Computing homology groups

Direct computations

$H_p(\{\ast \})\cong \mathbb{Z}$ for $p=0$ and $0$ otherwise

Primary note: Singular homology groups of the one-point space

The homology groups of the one-point space $\{\ast \}$ are

$$H_p(\{\ast \})\cong \{\begin{array}{ll}\mathbb{Z}& \text{for }p=0\\ 0& \text{otherwise}\end{array}.$$

This reflects the fact that $H_0$ is generated by a single path-component, while there are no non-trivial cycles (i.e., “holes”) in higher dimensions.

$H_0(X)\cong \mathbb{Z}[{\pi }_0(X)]$, the free abelian group generated by path-components

Primary note: Examples

Let ${\pi }_0(X)$ be the path components of some space $X$, meaning the quotient of $X$ by the relation $x\sim x’$ iff there is a path from $x$ to $x’$. Then there is a well-defined map of sets

$${\pi }_0(X)\to H_0(X),[x]\mapsto [x],$$

where the first $[x]$ is the path component of $x$ and the second is the homology class of the map ${\Delta }^0\to X$ ending the point ${\Delta }^0$ to $X$. This map from the non-group ${\pi }_0(X)$ induces a group homomorphism out of the free abelian group

$$\mathbb{Z}[{\pi }_0(X)]\to H_0(X).$$

In the case that $X$ is path-connected, meaning $[x]=[x’]$ for all $x,x’\in X$, this is reduces to claim that the map

$$\mathbb{Z}{\to }_{\cong }{H}_0(X),n\to n[x]$$

is an isomorphism. Hence the zeroth homology group for a path-connected space is infinite cyclic.

wip HW 6.4: Special case of quotient map inducing a homomorphism of free abelian groups.

$\text{ab}({\pi }_1(X,x))\cong H_1(X)$ when $X$ is path-connected

Primary note: (Theorem) The abelianization of the fundamental group of a path-connected space is isomorphic its first singular homology group

Define the homeomorphism

$$k:{\Delta }^1\to I,(t_0,t_1)\mapsto t_1$$

with inverse $t\mapsto (1-t,t)$. This gives a bijection

$$\lambda \mapsto \lambda \circ k$$

from a path $\lambda :I\to X$ to a $1$-simplex $\lambda \circ k:{\Delta }^1\to X$, where $X$ is any space. We can define a group homomorphism

$$\phi :{\pi }_1(X,x)\to H_1(X),[\lambda ]\mapsto [\lambda \circ k],$$

where $[\lambda ]$ is the homotopy class of a path and $[\lambda \circ k]$ is the homology class of a cycle. The homomorphism $\phi$ induces a homomorphism out of the abelianization of ${\pi }_1(X,x)$, which is the quotient by the commutator subgroup.

The fundamental group and first homology group of a path-connected space are isomorphic

If $X$ is path-connected, the homomorphism $\phi :{\pi }_1(X,x)\to H_1(X)$ sending the homotopy class of a path to the homology class of its composition with a singular $1$-simplex induces an isomorphism $\text{ab}({\pi }_1(X,x))\to H_1(X)$.

Exm

• Since ${\pi }_1(S^1,1)\cong \mathbb{Z}$ is infinite cyclic and already abelian, we have directly $H_1(S^1)\cong {\pi }_1(S^1,1)$. Similarly, the torus $S^1\times S^1$ has ${\pi }_1(S^1\times S^1)\cong H_1(S_1\times S_1)\cong {\mathbb{Z}}^2$.

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Homotopy equivalence and excision

$H_p(X,B)=0$ and $H_p(X,A)\cong H_{p-1}(A,B)$ when $B\hookrightarrow X$ is a homotopy equivalence

Primary note: The long exact sequence in relative homology

For a triple $B\subseteq A\subseteq X$ where the inclusion $B\hookrightarrow X$ is a homotopy equivalence, the long exact sequence

$$\cdots \to H_p(A,B){\to }_{i_{\ast }}{H}_p(X,B){\to }_{j_{\ast }}{H}_p(X,A){\to }_{\delta }{H}_{p-1}(A,B)\to \cdots$$

simplifies to short exact sequences of the form

$$0\to H_p(X,A){\to }_{\cong }{H}_{p-1}(A,B)\to 0,$$

implying that the connecting homomorphism is an isomorphism for all $p$.

$H_p(D^n,\partial D^n)\cong H_p(S^n,D^n)$ for all $p$

Consider the sphere as the union of upper and lower hemispheres

$$S^n=D_{+}^n\cup D_{-}^n,$$

and let $Z\subseteq D_{-}^n$ be a smaller sub-disk. The excision theorem with $X=S^n$, $A=D_{-}^n$, and $Z=Z$ implies that the inclusion

$$(S^n\backslash Z,D_{-}^n\backslash Z)\hookrightarrow (S^n,D^n)$$

induces an isomorphism on all homology groups. On the other hand, the inclusion

$$(D_{+}^n,\partial D_{+}^n)\hookrightarrow (S^n\backslash Z,D_{-}^n\backslash Z)$$

is a homotopy equivalence of pairs, and therefore also induces an isomorphism on all homology groups by homotopy invariance. Thus the map induced by inclusion is an isomorphism

$$H_p(D_{+}^n,\partial D_{+}^n){\to }_{\cong }{H}_p(S^n,D_{-}^n)$$

for all $p\ge 0$.

Homology of a disk relative to its boundary

Primary note: (Theorem) The homology of the n-disk relative to its boundary is infinite cyclic for the nth group and 0 otherwise

Using the fact that

$$H_p(D^n,\partial D^n)\cong H_{p-1}(D^{n-1},\partial D^{n-1})$$

for all $p$ and all $n\ge 1$ (this is shown via a composition of isomorphisms; see theorem note), we use induction to obtain

$$H_p(D^n,\partial D^n)\cong H_{p-n}(D^0,\partial D^0)=H_{p-n}(\{\ast \},\varnothing )=H_{p-n}(\{\ast \})$$ $$\cong \{\begin{array}{ll}\mathbb{Z}& \text{for }p=n\\ 0& \text{otherwise}\end{array}.$$

Homology of a sphere

$$H_p(S^n)\cong \{\begin{array}{ll}{\mathbb{Z}}^2& \text{if }p=n=0\\ \mathbb{Z}& \text{if }n>0\text{ and }n=p\\ \mathbb{Z}& \text{if }n>0\text{ and }p=0\\ 0& \text{otherwise}\end{array}$$

$H_p(X,A)\cong H_p(X\backslash A,A\backslash A)$ when $A\subseteq X$ is a deformation retract