17.4 Homology and cohomology

中文

In the previous section we considered regions that could be subdivided into ‘cubical’ parts. While this has practical advantages when it comes to integration, and makes the proof of Stokes’ theorem relatively straightforward, the subject of homology is more standardly based on triangular cells. There is no essential difference in this change since any $k$-cube is readily triangulated, as well as the converse. For example, a triangle in two dimensions is easily divided into squares (see Fig. 17.4). Dividing a tetrahedron into four cubical regions is harder to visualize, and is left as an excercise for the reader.

Ordered simplices and chains in Euclidean space

A set of points $\{{\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_p\}$ in Euclidean space ${\mathbb{R}}^n$ is said to be independent if the $p$ vectors ${\mathbf{x}}_i-{\mathbf{x}}_0(i=1,\dots ,p)$ are linearly independent. The ordered $p$-simplex with these points as vertices consists of their convex hull,

$$⟨{\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_p⟩=\left\{\mathbf{y}=\sum _{i=0}^p{t}^i{\mathbf{x}}_i\mid \text{ all }{t}^j\ge 0\text{ and }\sum _{i=0}^p{t}^i=1\right\},$$

together with a specific ordering $({\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_p)$ of the vertex points. We will often denote an ordered $r$-simplex by a symbol such as $\Delta$ or ${\Delta }_p$ . Two ordered $p$-simplices with the same set of vertices will be taken to be identical if the orderings are related by an even permutation, else they are given the opposite sign. For example,

Figure 17.4 Dividing the triangular into ‘cubical’ cells

$$⟨\mathbf{x},\mathbf{y},\mathbf{z}⟩=-⟨\mathbf{x},\mathbf{z},\mathbf{y}⟩=⟨\mathbf{y},\mathbf{z},\mathbf{x}⟩.$$

The standard $n$-simplex on ${\mathbb{R}}^n$ is ${\bar{\Delta }}_n=⟨0,{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n⟩$ where ${\mathbf{e}}_i$ is the $i$th basis vector ${\mathbf{e}}_i=(0,0,\dots ,1,\dots ,0)$ , i.e.

$${\bar{\Delta }}_n=\left\{\left(t_1,t_2,\dots ,t_n\right)\mid \sum _{j=1}^n{t}_j=1\text{ and }0\le t_i\le 1(i=1,\dots ,n)\right\}\subset {\mathbb{R}}^n.$$

A 0-simplex ${\mathbf{X}}_0$ is a single point ${\mathbf{X}}_0$ together with a plus or minus sign.

A 1-simplex ${\mathbf{X}}_0,{\mathbf{X}}_1$ is a closed directed line from ${\mathbf{X}}_0$ to ${\mathbf{X}}_1$ .

A 2-simplex ${\mathbf{X}}_0,{\mathbf{X}}_1,{\mathbf{X}}_2$ is an oriented triangle, where the vertices are taken in a definite order.

A 3-simplex is a tetrahedron in which the vertices are again given a specific order up to even permutations. These examples are depicted in Fig. 17.5.

A $p$-chain is a formal sum $\begin{array}{r}C={\sum }_{\mu =1}^M{c}^{\mu }{\Delta }_{\mu }\end{array}$ where $a_{\mu }$ are real numbers and ${\Delta }_{\mu }$ are $p$-simplices in ${\mathbb{R}}^n$ . The set of all $p$-chains on ${\mathbb{R}}^n$ is obviously a vector space, denoted ${\mathcal{C}}_p({\mathbb{R}}^n)$

The $i$th face of a $p$-simplex $\Delta =⟨{\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_p⟩$ is defined as the ordered $(p-1)$ simplex

$${\Delta }_i=(-1{)}^i⟨{\mathbf{x}}_0,\dots ,{\hat{\mathbf{x}}}_i,\dots ,{\mathbf{x}}_p⟩\equiv (-1{)}^i⟨{\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_{i-1},{\mathbf{x}}_{i+1},\dots ,{\mathbf{x}}_p⟩,$$

and the boundary of a $p$-simplex $\Delta$ is defined as the $(p-1)$ -chain

$$\partial \Delta =\sum _{i=0}^p{\Delta }_i=\sum _{i=0}^p(-1{)}^i⟨{\mathbf{x}}_0,\dots ,{\hat{\mathbf{x}}}_i,\dots ,{\mathbf{x}}_p⟩,$$

