3.4 General properties of homology groups

3.4.1 Connectedness and homology groups

Let $K=\{p_0\}$ and $L=\{p_0,p_1\}$ . From example 3.6 and exercise 3.1, we have $H_0(K)=\mathbb{Z}$ and $H_0(L)=\mathbb{Z}\oplus \mathbb{Z}$ . More generally, we have the following theorem.

Theorem 3.6

Let $K$ be a disjoint union of $N$ connected components, $K=K_1\cup K_2\cup \cdots \cup K_N$ where $K_i\cap K_j=\varnothing$ . Then

$$H_r(K)=H_r\left(K_1\right)\oplus H_r\left(K_2\right)\oplus \cdots \oplus H_r\left(K_N\right).\text{\hspace{1em}(3.52)}$$

Proof

We first note that an $r$ -chain group is consistently separated into a direct sum of $N$ $r$ -chain subgroups. Let

$$C_r(K)=\left\{\sum _{i=1}^{I_r}{c}_i{\sigma }_{r,i}|c_i\in \mathbb{Z}\right\}$$

where $I_r$ is the number of linearly independent $r$ -simplexes in $K$ . It is always possible to rearrange ${\sigma }_i$ so that those $r$ -simplexes in $K_1$ come first, those in $K_2$ next and so on. Then $C_r(K)$ is separated into a direct sum of subgroups,

$$C_r(K)=C_r\left(K_1\right)\oplus C_r\left(K_2\right)\oplus \cdots \oplus C_r\left(K_N\right).$$

This separation is also carried out for $Z_r(K)$ and $B_r(K)$ as

$$Z_r(K)=Z_r\left(K_1\right)\oplus Z_r\left(K_2\right)\oplus \cdots \oplus Z_r\left(K_N\right)$$

$$B_r(K)=B_r\left(K_1\right)\oplus B_r\left(K_2\right)\oplus \cdots \oplus B_r\left(K_N\right).$$

We now define the homology groups of each component $K_i$ by

$$H_r\left(K_i\right)=Z_r\left(K_i\right)/B_r\left(K_i\right).$$

This is well defined since $Z_r(K_i)\supset B_r(K_i)$ . Finally, we have

$$\begin{array}{rl}{H}_r(K)& =Z_r(K)/B_r(K)\\ & =Z_r\left(K_1\right)\oplus \cdots \oplus Z_r\left(K_N\right)/B_r\left(K_1\right)\oplus \cdots \oplus B_r\left(K_N\right)\\ & =\left\{Z_r\left(K_1\right)/B_r\left(K_1\right)\right\}\oplus \cdots \oplus \left\{Z_r\left(K_N\right)/B_r\left(K_N\right)\right\}\\ & =H_r\left(K_1\right)\oplus \cdots \oplus H_r\left(K_N\right).\end{array}$$

Corollary 3.1

(a) Let $K$ be a disjoint union of $N$ connected components, $K_1,\dots ,K_N$ . Then it follows that

$$H_0(K)\cong \underset{N\text{f a c t o r s}}{\underbrace{\mathbb{Z}\oplus \cdots \oplus \mathbb{Z}}}.\text{\hspace{1em}(3.53)}$$

(b) If $H_0(K)\cong \mathbb{Z}$ , $K$ is connected. [Together with theorem 3.5 we conclude that $H_0(K)\cong \mathbb{Z}$ if and only if $K$ is connected.]

3.4.2 Structure of homology groups

$Z_r(K)$ and $B_r(K)$ are free Abelian groups since they are subgroups of a free Abelian group $C_r(K)$ . It does not mean that $H_r(K)=Z_r(K)/B_r(K)$ is also free Abelian. In fact, according to theorem 3.2, the most general form of $H_r(K)$ is

$$H_r(K)\cong \underset{f}{\underbrace{\mathbb{Z}\oplus \cdots \oplus \mathbb{Z}}}\oplus {\mathbb{Z}}_{k_1}\oplus \cdots \oplus {\mathbb{Z}}_{k_p}.\text{\hspace{1em}(3.54)}$$

It is clear from our experience that the number of generators of $H_r(K)$ counts the number of $(r+1)$ -dimensional holes in $|K|$ . The first $f$ factors form a free Abelian group of rank $f$ and the next $p$ factors are called the torsion subgroup of $H_r(K)$ . For example, the projective plane has $H_1(K)\cong {\mathbb{Z}}_2$ and the Klein bottle has $H_1(K)\cong \mathbb{Z}\oplus {\mathbb{Z}}_2$ . In a sense, the torsion subgroup detects the ‘twisting’ in the polyhedron $|K|$ . We now clarify why the homology groups with $\mathbb{Z}$ -coefficients are preferable to those with ${\mathbb{Z}}_2$ - or $\mathbb{R}$ -coefficients. Since ${\mathbb{Z}}_2$ has no non-trivial subgroups, the torsion subgroup can never be recognized. Similarly, if $\mathbb{R}$ -coefficients are employed, we cannot see the torsion subgroup either, since $\mathbb{R}/m\mathbb{R}\cong \{0\}$ for any $m\in \mathbb{Z}-\{0\}$ . [For any $a,b\in \mathbb{R}$ , there exists a number $c\in \mathbb{R}$ such that $a-b=mc$ .] If $H_r(K;\mathbb{Z})$ is given by (3.54), $H_r(K;\mathbb{R})$ is

