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2014-08-11T15:17:26Z

CarsonPitcher:

{{Technical|date=August 2011}}
{{Development economics sidebar}}
'''Endogenous growth theory''' holds that [[economic growth]] is primarily the result of [[endogeny|endogenous]] and not external forces.<ref>{{cite journal |journal = [[The Journal of Economic Perspectives]] |volume= 8 |issue= 1 |year= 1994 |url= http://links.jstor.org/sici?sici=0895-3309%28199424%298%3A1%3C3%3ATOOEG%3E2.0.CO%3B2-H |first= P. M. |last= Romer |title= The Origins of Endogenous Growth |authorlink= Paul Romer |page = [http://www.iset.ge/old/upload/Romer%201994.pdf 3] }}</ref> Endogenous growth theory holds that investment in [[human capital]], [[innovation]], and knowledge are significant contributors to economic growth. The theory also focuses on [[positive externalities]] and [[spillover effects]] of a knowledge-based economy which will lead to economic development. The endogenous growth theory also holds that policy measures can have an impact on the long-run growth rate of an economy. For example, [[subsidies]] for [[research and development]] or [[education]] increase the growth rate in some endogenous growth models by increasing the incentive for innovation.

==Models in Endogenous Growth==

In the mid-1980s, a group of growth theorists became increasingly dissatisfied with common accounts of [[exogenous]] factors determining long-run growth. They favored a model that replaced the exogenous growth variable (unexplained technical progress) with a model in which the key determinants of growth were explicit in the model. The initial research was based on the work of [[Kenneth Arrow]] (1962), [[Hirofumi Uzawa]] (1965), and [[Miguel Sidrauski]] (1967).<ref>{{cite web |url= http://www.newschool.edu/nssr/het/essays/growth/moneygrowth.htm |title=Monetary Growth Theory |work=newschool.edu |year=2011 [last update] |accessdate=11 October 2011}}</ref> [[Paul Romer]] (1986), [[Robert Emerson Lucas, Jr. |Lucas]] (1988),<ref>{{cite journal |url= http://www.fordham.edu/economics/mcleod/LucasMechanicsEconomicGrowth.pdf |title= On the mechanics of Economic Development |first= R. E. |last= Lucas |authorlink= Robert Emerson Lucas, Jr. |journal = [[Journal of Monetary Economics]] |year=1988 |volume = 22 }}</ref> and Rebelo (1991)<ref>{{cite journal |url=http://www.nber.org/papers/w3325 |title= Long-Run Policy Analysis and Long-Run Growth |first= Sergio |last= Rebelo |journal = [[Journal of Political Economy]] |year=1991 |volume = 99 |issue= 3 |page= 500 }}</ref><ref>{{cite web |url= http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/Growth/RebeloAK.pdf |title= The Rebelo AK Growth Model |first= C.|last= Carroll |work=econ2.jhu.edu |year=2011 [last update] |accessdate=11 October 2011 |quote= the steady-state growth rate in a Rebelo economy is directly proportional to the saving rate.}}</ref> omitted technological change. Instead, growth in these models was due to indefinite investment in [[human capital]] which had [[spillover effect]] on economy and reduces the diminishing return to [[capital accumulation]].<ref name= "BX">{{cite book |first1= R. J. |last1= Barro |first2= Xavier |authorlink2= Xavier Sala-i-Martin |last2= Sala-i-Martin |title= Economic Growth |isbn= 978-0-262-02459-4 |date= 1998-11-20 |url= http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf }}</ref>

The [[AK model]], which is the simplest endogenous model, gives a constant-saving-rate of endogenous growth. It assumes a constant, exogenous saving rate and fixed level of the technology. It shows elimination of diminishing returns leading to endogenous growth. However, the endogenous growth theory is further supported with models in which agents optimally determined the consumption and saving, optimizing the resources allocation to research and development leading to technological progress. Romer (1987, 1990) and signiﬁcant contributions by Aghion and Howitt (1992) and Grossman and Helpman (1991), incorporated [[imperfect market]]s and R&D to the growth model.<ref name= "BX"/>

== The AK Model ==
{{main|AK model}}

The model works on the property of absence of diminishing returns to capital. The simplest form of production function with diminishing return is:
[[File:Ak model.png|thumb|figure 1.1]]
:<math>Y = AK\,</math>

where
:<math> A\,</math> , is a positive constant that reflects the level of the technology.

:<math> K \,</math> capital (broad sense to include human capital)

:<math>y = AK\,</math> , output per capita and the average and marginal product are constant at the level <math>A>0\,</math>

If we substitute <math>\frac{f(k)}{k}=A \,</math> in equation of transitional Dynamics of Solow-Swan model ([[Exogenous growth model]]) which shows how an economy’s per capita incomes converges toward its own steady-state value and to the per capita incomes of other nations.