Figure 17.5 Standard low dimensional $p$-simplexes

and extends to all $p$-chains by linearity. For example,

$$\begin{array}{c}\partial ⟨{\mathbf{x}}_0,{\mathbf{x}}_1⟩=⟨{\mathbf{x}}_1⟩-⟨{\mathbf{x}}_0⟩,\\ \partial ⟨{\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2⟩=⟨{\mathbf{x}}_1,{\mathbf{x}}_2⟩-⟨{\mathbf{x}}_0,{\mathbf{x}}_2⟩+⟨{\mathbf{x}}_0,{\mathbf{x}}_1⟩,\\ \partial ⟨{\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3⟩=⟨{\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3⟩-⟨{\mathbf{x}}_0,{\mathbf{x}}_2,{\mathbf{x}}_3⟩+⟨{\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_3⟩-⟨{\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2⟩.\end{array}$$

For each $p=0,\dots ,n$ the boundary operator generates a linear map $\partial :{\mathcal{C}}_p({\mathbb{R}}^n)\to {\mathcal{C}}_{p-1}({\mathbb{R}}^n)$ by setting

$$\partial \left(\sum _{\mu }{c}^{\mu }{\Delta }_{\mu }\right)=\sum _{\mu }{c}^{\mu }\partial {\Delta }_{\mu }.$$

If we set the boundary of any 0-form to be the zero chain, $\partial \mathbf{x}=0$ it is trivial to see that two successive applications of the boundary operator on any 1-simplex vanishes,

$${\partial }^2⟨{\mathbf{x}}_0,{\mathbf{x}}_1⟩=\partial (⟨{\mathbf{x}}_1⟩-⟨{\mathbf{x}}_0⟩)=0-0=0.$$

This identity generalizes to arbitrary $p$-simplices, for

$$\begin{array}{l}\partial \partial ⟨{\mathbf{x}}_0,{\mathbf{x}}_1,\dots ,{\mathbf{x}}_p⟩={\sum }_{i=0}^p\partial (-1{)}^i⟨{\mathbf{x}}_0,\dots ,{\hat{\mathbf{x}}}_i,\dots ,{\mathbf{x}}_p⟩\\ ={\sum }_{i=0}^p(-1{)}^i[{\sum }_{j<i}(-1{)}^j⟨{\mathbf{x}}_0,\dots ,{\hat{\mathbf{x}}}_j,\dots ,{\hat{\mathbf{x}}}_i,\dots ,{\mathbf{x}}_p⟩\\ +{\sum }_{j>i}(-1{)}^{j+1}⟨{\mathbf{x}}_0,\dots ,{\hat{\mathbf{x}}}_i,\dots ,{\hat{\mathbf{x}}}_j,\dots ,{\mathbf{x}}_p⟩]\\ =0\end{array}$$

since all terms cancel in pairs. The identity ${\partial }^2=0$ follows from the linearity of the boundary operator $\partial$ on ${\mathcal{C}}_p({\mathbb{R}}^n)$ ).

Exercise

Write out the cancellation of terms in this argument explicitly for a 2-simplex ${\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2$ and 3-simplex ${\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3$

A $p$-chain $C$ is said to be a cycle if it has no boundary, $\partial C=0.$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\prime }$ such that $C=\partial C^{\prime }$ . Clearly every boundary is a $p$-cycle since $\partial C={\partial }^2{C}^{\prime }=0$ , but the converse need not be true.

Simplicial homology on manifolds

Let $M$ be an $n$-dimensional differentiable manifold. A (singular) $p$-simplex ${\sigma }_p$ on $M$ is a smooth map $\varphi :U\to M$ where $U$ is an open subset of ${\mathbb{R}}^p$ containing the standard $p$-simplex ${\bar{\Delta }}_p$ . A $p$-chain on $M$ is a formal linear combination of $p$-simplices on $M$,

$$C=\sum _{\mu =1}^M{c}^{\mu }{\sigma }_{p\mu }(c^{\mu }\in \mathbb{R}),$$

and let ${\mathcal{C}}_p(M)$ be the real vector space generated by all $p$-simplices on $M$