$$H_r(K;\mathbb{R})\cong \underset{f}{\underbrace{\mathbb{R}\oplus \mathbb{R}\oplus \cdots \oplus \mathbb{R}}}.\text{\hspace{1em}(3.55)}$$

3.4.3 Betti numbers and the Euler-Poincaré theorem

Definition 3.6

Let $K$ be a simplicial complex. The $r$ th Betti number $b_r(K)$ is defined by

$$b_r(K)\equiv \dim H_r(K;\mathbb{R}).\text{\hspace{1em}(3.56)}$$

In other words, $b_r(K)$ is the rank of the free Abelian part of $H_r(K;\mathbb{Z})$ .

For example, the Betti numbers of the torus $T^2$ are (see example 3.12)

$$b_0(K)=1,b_1(K)=2,b_2(K)=1$$

and those of the sphere $S^2$ are (exercise 3.3)

$$b_0(K)=1,b_1(K)=0,b_2(K)=1.$$

The following theorem relates the Euler characteristic to the Betti numbers.

Theorem 3.7 · The Euler-Poincaré theorem

Let $K$ be an $n$ -dimensional simplicial complex and let $I_r$ be the number of $r$ -simplexes in $K$ . Then

$$\chi (K)\equiv \sum _{r=0}^n(-1{)}^r{I}_r=\sum _{r=0}^n(-1{)}^r{b}_r(K).\text{\hspace{1em}(3.57)}$$

[Remark: The first equality defines the Euler characteristic of a general polyhedron $|K|$ . Note that this is the generalization of the Euler characteristic defined for surfaces in section 2.4.]

Proof

Consider the boundary homomorphism,

$${\partial }_r:C_r(K;\mathbb{R})\to C_{r-1}(K;\mathbb{R})$$

where $C_{-1}(K;\mathbb{R})$ is defined to be $\{0\}$ . Since both $C_{r-1}(K;\mathbb{R})$ and $C_r(K;\mathbb{R})$ are vector spaces, theorem 2.1 can be applied to yield

$$\begin{array}{rl}{I}_r& =\dim C_r(K;\mathbb{R})=\dim (\ker {\partial }_r)+\dim (\mathrm{im}{\partial }_r)\\ & =\dim Z_r(K;\mathbb{R})+\dim B_{r-1}(K;\mathbb{R})\end{array}$$

where $B_{-1}(K)$ is defined to be trivial. We also have

$$\begin{array}{rl}{b}_r(K)& =\dim H_r(K;\mathbb{R})=\dim \left(Z_r(K;\mathbb{R})/B_r(K;\mathbb{R})\right)\\ & =\dim Z_r(K;\mathbb{R})-\dim B_r(K;\mathbb{R}).\end{array}$$

From these relations, we obtain

$$\begin{array}{rl}\chi (K)& =\sum _{r=0}^n(-1{)}^r{I}_r=\sum _{r=0}^n(-1{)}^r(\dim Z_r(K;\mathbb{R})+\dim B_{r-1}(K;\mathbb{R}))\\ & =\sum _{r=0}^n\left\{(-1{)}^r\dim Z_r(K;\mathbb{R})-(-1{)}^r\dim B_r(K;\mathbb{R})\right\}\\ & =\sum _{r=0}^n(-1{)}^r{b}_r(K).\end{array}$$

Since the Betti numbers are topological invariants, $\chi (K)$ is also conserved under a homeomorphism. In particular, if $f:|K|\to X$ and $g:|K^{\prime }|\to X$ are two triangulations of $X$ , we have $\chi (K)=\chi (K^{\prime })$ . Thus, it makes sense to define the Euler characteristic of $X$ by $\chi (K)$ for any triangulation $(K,f)$ of $X$ .

3.1 The most general orientable two-dimensional surface is a 2-sphere with $h$ handles and $q$ holes. Compute the homology groups and the Euler characteristic of this surface. 3.2 Consider a sphere with a hole and identify the edges of the hole as shown in figure 3.13. The surface we obtained was simply the projective plane $\mathbb{R}{P}^2$ . More generally, consider a sphere with $q$ such ‘crosscaps’ and compute the homology groups and the Euler characteristic of this surface.

Figure 3.13. A hole in $S^2$ , whose edges are identified as shown. We may consider $S^2$ with $q$ such holes.

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