Transitional Dynamics equation, where Growth rate on <math> k\,</math> is given by,

:<math>\gamma_K=\dot{k}/k = s.f(k)/ k - (n+\delta)\ ,</math>

on substituting <math> A\,</math>, we get,
:<math>\gamma_K= sA -(n+\delta)\ ,</math>

We return here to the case of zero technological progress, <math> x=0\,</math>, because we want to show that per capita growth can now occur in the long-run even without exogenous technological change. The figure 1.1 explains the perpetual growth, with exogenous technical progress. The vertical distance between the two line, <math> sA\,</math>and n+δ gives the<math>\gamma_K\,</math>

As, <math> sA>\, </math>n+δ, so that<math>\gamma_K > 0\,</math>. Since the two line are parallel, <math>\gamma_K\,</math>is constant; in particular, it is independent of <math>K\,</math>. In other words,<math>K\,</math> always grows at steady states rate,<math>\gamma_K^*= sA -(n+\delta)\ ,</math>.

Since
:<math>y = AK\,</math>,<math>\gamma_K\,</math> equals <math>\gamma_K^*\,</math>

at every point of time. In addition, since
:<math>c= (1-s) y\,</math>,

the growth rate of
:<math>c\,</math> equals <math>\gamma_K^*\,</math>.

Hence, the entire per capita variable in the model grows at same rate, given by
:<math>\gamma^*= sA -(n+\delta)\ ,</math>

However, we can observe that<math>y = AK\,</math> technology displays a positive long-run per capita growth without any exogenous technological development. The per capita growth depends on behavioural factors of the model as the saving rate and population. It is unlike neoclassical model, which is higher saving, s, promotes higher long-run per capita growth <math>\gamma^*\,</math>.<ref>Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin</ref>

== Endogenous versus exogenous growth theory ==
In neo-classical growth models, the long-run rate of growth is [[exogeny|exogenous]]ly determined by either the savings rate (the [[Harrod–Domar model]]) or the rate of technical progress ([[Solow model]]). However, the savings rate and rate of technological progress remain unexplained. Endogenous growth theory tries to overcome this shortcoming by building macroeconomic models out of [[Microfoundations|microeconomic foundations]]. Households are assumed to maximize utility subject to budget constraints while firms maximize profits. Crucial importance is usually given to the production of new technologies and [[human capital]]. The engine for growth can be as simple as a constant return to scale production function (the AK model) or more complicated set ups with [[Knowledge spillover|spillover]] effects (spillovers are positive externalities, benefits that are attributed to costs from other firms), increasing numbers of goods, increasing qualities, etc.

Often endogenous growth theory assumes constant marginal product of capital at the aggregate level, or at least that the limit of the marginal product of capital does not tend towards zero. This does not imply that larger firms will be more productive than small ones, because at the firm level the marginal product of capital is still diminishing. Therefore, it is possible to construct endogenous growth models with [[perfect competition]]. However, in many endogenous growth models the assumption of perfect competition is relaxed, and some degree of [[monopoly]] power is thought to exist. Generally monopoly power in these models comes from the holding of patents. These are models with two sectors, producers of final output and an R&D sector. The R&D sector develops ideas that they are granted a monopoly power. R&D firms are assumed to be able to make monopoly profits selling ideas to production firms, but the [[free entry]] condition means that these profits are dissipated on R&D spending.

==Implications==
An Endogenous growth theory implication is that policies which embrace openness, competition, change and innovation will promote growth.<ref>{{cite journal|last=Fadare|first=Samuel O.|title=Recent Banking Sector Reforms and Economic Growth in Nigeria|journal=Middle Eastern Finance and Economics|issue=Issue 8 (2010)|url=http://www.eurojournals.com/MEFE_8_12.pdf}}</ref> Conversely, policies which have the effect of restricting or slowing change by protecting or favouring particular existing industries or firms are likely over time to slow growth to the disadvantage of the community. [[Peter Howitt (economist)|Peter Howitt]] has written:
<blockquote>
Sustained economic growth is everywhere and always a process of continual transformation. The sort of economic progress that has been enjoyed by the richest nations since the Industrial Revolution would not have been possible if people had not undergone wrenching changes. Economies that cease to transform themselves are destined to fall off the path of economic growth. The countries that most deserve the title of “developing” are not the poorest countries of the world, but the richest. [They] need to engage in the never-ending process of economic development if they are to enjoy continued prosperity. (Conclusion, "Growth and development: a Schumpeterian perspective", 2006 [http://www.cdhowe.org/pdf/commentary_246.pdf]).
</blockquote>

== Criticisms ==

One of the main failings of endogenous growth theories is the collective failure to explain conditional convergence reported in the empirical literature.<ref> See {{cite journal |last=Sachs |first=Jeffrey D. |first2=Andrew M. |last2=Warner |year=1997 |title=Fundamental Sources of Long-Run Growth |journal=[[American Economic Review]] |volume=87 |issue=2 |pages=184–188 |doi= |jstor=2950910 }}</ref> Another frequent critique concerns the cornerstone assumption of diminishing returns to capital. Some contend that ''new growth theory'' has proven no more successful than [[exogenous growth model|exogenous growth theory]] in explaining the income divergence between the [[developing nation|developing]] and [[developed nation|developed]] worlds (despite usually being more complex).<ref>See for instance, Professor Stephen Parente's 2001 review, ''The Failure of Endogenous Growth'' ([https://netfiles.uiuc.edu/parente/The%20Failure%20of%20Endogenous%20Growth.pdf Online] at the [[University of Illinois at Urbana-Champaign]]). (Published in [http://www.metapress.com/(gqdg4dzadovmv5fnfj5jki2x)/home/main.mpx Knowledge Technology & Policy] Volume XIII, Number 4.)</ref>