For each $i=1,2,\dots ,p-1$ denote by ${\phi }_i:{\mathbb{R}}^{p-1}\to {\mathbb{R}}^p$ the map that embeds ${\mathbb{R}}^{p-1}$ into the plane $x^i=0$ of ${\mathbb{R}}^p$ ,

$${\phi }_i(x^1,\dots ,x^{p-1})=(x^1,\dots ,x^i,0,x^{i+1},\dots ,x^{p-1}),$$

and for $i=0$ set

$${\phi }_0(x^1,\dots ,x^{p-1})=(1-\sum _{i=1}^{p-1}{x}^i,x^1,\dots ,x^{p-1}).$$

The maps ${\phi }_0,{\phi }_1,\dots ,{\phi }_{p-1}$ are $(p-1)$ -simplices in ${\mathbb{R}}^p$ , whose supports are the various faces of the standard $p$-simplex, ${\bar{\Delta }}_p$ ,

$${\phi }_i({\bar{\Delta }}_{p-1})={\bar{\Delta }}_{pi}(i=0,1,\dots ,p-1).$$

If $\sigma$ is a $p$-simplex in $M$, define its $i$th face to be the $(p-1)$-simplex

$${\sigma }_i=\sigma \circ {\phi }_i:{\bar{\Delta }}_{p-1}\to M,$$

and its boundary to be the $(p-1)$-chain

$${\partial }_p\sigma =\sum _{i=0}^{p-1}(-1{)}^i{\sigma }_i.$$

Extend by linearity to all chains $C\in {\mathcal{C}}_p(M)$ ,

$${\partial }_{p}C=\partial \sum _{\mu =1}^M{c}^{\mu }{\sigma }_{p\mu }=\sum _{\mu =1}^M{c}^{\mu }\partial {\sigma }_{p\mu }.$$

A $p$-boundary $B$ is a singular $p$-cycle on $M$ that is the boundary of a $(p+1)$-chain, $B={\partial }_{p+1}C.\mathrm{A}$ $p$-cycle $C$ is a singular $p$-chain on $M$ whose boundary vanishes, ${\partial }_{p}C=0$ Since ${\partial }^2=0$ it is clear that every $p$-boundary is a $p$-cycle.

If we let $B_p(M)$ be the set of all $p$-boundaries on $M,$ , and $Z_p(M)$ all $p$-cycles, these are both vector subspaces of ${\mathcal{C}}_p(M)$

$$\begin{array}{l}{B}_p(M)=\mathrm{im}{\partial }_{p+1},\\ Z_p(M)=\ker {\partial }_p\supseteq B_p(M).\end{array}$$

We define the $p$th homology space to be the factor space

$$H_p(M)=Z_p(M)/B_p(M).$$

Commonly this is called the $p$th homology group, only the abelian group property being relevant. Two cycles $C_1$ and $C_2$ are said to be homologous if they belong to the same homology class – that is, if there exists a chain $C$ such that $C_1-C_2=\partial C$ . The dimension of the $p$th homology space is known as the $p$ th Betti number,

$$b_p=\dim H_p(M),$$

and the quantity

$$\chi (M)=\sum _{p=0}^n(-1{)}^p{b}_p$$

is known as the Euler characteristic of the manifold $M$. A non-trivial result that we shall not attempt to prove is that the Betti numbers are topological invariants – two manifolds that are topologically homeomorphic have the same Betti numbers and Euler characteristic [13].

Example 17.4

Since $\partial 0=0$ every 0-simplex in a manifold $M$ has boundary 0. Hence every 0-chain in $M$ is a 0-cycle, and $Z_0(M)={\mathcal{C}}_0(M)$ . The zeroth homology space $H_0(M)=Z_0(M)/B_0(M)$ counts the number of 0-chains that are not boundaries of 1-chains. Since a 1-simplex is essentially a smooth curve $\sigma :[0,1]\to M$ it has boundary $\sigma (1)-\sigma (0)$ ), where we represent the 0-simplex map $\bullet \to p\in M$ simply by its image point $p$ . Two 0-simplices $p$ and $q$ are homologous if $p-q$ is a boundary; that is, if they are the end points of a smooth curve connecting them. This is true if and only ifthey belong to the same connected component of $M$. Thus $H_0(M)$ is spanned by a set ofsimplices $\{p_0,p_1,\dots \}$ , one from each connected component of $M$, and the zeroth Betti number ${\beta }_0$ is the number of connected components of the topological space $M$.

De Rham cohomology groups and duality

Let ${\mathcal{C}}^r(M)={\Lambda }_r(M)$ be the real vector space consisting of all differential $r$-forms on $M$. Its elements are also known as $r$-cochains on $M$. The exterior derivative $d$ is a linear operator $d$ : $\begin{array}{r}\begin{array}{r}{\mathcal{C}}^r(M)\to {\mathcal{C}}^{r+1}(M)\end{array}\end{array}$ for each $r=0,1,\dots ,n$ , with the property ${\mathrm{d}}^2=0$ ). We write its restriction to ${\mathcal{C}}^r(M)$ as $d$ .

