Fundamental class

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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 if $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ is an $n$$n$-dimensional topological manifold (possibly with boundary), then for each $x$$x$ in the interior of $M$$M$, one has $H_n(M,M-x;\Zz)\cong \Zz$$H_n(M,M-x;\Zz)\cong \Zz$, [Greenberg&Harper1981, 22.1].

Theorem 1.1. Let $M$$M$ be an $n$$n$-dimensional compact topological manifold (possibly with boundary). If $M$$M$ is connected and orientable then for each x in the interior of $M$$M$, the map induced by the inclusion

$\displaystyle H_n(M,\partial M;\Zz) \to H_n(M,M-x;\Zz),$ $$\displaystyle H_n(M,\partial M;\Zz) \to H_n(M,M-x;\Zz),$$

is an isomorphism. In particular, $H_n(M,\partial M;\Zz)\cong \Zz$$H_n(M,\partial M;\Zz)\cong \Zz$. If $M$$M$ is connected and non-orientable then $H_n(M,\partial M;Z)$$H_n(M,\partial M;Z)$ is zero.

Proof. If $M$$M$ is closed then this is part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. If $M$$M$ has a boundary $\partial M$$\partial M$, then the inclusion $\partial M=\partial M \times \{1\} \subset M$$\partial M=\partial M \times \{1\} \subset M$ extends to an embedding $\partial M \times I \subset M$$\partial M \times I \subset M$ of a collar, where $I=[0,1]$$I=[0,1]$ [Hatcher2002, Proposition 3.42]. Let

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$M_0={\rm cl.}(M-\partial M \times I)$, so that $M = M_0 \cup_{\partial M} (\partial M \times I)$$M = M_0 \cup_{\partial M} (\partial M \times I)$. By excision

$\displaystyle H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$ $$\displaystyle H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$$

Now apply [Dold1995, VIII Corollary 3.4] to the open manifold $X : = M - \partial M \times \{ 1 \}$$X : = M - \partial M \times \{ 1 \}$ and the closed subset $M_0 \subset X$$M_0 \subset X$.

$\square$$\square$

Theorem 1.1 implies that a connected compact manifold $M$$M$ is orientable if and only if

$\displaystyle H_n(M,\partial M;\mathbb Z) \cong \mathbb Z.$ $$\displaystyle H_n(M,\partial M;\mathbb Z) \cong \mathbb Z.$$

A choice of a generator is then called a fundamental class $[M,\partial M] \in H_n(M,\partial M;\mathbb Z)$$[M,\partial M] \in H_n(M,\partial M;\mathbb Z)$ for $M$$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$$M$. In other words a connected compact manifold together with the choice of a fundamental class $[M,\partial M]$$[M,\partial M]$ is the same as an oriented manifold. If $M$$M$ is not connected, then $M$$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$$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 1.2 [Greenberg&Harper1981, 22.24] . Let $M$$M$ be a connected oriented $n$$n$-dimensional manifold. Then for each compact subset $K \subset M$$K \subset M$ there is a class $[M]_K \in H_n(M, M-K)$$[M]_K \in H_n(M, M-K)$ such that the following hold.

1. If $K \subset K'$$K \subset K'$ is another compact subset, then $[M]_K'$$[M]_K'$ maps to $M_K$$M_K$ under the map induced by the inclusion.
2. For each $x \in M$$x \in M$ the class $[M]_x$$[M]_x$ is the local orientation of $M$$M$.
3. The classes $M_K$$M_K$ are uniquely characterized by these properties.

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$$M_K$. The inductive construction is rather indirect. If one defines the homology of a space $X$$X$ as the bordism classes of certain stratified spaces $\mathcal S$$\mathcal S$ together with a continuous map to $X$$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

$\displaystyle id:M \to M.$ $$\displaystyle id:M \to M.$$

For this see [Kreck2010, Chapter 7, Section 1].

2 The Z/2-fundamental class

For all $n$$n$-dimensional connected compact manifolds - even if they are not orientable - one has

$\displaystyle H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2,$ $$\displaystyle H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2,$$

and one calls the non-trivial element the $\mathbb Z/2$$\mathbb Z/2$-fundamental class [Dold1995, VIII Definition 4.1]. As for the integral fundamental class (if $M$$M$ is oriented) one gets from these classes the $\mathbb Z/2$$\mathbb Z/2$-fundamental class of a non-connected compact manifold. Also one has a generalization of Theorem 1.2 to non-compact connected manifolds, i.e. for each compact subset $K$$K$ one has $H_n(M,M-K;\mathbb Z/2) \cong \mathbb Z/2)$$H_n(M,M-K;\mathbb Z/2) \cong \mathbb Z/2)$ and for $K \subset K'$$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)$$H_n(M,M-K';\mathbb Z/2) \to H_n(N,M-K;\mathbb Z/2)$.

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.