== See also ==
* [[Economic growth]]
* [[Human capital]]
* [[Paul Romer]]
* [[Neoclassical growth model|Exogenous growth model]]
* [[Mahalanobis model]]
* [[Ramsey–Cass–Koopmans model]]

==Notes==
{{reflist}}

== External links ==
* [http://www.stanford.edu/~promer/EconomicGrowth.pdf Economic Growth] by [[Paul Romer]].
* [http://www.eda.gov/ImageCache/EDAPublic/documents/pdfdocs/1g3lr_5f7_5fcortright_2epdf/v1/1g3lr_5f7_5fcortright.pdf New Growth Theory, Technology and Learning: A Practitioner's Guide], [[Economic Development Administration|U.S. Economic Development Administration]].
* [http://tcdc.undp.org/CoopSouth/1998_2/cop9829.pdf Technological Implications of New Growth Theory for the South], [[United Nations Development Programme]].
*[http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf The AK Model] by Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin
*The Origins of Endogenous Growth, Romer.M Paul,The Journal of Economic Perspectives, Vol. 8, No. 1. (Winter, 1994), pp. 3-22.

[[Category:Economic theories]]
[[Category:Economic growth]]

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[[de:Endogene Wachstumstheorie]]
[[fr:Théorie de la croissance endogène]]
[[it:Teoria della crescita endogena]]
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2014-08-11T15:17:16Z

CarsonPitcher:

In [[mathematics]], particularly [[algebraic topology]] and [[homology theory]], the '''Mayer–Vietoris sequence''' is an [[algebra]]ic tool to help compute [[algebraic invariant]]s of [[topological space]]s, known as their [[Homology group|homology]] and [[cohomology group]]s. The result is due to two [[Austria]]n mathematicians, [[Walther Mayer]] and [[Leopold Vietoris]]. The method consists of splitting a space into pieces, called [[Subspace topology|subspaces]], for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a [[Natural (category theory)|natural]] [[long exact sequence]], whose entries are the (co)homology groups of the whole space, the [[direct sum of abelian groups|direct sum]] of the (co)homology groups of the subspaces, and the (co)homology groups of the [[intersection (set theory)|intersection]] of the subspaces.

The Mayer–Vietoris sequence holds for a variety of [[cohomology theory|cohomology]] and [[homology theory|homology theories]], including [[singular homology]] and [[singular cohomology]]. In general, the sequence holds for those theories satisfying the [[Eilenberg–Steenrod axioms]], and it has variations for both [[Reduced homology|reduced]] and [[Relative homology|relative]] (co)homology. Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in [[topology]] are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the [[Seifert–van Kampen theorem]] for the [[fundamental group]], and a precise relation exists for homology of dimension one.

==Background, motivation, and history==

[[Image:Vietoris4343.jpg|Right|thumb|Leopold Vietoris on his 110th birthday]]

Like the [[fundamental group]] or the higher [[homotopy group]]s of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of [[linear algebra]], many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection.

The most natural and convenient way to express the relation involves the algebraic concept of [[exact sequence]]s: sequences of [[Object (category theory)|objects]] (in this case [[Group (mathematics)|groups]]) and [[morphism]]s (in this case [[group homomorphism]]s) between them such that the [[Image (mathematics)|image]] of one morphism equals the [[Kernel (algebra)|kernel]] of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are [[topological manifold]]s, [[simplicial complex]]es, or [[CW complex]]es, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.

Mayer was introduced to topology by his colleague Vietoris when attending his lectures in 1926 and 1927 at a local university in [[Vienna]].<ref>{{harvnb|Hirzebruch|1999}}</ref> He was told about the conjectured result and a way to its solution, and solved the question for the [[Betti number]]s in 1929.<ref>{{harvnb|Mayer|1929}}</ref> He applied his results to the [[torus]] considered as the union of two cylinders.<ref>{{harvnb|Dieudonné|1989|p=39}}</ref><ref>{{harvnb|Mayer|1929|p=41}}</ref> Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence.<ref>{{harvnb|Vietoris|1930}}</ref> The concept of an exact sequence only appeared in print in the 1952 book ''Foundations of Algebraic Topology'' by [[Samuel Eilenberg]] and [[Norman Steenrod]]<ref>{{harvnb|Corry|2004|p=345}}</ref> where the results of Mayer and Vietoris were expressed in the modern form.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.3}}</ref>
{{-}}

==Basic versions for singular homology==
Let ''X'' be a [[topological space]] and ''A'', ''B'' be two subspaces whose [[Interior (topology)|interiors]] cover ''X''. (The interiors of ''A'' and ''B'' need not be disjoint.) The Mayer–Vietoris sequence in [[singular homology]] for the triad (''X'', ''A'', ''B'') is a [[long exact sequence]] relating the singular homology groups (with coefficient group the integers '''Z''') of the spaces ''X'', ''A'', ''B'', and the [[intersection (set theory)|intersection]] ''A''∩''B''.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=§15}}</ref> There is an unreduced and a reduced version.

