Fundamental class
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− | + | {{Authors|Matthias Kreck}} | |
− | == The integral fundamental class== | + | == The integral fundamental class == |
<wikitex include="TeXInclude:">; | <wikitex include="TeXInclude:">; | ||
− | For compact manifolds one can characterize the orientability by the existence of a certain homology class called the fundamental class. The background is the following theorem | + | For compact manifolds one can characterize the orientability by the existence of a certain homology class called the fundamental class. The background is the following theorem. Recall that |
+ | if $M$ is an $n$-dimensional topological manifold (possibly with boundary), then for each $x$ in the interior of $M$, one has $H_n(M,M-x;\Zz)\cong \Zz$, \cite{Greenberg&Harper1981|22.1}. | ||
− | {{beginthm|Theorem}}\label{thm:fundamental_class}Let $M$ be | + | {{beginthm|Theorem}}\label{thm:fundamental_class} |
+ | Let $M$ be an $n$-dimensional compact topological manifold (possibly with boundary). | ||
+ | If $M$ is connected and orientable then for each x in the interior of $M$, the map induced by the inclusion | ||
+ | $$ | ||
+ | H_n(M,\partial M;\Zz) \to H_n(M,M-x;\Zz), | ||
+ | $$ | ||
+ | is an isomorphism. In particular, $H_n(M,\partial M;\Zz)\cong \Zz$. | ||
+ | If $M$ is connected and non-orientable then $H_n(M,\partial M;Z)$ is zero. | ||
{{endthm}} | {{endthm}} | ||
− | {{beginproof}} If $M$ is closed then this part of \cite | + | {{beginproof}} If $M$ is closed then this is part of \cite[VIII Corollary 3.4]{Dold1995}; see also \cite[22.26]{Greenberg&Harper1981}. If $M$ has a boundary $\partial M$, then the inclusion $\partial M=\partial M \times \{1\} \subset M$ extends to an embedding $\partial M \times I \subset M$ of a collar, where $I=[0,1]$ \cite[Proposition 3.42]{Hatcher2002}. Let $M_0={\rm cl.}(M-\partial M \times I)$, so that $M = M_0 \cup_{\partial M} (\partial M \times I)$. By excision |
− | $M = | + | |
− | + | ||
$$H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong | $$H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong | ||
H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$$ | H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$$ | ||
− | Now apply \cite | + | Now apply \cite[VIII Corollary 3.4]{Dold1995} to the open manifold $X : = M - \partial M \times \{ 1 \}$ and the closed subset $M_0 \subset X$. |
{{endproof}} | {{endproof}} | ||
− | Theorem \ref{thm:fundamental_class} implies that a connected compact manifold is orientable if and only if $$H_n(M,\partial M;\mathbb Z) \cong \mathbb Z. $$ A choice of a generator is then called a '''fundamental class''' $[M,\partial M] \in H_n(M,\partial M;\mathbb Z)$ for $M$. The fundamental class determines by the isomorphism above a continuous choice of local orientations and in turn the fundamental class is determined by a homological orientation of $M$. | + | Theorem \ref{thm:fundamental_class} implies that a connected compact manifold $M$ is orientable if and only if $$H_n(M,\partial M;\mathbb Z) \cong \mathbb Z. $$ A choice of a generator is then called a '''fundamental class''' $[M,\partial M] \in H_n(M,\partial M;\mathbb Z)$ for $M$. The fundamental class determines by the isomorphism above a continuous choice of local orientations and in turn the fundamental class is determined by a homological orientation of $M$. In other words a connected compact manifold together with the choice of a fundamental class $[M,\partial M]$ is the same as an oriented manifold. If $M$ is not connected, then $M$ is orientable if and only all components are orientable. If the components are oriented the fundamental classes of the components give the '''fundamental class of $M$''' under the isomorphism which decomposes the homology groups into the homology groups of the components. Thus for oriented manifolds again one has a fundamental class which corresponds to a orientation as in the connected case. The construction of the fundamental class of an oriented closed manifold is done inductively over an atlas (similarly for manifolds with boundary). Namely one has the following generalization of Theorem 1: |
{{beginthm|Theorem|\cite{Greenberg&Harper1981|22.24} }}\label{thm:K_fundamental_class} Let $M$ be a connected oriented $n$-dimensional manifold. Then for each compact subset $K \subset M$ there is a class $[M]_K \in H_n(M, M-K)$ such that the following hold. | {{beginthm|Theorem|\cite{Greenberg&Harper1981|22.24} }}\label{thm:K_fundamental_class} Let $M$ be a connected oriented $n$-dimensional manifold. Then for each compact subset $K \subset M$ there is a class $[M]_K \in H_n(M, M-K)$ such that the following hold. | ||
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# The classes $M_K$ are uniquely characterized by these properties. | # The classes $M_K$ are uniquely characterized by these properties. | ||
{{endthm}} | {{endthm}} | ||
− | Using this one can use the Mayer-Vietoris sequence to "glue" the local orientations inductively over a finite oriented atlas together to construct $M_K$. The inductive construction is rather indirect. If one defines homology of a space $X$ as bordism classes of certain stratified spaces $\mathcal S$ together with a continuous map to $X$, e.g. stratifolds, then the the fundamental class is easy to obtain, it is a tautology. Then the fundamental class of a closed manifold is the bordism class represented by the identity map $$id:M \to M. $$ For this see {{cite|Kreck2010|Chapter 7, Section 1}}. | + | Using this one can use the Mayer-Vietoris sequence to "glue" together the local orientations inductively over a finite oriented atlas together to construct $M_K$. The inductive construction is rather indirect. If one defines the homology of a space $X$ as the bordism classes of certain stratified spaces $\mathcal S$ together with a continuous map to $X$, e.g. stratifolds, then the the fundamental class is easy to obtain, it is a tautology. Then the fundamental class of a closed manifold is the bordism class represented by the identity map $$id:M \to M. $$ For this see {{cite|Kreck2010|Chapter 7, Section 1}}. |