A differential $r$-form $\alpha$ is said to be closed if $\mathrm{d}\alpha =0$, and it is said to be exact if there exists an $(r-1)$-form $\beta$ such that $\alpha =\mathrm{d}\beta$ . Clearly every exact $r$-form is closed since ${\mathbf{d}}^2\mathbf{\beta }=0$ . In the language of cochains these definitions can be expressed as follows: an $r$-cochain $\alpha$ is called an $r$-cocycle if it is a closed differential form, while it is an $r$-coboundary if it is exact. We denoted the vector subspace of $r$-cocycles by $Z^r(M)$ , and the subspace of $r$-coboundaries by $B^r(M)$

$$\begin{array}{l}{Z}^r(M)=\{\alpha \in {\mathcal{C}}^r(M)|\mathrm{d}\alpha =0\}=\ker {\mathrm{d}}_r\subset {\mathcal{C}}^r(M)\\ B^r(M)=\{\alpha \in {\mathcal{C}}^r(M)|\alpha =\mathrm{d}\beta ,\beta \in {\mathcal{C}}^{r-1}(M)\}=\mathrm{im}{\mathrm{d}}_{r-1}\subset Z^r(M).\end{array}$$

The $r$th de Rham cohomology space (group) is defined as the factor space $H^r(M)=Z^r(M)/B^r(M)$ , and any two $r$-cocycles $\alpha$ and $\beta$ are said to be cohomologous if they belong to the same coset, $\alpha -\beta =\mathrm{d}\gamma$ for some $(r-1)$-cochain $\gamma$ . The dimensions of the vector spaces $H^r(M)$ are denoted $b^r$ .

Example 17.5

Since there are no differential forms of degree 1 we always set $B^0(M)=$ 0. Hence $H^0(M)=Z^0(M)$ . A 0-form $f$ is closed and belongs to $Z^0(M)$ if and only if $\mathrm{d}f=0$ . Hence $f=\mathrm{const}$ . on each connected component of $M$, and $H^0(M)=\mathbb{R}+\mathbb{R}+\cdots +\mathbb{R},$ , one contribution from each such component. Hence $b^0=b_0$ is the number of connected components of $M$ (see Example 17.4).

Example 17.6

If $M=\mathbb{R}$ , then from the previous example $H^0(\mathbb{R})=\mathbb{R}.\mathrm{A}1-\mathrm{form}\omega \in {\mathcal{C}}^1(M)$ is closed if $\mathrm{d}\omega =0$. Setting $\omega =f(x)$ $dx$ we can clearly always write

$$\omega =\mathrm{d}f=\frac{\mathrm{d}F}{x}\text{ where }F(x)={\int }_0^{x}f(y)\mathrm{d}y.$$

Hence every closed 1-form is exact and $H^1(\mathbb{R})=0,b^1=0$ . It is not difficult to verify that this is also the value of the Betti numbers, $b_1=0$

Define a bracket $⟨,⟩:{\mathcal{C}}_r(M)\times {\mathcal{C}}^r(M)\to \mathbb{R}$ by setting

$$⟨C,\alpha ⟩={\int }_C\alpha .$$

for every $r$-chain $C\in {\mathcal{C}}_r(M)$ and $r$-cochain $\alpha \in {\mathcal{C}}^r(M)={\Lambda }_r(M)$ . For every $\alpha$ the map $C\mapsto ⟨C,\alpha ⟩$ is evidently linear on ${\mathcal{C}}_r(M)$ and for every $C\in {\mathcal{C}}_r(M)$ the map $\alpha \mapsto ⟨C,\alpha ⟩$ is linear on ${\mathcal{C}}^r(M)$ . By Stokes’ theorem the exterior derivative $d$ is the adjoint of the boundary operator $\partial$ with respect to this bracket, in the sense that

$$⟨C,\mathrm{d}\alpha ⟩=⟨\partial C,\alpha ⟩.$$

The bracket induces a bracket , on $H_r(M)\times H^r(M)$ by setting

$$⟨[C],[\alpha ]⟩=⟨C,\alpha ⟩={\int }_C\alpha$$

for any pair $[C]\in H_r(M)\mathrm{and}[\alpha ]\in H^r(M)$ . It is independent of the choice of representative from the homology and cohomology classes, for if $C^{\prime }=C+\partial C_1$ and ${\alpha }^{\prime }=\alpha +\mathrm{d}{\alpha }_1$ , where