===Unreduced version===
For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:<ref name="Hatcher149">{{harvnb|Hatcher|2002|p=149}}</ref>

 <math>\begin{align}
\cdots\rightarrow H_{n+1}(X)\,&\xrightarrow{\partial_*}\,H_{n}(A\cap B)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A)\oplus H_{n}(B)\,\xrightarrow{k_* - l_*}\,H_{n}(X)\xrightarrow{\partial_*}\\
&\quad\xrightarrow{\partial_*}\,H_{n-1} (A\cap B)\rightarrow \cdots\rightarrow H_0(A)\oplus H_0(B)\,\xrightarrow{k_* - l_*}\,H_0(X)\rightarrow\,0.
\end{align}</math>

Here the maps ''i'' : ''A''∩''B'' ↪ ''A'', ''j'' : ''A''∩''B'' ↪ ''B'', ''k'' : ''A'' ↪ ''X'', and ''l'' : ''B'' ↪ ''X'' are [[inclusion map]]s and <math>\oplus</math> denotes the [[direct sum of abelian groups]].

===Boundary map===
[[Image:Mayer Vietoris sequence boundary map on torus.png|thumb|280px|right|Illustration of the boundary map ∂* on the torus where the 1-cycle ''x'' = ''u'' + ''v'' is the sum of two 1-chains whose boundary lies in the intersection of ''A'' and ''B''.]]
The boundary maps ∂* lowering the dimension may be made explicit as follows.<ref name="Hatcher 2002 150">{{harvnb|Hatcher|2002|p=150}}</ref> An element in ''H''n(''X'') is the homology class of an ''n''-cycle ''x'' which, by [[barycentric subdivision]] for example, can be written as the sum of two ''n''-chains ''u'' and ''v'' whose images lie wholly in ''A'' and ''B'', respectively. Thus ∂''x'' = ∂(''u'' + ''v'') = 0 so that ∂''u'' = −∂''v''. This implies that the images of both these boundary (''n'' − 1)-cycles are contained in the intersection ''A''∩''B''. Then ∂*([''x'']) is the class of ∂''u'' in ''H''n−1(''A''∩''B''). Choosing a different representative ''x′'' does not affect ∂''u'' since ∂''x′'' = ∂''x'' = 0; nor does choosing another decomposition ''x'' = ''u′'' + ''v′'' since then ∂''u'' + ∂''v'' − ∂''u′'' − ∂''v′'' = 0 which implies ∂''u'' = ∂''u′'' and ∂''v'' = ∂''v′''. Notice that the maps in the Mayer–Vietoris sequence depend on choosing an order for ''A'' and ''B''. In particular, the boundary map changes sign if ''A'' and ''B'' are swapped.

===Reduced version===
For [[reduced homology]] there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have [[non-empty]] intersection.<ref>{{harvnb|Spanier|1966|p=187}}</ref> The sequence is identical for positive dimensions and ends as:

 <math>\cdots\rightarrow\tilde{H}_0(A\cap B)\,\xrightarrow{(i_*,j_*)}\,\tilde{H}_0(A)\oplus\tilde{H}_0(B)\,\xrightarrow{k_* - l_*}\,\tilde{H}_0(X)\rightarrow\,0.</math>

===Analogy with the Seifert–van Kampen theorem===
There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the [[Seifert–van Kampen theorem]].<ref name="Hatcher 2002 150"/><ref>{{harvnb|Massey|1984|p=240}}</ref> Whenever ''A''∩''B'' is [[path-connected]] the reduced Mayer–Vietoris sequence yields the isomorphism

:<math>H_1(X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*)</math>

where, by exactness,

:<math>\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).</math>

This is precisely the [[Commutator subgroup#Abelianization|abelianized]] statement of the Seifert–van Kampen theorem. Compare with the fact that ''H''1(''X'') is the abelianization of the [[fundamental group]] π1(''X'') when ''X'' is path-connected.<ref>{{harvnb|Hatcher|2002|loc=Theorem 2A.1, p. 166}}</ref>

==Basic applications==

===''k''-sphere===
[[Image:SphereCoverStriped.png|thumb|250px|right|The decomposition for ''X'' = ''S''2]]
To completely compute the homology of the [[n-sphere|''k''-sphere]] ''X'' = ''S''''k'', let ''A'' and ''B'' be two hemispheres of ''X'' with intersection [[homotopy equivalent]] to a (''k'' − 1)-dimensional equatorial sphere. Since the ''k''-dimensional hemispheres are [[homeomorphic]] to ''k''-discs, which are [[contractible]], the homology groups for ''A'' and ''B'' are [[Trivial group|trivial]]. The Mayer–Vietoris sequence for [[reduced homology]] groups then yields