</wikitex> | </wikitex> | ||
== The Z/2-fundamental class== | == The Z/2-fundamental class== | ||
<wikitex include="TeXInclude:">; | <wikitex include="TeXInclude:">; | ||
− | For all $n$-dimensional connected compact manifolds - even if they are not orientable - one has $$ H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2, $$ and one calls the non-trivial element the $\mathbb Z/2$-fundamental class {{cite| | + | For all $n$-dimensional connected compact manifolds - even if they are not orientable - one has $$ H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2, $$ and one calls the non-trivial element the $\mathbb Z/2$-fundamental class {{cite|Dold1995|VIII Definition 4.1}}. As for the integral fundamental class (if $M$ is oriented) one gets from these classes the '''$\mathbb Z/2$-fundamental class''' of a non-connected compact manifold. Also one has a generalization of Theorem \ref{thm:K_fundamental_class} to non-compact connected manifolds, i.e. for each compact subset $K$ one has $H_n(M,M-K;\mathbb Z/2) \cong \mathbb Z/2)$ and for $K \subset K'$ the map induced by the inclusion is an ismorphism $ H_n(M,M-K';\mathbb Z/2) \to H_n(N,M-K;\mathbb Z/2)$. |
</wikitex> | </wikitex> | ||
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{{#RefList:}} | {{#RefList:}} | ||
== External links == | == External links == | ||
− | * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Fundamental_class]. | + | * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Fundamental_class the fundamental class]. |
− | * The Wikipedia page on the [[Wikipedia: | + | * The Wikipedia page on the [[Wikipedia:Fundamental_class|fundamental class]]. |
− | + | [[Category:Definitions]] | |
− | [[Category: | + |
Latest revision as of 17:25, 6 March 2014
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 15:25, 6 March 2014 and the class&diff=cur&oldid=11505 changes since publication. |
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
Contents |
1 The integral fundamental class
For compact manifolds one can characterize the orientability by the existence of a certain homology class called the fundamental class. The background is the following theorem. Recall that
ifTex syntax erroris an -dimensional topological manifold (possibly with boundary), then for each in the interior of
Tex syntax error, one has , [Greenberg&Harper1981, 22.1].
Theorem 1.1.
LetTex syntax errorbe an -dimensional compact topological manifold (possibly with boundary). If
Tex syntax erroris connected and orientable then for each x in the interior of
Tex syntax error, the map induced by the inclusion
is an isomorphism. In particular, .
IfTex syntax erroris connected and non-orientable then is zero.
Proof. If
Theorem 1.1 implies that a connected compact manifold Tex syntax erroris closed then this is part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. If
Tex syntax errorhas a boundary , then the inclusion extends to an embedding of a collar, where [Hatcher2002, Proposition 3.42]. Let
Tex syntax error, so that . By excision
Now apply [Dold1995, VIII Corollary 3.4] to the open manifold and the closed subset .
Tex syntax erroris orientable if and only if
A choice of a generator is then called a fundamental class for
Tex syntax error. The fundamental class determines by the isomorphism above a continuous choice of local orientations and in turn the fundamental class is determined by a homological orientation of
Tex syntax error. In other words a connected compact manifold together with the choice of a fundamental class is the same as an oriented manifold. If
Tex syntax erroris not connected, then
Tex syntax erroris orientable if and only all components are orientable. If the components are oriented the fundamental classes of the components give the fundamental class of
Tex syntax errorunder the isomorphism which decomposes the homology groups into the homology groups of the components. Thus for oriented manifolds again one has a fundamental class which corresponds to a orientation as in the connected case. The construction of the fundamental class of an oriented closed manifold is done inductively over an atlas (similarly for manifolds with boundary). Namely one has the following generalization of Theorem 1:
Theorem 1.2 [Greenberg&Harper1981, 22.24] . Let
Using this one can use the Mayer-Vietoris sequence to "glue" together the local orientations inductively over a finite oriented atlas together to construct . The inductive construction is rather indirect. If one defines the homology of a space as the bordism classes of certain stratified spaces together with a continuous map to , e.g. stratifolds, then the the fundamental class is easy to obtain, it is a tautology. Then the fundamental class of a closed manifold is the bordism class represented by the identity map Tex syntax errorbe a connected oriented -dimensional manifold. Then for each compact subset there is a class such that the following hold.
- If is another compact subset, then maps to under the map induced by the inclusion.
- For each the class is the local orientation of
Tex syntax error
. - The classes are uniquely characterized by these properties.
For this see [Kreck2010, Chapter 7, Section 1].
2 The Z/2-fundamental class
and one calls the non-trivial element the -fundamental class [Dold1995, VIII Definition 4.1]. As for the integral fundamental class (if
Tex syntax erroris oriented) one gets from these classes the -fundamental class of a non-connected compact manifold. Also one has a generalization of Theorem 1.2 to non-compact connected manifolds, i.e. for each compact subset one has and for the map induced by the inclusion is an ismorphism .
3 References
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Kreck2010] M. Kreck, Differential algebraic topology, Graduate Studies in Mathematics, 110, American Mathematical Society, 2010. MR2641092 (2011i:55001) Zbl 05714474
4 External links
- The Encylopedia of Mathematics article on the fundamental class.
- The Wikipedia page on the fundamental class.