= http://www.math.rochester.edu/people/faculty/doug/mu.html
| |
| |publisher= AMS Chelsea
| |
| |year= 2003
| |
| |isbn= 978-0-8218-2967-7
| |
| |mr= 0860042
| |
| }}.
| |
| | |
| ==External links==
| |
| *{{citation|url=http://www.math.cornell.edu/~hatcher/SSAT/SSch2.pdf|title= Book chapter (PDF)|author=Hatcher}}
| |
| | |
| [[Category:Homotopy theory]]
| |
| [[Category:Spectral sequences]]
| |
Nice to meet you, my name is Ling and I completely dig that name. I currently live in Alabama. Interviewing is what she does but quickly she'll be on her own. What she enjoys performing is bottle tops collecting and she is attempting to make it a profession.
My webpage: extended auto warranty