: <math> \cdots\rightarrow 0 \rightarrow \tilde{H}_{n}\left(S^k\right) \xrightarrow{\partial_*}\, \tilde{H}_{n-1}\left(S^{k-1}\right) \rightarrow 0 \rightarrow \cdots \! </math>

Exactness immediately implies that the map ∂* is an isomorphism. Using the [[reduced homology]] of the [[0-sphere]] (two points) as a [[Mathematical induction|base case]], it follows<ref>{{harvnb|Hatcher|2002|loc=Example 2.46, p. 150}}</ref>

: <math>\tilde{H}_n\left(S^k\right)\cong\delta_{kn}\,\mathbb{Z}=\left\{\begin{matrix}
\mathbb{Z} & \mbox{if } n=k \\
0 & \mbox{if } n \ne k \end{matrix}\right.</math>

where δ is the [[Kronecker delta]]. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of [[homotopy groups of spheres]], especially for the case ''n'' > ''k'' about which little is known.<ref>{{harvnb|Hatcher|2002|p=384}}</ref>
{{-}}

===Klein bottle===
[[Image:KleinBottle2D covered by Möbius strips.svg|thumb|200px|right|The Klein bottle ([[fundamental polygon]] with appropriate edge identifications) decomposed as two Möbius strips ''A'' (in blue) and ''B'' (in red).]]
A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the [[Klein bottle]] ''X''. One uses the decomposition of ''X'' as the union of two [[Möbius strip]]s ''A'' and ''B'' [[Quotient space|glued]] along their boundary circle (see illustration on the right). Then ''A'', ''B'' and their intersection ''A''∩''B'' are [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalent]] to circles, so the nontrivial part of the sequence yields<ref>{{harvnb|Hatcher|2002|p=151}}</ref>

: <math> 0 \rightarrow H_{2}(X) \rightarrow\, \mathbb{Z}\ \xrightarrow{\alpha} \ \mathbb{Z} \oplus \mathbb{Z} \rightarrow \, H_1(X) \rightarrow 0 \! </math>

and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, −2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is [[Injective function|injective]] so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for '''Z'''2, it follows

: <math>\tilde{H}_n\left(X\right)\cong\delta_{1n}\,(\mathbb{Z}\oplus\mathbb{Z}_2)=\left\{\begin{matrix}
\mathbb{Z}\oplus\mathbb{Z}_2 & \mbox{if } n=1\\
0 & \mbox{if } n\ne1 \end{matrix}\right.
</math>
{{-}}

===Wedge sums===
[[Image:WedgeSumSpheres.png|right|300px|thumb|This decomposition of the wedge sum ''X'' of two 2-spheres ''K'' and ''L'' yields all the homology groups of ''X''.]]
Let ''X'' be the [[wedge sum]] of two spaces ''K'' and ''L'', and suppose furthermore that the identified [[basepoint]] is a [[deformation retract]] of [[Neighbourhood (mathematics)|open neighborhoods]] ''U'' ⊂ ''K'' and ''V'' ⊂ ''L''. Letting ''A'' = ''K''∪''V'' and ''B'' = ''U''∪''L'' it follows that ''A''∪''B'' = ''X'' and ''A''∩''B'' = ''U''∪''V'', which is [[contractible]] by construction. The reduced version of the sequence then yields (by exactness)<ref>{{harvnb|Hatcher|2002|loc=Exercise 31}}</ref>
:<math>\tilde{H}_n(K\vee L)\cong \tilde{H}_n(K)\oplus\tilde{H}_n(L)</math>
for all dimensions ''n''. The illustration on the right shows ''X'' as the sum of two 2-spheres ''K'' and ''L''. For this specific case, using the result [[Mayer–Vietoris sequence#k-sphere|from above]] for 2-spheres, one has
:<math>\tilde{H}_n\left(S^2\vee S^2\right)\cong\delta_{2n}\,(\mathbb{Z}\oplus\mathbb{Z})=\left\{\begin{matrix}
\mathbb{Z}\oplus\mathbb{Z} & \mbox{if } n=2 \\
0 & \mbox{if } n \ne 2 \end{matrix}\right.</math>
{{-}}

===Suspensions===
[[Image:0-Sphere Suspension - Mayer-Vietoris Cover.svg|right|500px|thumb|This decomposition of the suspension ''X'' of the 0-sphere ''Y'' yields all the homology groups of ''X''.]]
If ''X'' is the [[Suspension (topology)|suspension]] ''SY'' of a space ''Y'', let ''A'' and ''B'' be the [[Complement (set theory)|complements]] in ''X'' of the top and bottom 'vertices' of the double cone, respectively. Then ''X'' is the union ''A''∪''B'', with ''A'' and ''B'' contractible. Also, the intersection ''A''∩''B'' is homotopy equivalent to ''Y''. Hence the Mayer–Vietoris sequence yields, for all ''n'',<ref>{{harvnb|Hatcher|2002|loc=Exercise 32}}</ref>
:<math>\tilde{H}_n(SY)\cong \tilde{H}_{n-1}(Y)</math>