$\partial C=0$ and $\mathrm{d}\alpha =0$, then

$$\begin{array}{l}⟨C^{\prime },{\alpha }^{\prime }⟩={\int }_{C+\partial C_1}\alpha +\mathrm{d}{\alpha }_1\\ ={\int }_C\alpha +{\int }_{\partial C_1}\alpha +{\int }_C\mathrm{d}{\alpha }_1+{\int }_{\partial C_1}\mathrm{d}{\alpha }_1\\ ={\int }_C\alpha +{\int }_{C_1}\mathrm{d}\alpha +{\int }_{\partial C}{\alpha }_1+{\int }_{{\partial }^2{C}_1}{\alpha }_1\text{ by Stokes’ theorem }\\ ={\int }_C\alpha =⟨C,\alpha ⟩.\end{array}$$

For any fixed $r$-cohomology class [$\alpha$], the map $f_{[\alpha ]}:H_p(M)\to \mathbb{R}$ given by

$$f_{[\alpha ]}([C])=⟨[C],[\alpha ]⟩$$

is a well-defined linear functional on $H_r(M)$ . De Rham’s theorem asserts that this corre spondence between linear functionals on $H_r(M)$ and cohomology classes $[\alpha ]\in H^r(M)$ is bijective.

Theorem 17.4 · de Rham’s theorem

(de Rham) The bilinear map on $H_r(M)\times H^r(M)\to \mathbb{R}$ defined by $([C],[\alpha ])\mapsto ⟨[C],[\alpha ]⟩$ is non-degenerate in both arguments. That is, every linear func tional on $H$ ($M$) has theform $f_{[\alpha ]}for$ a uniquely defined $r$-cohomology class [$\alpha$].

The proof lies beyond the scope of this book, and may be found in [6, 10]. There are a variety of ways ofexpressing de Rham’s theorem. Essentially it says that the $r$th cohomology group is isomorphic with the dual space of the $r$th homology group,

$$H^r(M)\cong {\left(H_r(M)\right)}^{\ast }.$$

If the Betti numbers are finite then $b^r=b_r$

The integral of a closed $r$-form $\alpha$ over an $r$-cycle $C$

$$⟨C,\alpha ⟩={\int }_C\alpha$$

is sometimes called a period of $\alpha$. By Stokes’ theorem all periods of $\alpha$ vanish if $\alpha$ is an exact form, and the period of any closed $r$-form vanishes over a boundary $r$-cycle $C=\partial C^{\prime }$ Let $C_1,\dots ,C_k$ be $k=b_r$ linearly independent cycles in $Z_r(M)$ , such that $[C_i]\ne [C_j]$ for $i\ne j$ . De Rham’s theorem implies that an $r$-form $\alpha$ is exact if and only if all the periods ${\int }_{C_i}\alpha =0$ . If $\alpha$ is exact then we have already remarked that all its periods vanish. The converse follows from the fact that $⟨[C],[\alpha ]⟩=0$ for every $[C]\in H_r(M)$ , since [$C$] can be expanded to $[C]={\sum }_{i=1}^k[C_i]$ . By non-degeneracy of the product , we must have $[\alpha ]=0$ so that $\alpha =\mathrm{d}\beta$ for some $(r-1)$-form $\beta .$

Problems

Problem 17.9

Show that any tetrahedron may be divided into ‘cubical’ regions. Describe a procedure for achieving the same result for a general $k$-simplex.

Problem 17.10

For any pair of subspaces $H$ and $K$ of the exterior algebra ${\Lambda }^{\ast }(M)$ , set $H\wedge K$ to be the vector subspace spanned by all $\alpha \wedge \beta$ where $\alpha \in H,\beta \in K$ . Show that

(a) $Z^p(M)\wedge Z^q(M)\subseteq Z^{p+q}(M),$

(b) $Z^p(M)\wedge B^q(M)\subseteq B^{p+q}(M).$

(c) $B^p(M)\wedge B^q(M)\subseteq B^{p+q}(M)$

Problem 17.11

Show that for any set of real numbers $a_1,\dots ,a_k$ there exists a closed $r$-form $\alpha$ whose periods ${\int }_{C_i}\alpha =a_i$

Problem 17.12

$\mathrm{If}{S}^1$ is the unit circle, show that $b^0=b^1=1$