Fundamental class
(→The Z/2-fundamental class) |
|||
Line 6: | Line 6: | ||
{{beginthm|Theorem}}\label{thm:fundamental_class}Let $M$ be a $n$-dimensional compact topological manifold (possibly with boundary). If $M$ is connected and [[Orientations_of_manifolds#Orientation_of_topological_manifolds|non-orientable]], then $H_n(M,\partial M;\mathbb Z)$ is zero. If $M$ is connected and orientable then $H_n(M, \partial M;\mathbb Z) \cong \mathbb Z$ and for each $x$ in the interior of $M$ the map induced by the inclusion $$ H_n(M,\partial M;\mathbb Z) \to H_n(M,M-x;\mathbb Z) $$ is an isomorphism. | {{beginthm|Theorem}}\label{thm:fundamental_class}Let $M$ be a $n$-dimensional compact topological manifold (possibly with boundary). If $M$ is connected and [[Orientations_of_manifolds#Orientation_of_topological_manifolds|non-orientable]], then $H_n(M,\partial M;\mathbb Z)$ is zero. If $M$ is connected and orientable then $H_n(M, \partial M;\mathbb Z) \cong \mathbb Z$ and for each $x$ in the interior of $M$ the map induced by the inclusion $$ H_n(M,\partial M;\mathbb Z) \to H_n(M,M-x;\mathbb Z) $$ is an isomorphism. | ||
{{endthm}} | {{endthm}} | ||
− | {{beginproof}} If $M$ is closed then this part of \cite{ | + | {{beginproof}} If $M$ is closed then this part of \cite{Dold1995|VIII Corollary 3.4}; see also \cite{Greenberg&Harper1981|22.26}. If $M$ has a boundary $\partial M$, then write |
$M = M_0 \cup_{\partial M} (\partial M \times I)$ where $I = [0, 1]$ and $\partial M \times I$ is a | $M = M_0 \cup_{\partial M} (\partial M \times I)$ where $I = [0, 1]$ and $\partial M \times I$ is a | ||
collar of the boundary of $M$. By excision | collar of the boundary of $M$. By excision | ||
$$H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong | $$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{Dold1995|VIII Corollary 3.4} to the 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 $M$. With 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: | 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 $M$. With 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: |
Revision as of 18:56, 16 December 2012
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. |
This page has not been refereed. The information given here might be incomplete or provisional. |
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:
Proof. If is closed then this part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. If has a boundary , then write where and is a collar of the boundary of . By excision
Now apply [Dold1995, VIII Corollary 3.4] to the manifold and the closed subset .
Theorem 1.2 [Greenberg&Harper1981, 22.24] . Let be 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 .
- The classes are uniquely characterized by these properties.
2 The Z/2-fundamental class
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
- [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 [1].
- The Wikipedia page on the fundamental class.