The illustration on the right shows the 1-sphere ''X'' as the suspension of the 0-sphere ''Y''. Noting in general that the ''k''-sphere is the suspension of the (''k'' − 1)-sphere, it is easy to derive the homology groups of the ''k''-sphere by induction, [[Mayer–Vietoris sequence#k-sphere|as above]].
{{-}}

==Further discussion==

===Relative form===
A [[relative homology|relative]] form of the Mayer–Vietoris sequence also exists. If ''Y'' ⊂ ''X'' and is the union of ''C'' ⊂ ''A'' and ''D'' ⊂ ''B'', then the exact sequence is:<ref>{{harvnb|Hatcher|2002|p=152}}</ref>

 <math>\cdots\rightarrow H_{n}(A\cap B,C\cap D)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A,C)\oplus H_{n}(B,D)\,\xrightarrow{k_* - l_*}\,H_{n}(X,Y)\,\xrightarrow{\partial_*}\,H_{n-1}(A\cap B,C\cap D)\rightarrow\cdots</math>

===Naturality===
The homology groups are [[Natural (category theory)|natural]] in the sense that if ''ƒ'' is a [[Continuous function (topology)|continuous]] map from ''X''1 to ''X''2, then there is a canonical [[pushforward (homology)|pushforward]] map ''ƒ''∗ of homology groups ''ƒ''∗ : ''H''''k''(''X''1) → ''H''''k''(''X''2), such that the composition of pushforwards is the pushforward of a composition: that is, <math>(g\circ h)_* = g_*\circ h_*</math>. The Mayer–Vietoris sequence is also natural in the sense that if ''X''1 = ''A''1∪''B''1 to ''X''2 = ''A''2∪''B''2 and the mapping ''ƒ'' satisfies ''ƒ''(''A''1) ⊂ ''A''2 and ''ƒ''(''B''1) ⊂ ''B''2, then the connecting morphism ∂∗ of the Mayer–Vietoris sequence commutes with ''ƒ''∗.<ref>{{harvnb|Massey|1984|p=208}}</ref> That is,<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.4}}</ref> the following diagram [[Commutative diagram|commutes]] (the horizontal maps are the usual ones):
[[Image:Mayer-Vietoris naturality.png|center|740px]]

===Cohomological versions===

The Mayer–Vietoris long exact sequence for [[singular cohomology]] groups with coefficient [[group (mathematics)|group]] ''G'' is [[Duality (mathematics)|dual]] to the homological version. It is the following:<ref>{{harvnb|Hatcher|2002|p=203}}</ref>

 <math>\cdots\rightarrow H^{n}(X;G)\rightarrow H^{n}(A;G)\oplus H^{n}(B;G)\rightarrow H^{n}(A\cap B;G)\rightarrow H^{n+1}(X;G)\rightarrow\cdots</math>

where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation.

As an important special case when ''G'' is the group of [[real number]]s '''R''' and the underlying topological space has the additional structure of a [[smooth manifold]], the Mayer–Vietoris sequence for [[de Rham cohomology]] is

 <math>\cdots\rightarrow H^{n}(X)\,\xrightarrow{\rho}\,H^{n}(U)\oplus H^{n}(V)\,\xrightarrow{\Delta}\,H^{n}(U\cap V)\,\xrightarrow{d^*}\,H^{n+1}(X)\rightarrow\cdots</math>

where {''U'', ''V''} is an [[open cover]] of ''X'', ''ρ'' denotes the restriction map, and Δ is the difference. The map ''d*'' is defined similarly as the map ''∂''* from above. It can be briefly described as follows. For a cohomology class [''ω''] represented by [[closed and exact differential forms|closed form]] ''ω'' in ''U''∩''V'', express ''ω'' as a difference of forms ''ωU'' - ''ωV'' via a [[partition of unity]] subordinate to the open cover {''U'', ''V''}, for example. The exterior derivative ''dωU'' and ''dωV'' agree on ''U''∩''V'' and therefore together define an ''n'' + 1 form ''σ'' on ''X''. One then has ''d*''([''ω'']) = [''σ''].

===Derivation===
Consider the [[Homological algebra#Functoriality|long exact sequence associated to]] the [[short exact sequence]]s of [[chain group]]s (constituent groups of [[chain complex]]es)

:<math>0 \rightarrow C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \rightarrow 0 </math>

where α(''x'') = (''x'', −''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''''n''(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''.<ref name="Hatcher149"/> It is a fact that the singular ''n''-simplices of ''X'' whose images are contained in either ''A'' or ''B'' generate all of the homology group ''H''''n''(''X'').<ref>{{harvnb|Hatcher|2002|loc=Proposition 2.21, p. 119}}</ref> In other words, ''H''''n''(''A'' + ''B'') is isomorphic to ''H''''n''(''X''). This gives the Mayer–Vietoris sequence for singular homology.

The same computation applied to the short exact sequences of vector spaces of [[differential form]]s

:<math>
0\rightarrow\Omega^{n}(X)\rightarrow\Omega^{n}(U)\oplus\Omega^{n}(V)\rightarrow\Omega^{n}(U\cap V)\rightarrow0
</math>

yields the Mayer–Vietoris sequence for de Rham cohomology.<ref>{{harvnb|Bott|Tu|1982|loc=§I.2}}</ref>

From a formal point of view, the Mayer–Vietoris sequence can be derived from the [[Eilenberg–Steenrod axioms]] for [[homology theory|homology theories]] using the [[long exact sequence in homology]].<ref>{{harvnb|Hatcher|2002|p=162}}</ref>

===Other homology theories===
The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the [[dimension axiom]],<ref>{{harvnb|Kōno|Tamaki|2006|pp=25–26}}</ref> so in addition to existing in [[List of cohomology theories#Ordinary homology theories|ordinary cohomology theories]], it holds in [[extraordinary cohomology theories]] (such as [[topological K-theory]] and [[cobordism]]).

===Sheaf cohomology===
From the point of view of [[sheaf cohomology]], the Mayer–Vietoris sequence is related to [[Čech cohomology]]. Specifically, it arises from the [[Spectral sequence|degeneration]] of the [[spectral sequence]] that relates Čech cohomology to sheaf cohomology (sometimes called the [[Mayer–Vietoris spectral sequence]]) in the case where the open cover used to compute the Čech cohomology consists of two open sets.<ref>{{harvnb|Dimca|2004|pp=35–36}}</ref> This spectral sequence exists in arbitrary [[Topos|topoi]].<ref>{{harvnb|Verdier|1972}} (SGA 4.V.3)</ref>

==See also==
*[[Excision theorem]]
*[[Zig-zag lemma]]

==Notes==
{{reflist|colwidth=30em}}

==References==

*{{citation
|last1=Bott
|first1=Raoul
|author1-link=Raoul Bott
|last2=Tu
|first2=Loring W.
|title=Differential Forms in Algebraic Topology
|publisher=[[Springer Science+Business Media|Springer-Verlag]]
|location=Berlin, New York
|isbn=978-0-387-90613-3
|year=1982}}.

*{{citation
|first= Leo
|last= Corry
|authorlink= Leo Corry
|title= Modern Algebra and the Rise of Mathematical Structures
|publisher= Birkhäuser
|year= 2004
|page= 345
|isbn= 3-7643-7002-5
}}.

*{{citation
|first= Jean
|last= Dieudonné
|authorlink= Jean Dieudonné
|title= A History of Algebraic and Differential Topology 1900–1960
|publisher= Birkhäuser
|year= 1989
|page= 39
|isbn= 0-8176-3388-X
}}.

*{{citation
| last=Dimca
| first=Alexandru
| title=Sheaves in topology
| publisher=[[Springer-Verlag]]
| year=2004
| location=Berlin
| series=Universitext
| isbn=978-3-540-20665-1
| mr=2050072
}}

* {{citation
|last1=Eilenberg
|first1=Samuel
|authorlink1=Samuel Eilenberg
|last2=Steenrod
|first2=Norman
|authorlink2=Norman Steenrod
|title=Foundations of Algebraic Topology
|year=1952
|isbn=978-0-691-07965-3
|publisher= [[Princeton University Press]]
}}.

*{{citation
|first= Allen
|last= Hatcher
|author-link= Allen Hatcher
|title= Algebraic Topology
|url= http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html
|year= 2002
|publisher= [[Cambridge University Press]]
|isbn= 978-0-521-79540-1
|mr= 1867354
}}.

*{{citation
|title= The Heritage of Emmy Noether
|last=Hirzebruch
|first=Friedrich
|authorlink=Friedrich Hirzebruch
|contribution=Emmy Noether and Topology
|pages=61–63
|editor= Teicher, M.
|series= Israel Mathematical Conference Proceedings
|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]
|year= 1999
|isbn= 978-0-19-851045-1
|oclc= 223099225
}}.

*{{citation
|last=Kōno
|first=Akira
|last2=Tamaki
|first2=Dai
|title=Generalized cohomology
|publisher=[[American Mathematical Society]]
|location=Providence, RI
|series=Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs
|volume=230
|year=2006
|origyear=2002
|edition=Translated from the 2002 Japanese edition by Tamaki
|isbn=978-0-8218-3514-2
|mr=2225848
}}

*{{citation
|first= William
|last= Massey
|author-link= William S. Massey
|title= Algebraic Topology: An Introduction
|year= 1984
|publisher= [[Springer Science+Business Media|Springer-Verlag]]
|isbn= 978-0-387-90271-5
}}.

*{{citation
|first= Walther
|last= Mayer
|author-link= Walther Mayer
|title= Über abstrakte Topologie
|year= 1929
|journal= [[Monatshefte für Mathematik]]
|url= http://www.springerlink.com/content/x33611021p942518/
|doi= 10.1007/BF02307601
|issn= 0026-9255
|volume= 36
|issue= 1
|pages= 1–42
}}. {{de icon}}

*{{citation
|first= Edwin
|last= Spanier
|author-link= Edwin Spanier
|title= Algebraic Topology
|year= 1966
|publisher= [[Springer Science+Business Media|Springer-Verlag]]
|isbn= 0-387-94426-5
}}.

*{{citation
|first=Jean-Louis
|last=Verdier
|author-link=Jean-Louis Verdier
|contribution=Cohomologie dans les topos
|editor1-first=Michael
|editor1-last=Artin
|editor1-link=Michael Artin
|editor2-first=Alexander
|editor2-last=Grothendieck
|editor2-link=Alexander Grothendieck
|editor3-first=Jean-Louis
|editor3-last=Verdier
|editor3-link=Jean-Louis Verdier
|title=Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – Tome 2
|year=1972
|publisher = [[Springer Science+Business Media|Springer-Verlag]]
|location = Berlin; Heidelberg
|language = French
|series=[[Lecture Notes in Mathematics]]
|volume=270
|isbn=978-3-540-06012-3
|doi=10.1007/BFb0061320
|pages=1
}}

*{{citation
|first= Leopold
|last= Vietoris
|author-link= Leopold Vietoris
|title= Über die Homologiegruppen der Vereinigung zweier Komplexe
|year= 1930
|journal= [[Monatshefte für Mathematik]]
|volume= 37
|pages= 159–62
|doi=10.1007/BF01696765
}}. {{de icon}}

==Further reading==
* {{citation
|last1=Reitberger
|first1=Heinrich
|title=Leopold Vietoris (1891–2002)
|url=http://www.ams.org/notices/200210/fea-vietoris.pdf
|format=PDF|year=2002
|journal=[[Notices of the American Mathematical Society]]
|issn=0002-9920
|volume=49
|issue=20
}}.

{{good article}}

{{DEFAULTSORT:Mayer-Vietoris Sequence}}
[[Category:Homology theory]]

Main Page

2014-08-11T11:45:32Z

CarsonPitcher:

{{about|the rotation of an object around a single axis (a one-dimensional rotation)|the kinetic energy of an object that rotates in three dimensions|rigid rotor}}

The '''rotational energy''' or '''angular kinetic energy''' is the [[kinetic energy]] due to the rotation of an object and is part of its [[Kinetic energy#Rotation in systems|total kinetic energy]]. Looking at rotational energy separately around an object's [[axis of rotation]], one gets the following dependence on the object's [[moment of inertia]]:

:<math>E_\mathrm{rotational} = \frac{1}{2} I \omega^2 </math>
where
: <math> \omega \ </math> is the [[angular velocity]]
: <math> I \ </math> is the [[moment of inertia]] around the axis of rotation
: <math> E \ </math> is the [[kinetic energy]]
The [[mechanical work]] required for / applied during rotation is the torque times the rotation angle. The instantaneous [[power (physics)|power]] of an angularly accelerating body is the torque times the angular velocity. For free-floating (unattached) objects, the axis of rotation is commonly around its [[center of mass]].

Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion:

:<math>E_\mathrm{translational} = \frac{1}{2} m v^2 </math>

In the rotating system, the [[moment of inertia]], ''I'', takes the role of the mass, ''m'', and the [[angular velocity]], <math> \omega </math>, takes the role of the linear velocity, ''v''. The ''rotational energy'' of a [[wheel|rolling]] [[cylinder (geometry)|cylinder]] varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow).

As an example, let us calculate the rotational kinetic energy of the Earth. As the Earth has a period of about 23.93 hours, it has an angular velocity of 7.29×10−5 rad/s. The Earth has a moment of inertia, I = 8.04×1037 kg·m2.<ref>[http://scienceworld.wolfram.com/physics/MomentofInertiaEarth.html Moment of inertia--Earth], Wolfram</ref> Therefore, it has a rotational kinetic energy of 2.138×1029 J.

Part of it can be tapped using [[tidal power]]. Additional friction of the two global tidal waves creates energy in a physical manner, infinitesimally slowing down Earth's angular velocity ''ω''. Due to the [[Angular momentum#Conservation of angular momentum|conservation]] of [[angular momentum]], this process transfers angular momentum to the [[Moon]]'s [[orbit]]al motion, increasing its distance from Earth and its orbital period (see [[tidal locking]] for a more detailed explanation of this process).

==See also==

*[[Flywheel]]
*[[List of energy storage projects]]
*[[Rigid rotor]]
*[[Rotational spectroscopy]]

==References==
{{reflist}}
{{Footer energy}}

{{DEFAULTSORT:Rotational Energy}}
[[Category:Forms of energy]]
[[Category:Rotation]]