When is $H^1(X, \mathbb Z)$ isomorphic to $\text{Hom}(\pi_1(X), \mathbb Z)$, and how?

Question

In Mumford's 'Abelian Varieties', he says that for a good topological space $X$, we have an isomorphism $H^1(X,\mathbb{Z}) \cong \operatorname{Hom}(\pi_1(X),\mathbb{Z})$. What does he mean by a 'good topological space' and what does this isomorphism look like?

Answer 1

I'll elaborate on A.Rod's comment. In fact, we can show that for any abelian group $G$, and path connected space $X$, $$H^1(X;G)\cong \operatorname{Hom}(\pi_1(X,x_0),G).$$ First, by the universal coefficient theorem, there is an exact sequence $$0\xrightarrow{\ \ \ }\operatorname{Ext}_{\mathbb{Z}}^1(H_0(X),G)\xrightarrow{\ \ \ }H^1(X;G)\xrightarrow{\ \tau \ }\operatorname{Hom}(H_1(X),G)\xrightarrow{\ \ \ }0.$$ Since $X$ is path connected, $H_0(X)\cong\mathbb{Z}$, and therefore $$\operatorname{Ext}_{\mathbb{Z}}^1(H_0(X),G) \cong \operatorname{Ext}_{\mathbb{Z}}^1(\mathbb{Z},G) = 0.$$ Thus, by exactness we have an isomorphism $$H^1(X;G)\cong \operatorname{Hom}(H_1(X),G).$$ As A.Rod explained, since $X$ is path connected, there is another isomorphism provided by the Hurewicz theorem: $$\bar{h}:\pi_1(X,x_0)^{ab}:= \frac{\pi_1(x,x_0)}{[\pi_1(X,x_0),\pi_1(X,x_0)]}\xrightarrow{\ \cong\ } H_1(X).$$ Now, recall the universal property that abelianization satisfies:

Let $A$ and $B$ be any groups, with $B$ abelian. Then, any map $\varphi:A\to B$ factors uniquely through a map $\bar{\varphi}:A/[A,A]\to B$ such that $$\varphi = \bar{\varphi}\circ\pi$$ where $\pi:A\to A/[A,A]$ is the usual projection (so this is really just the universal property of quotients).

The above is a fancy way of saying that pre-composition with $\pi$ gives an isomorphism of abelian groups: $$-\circ\pi:\operatorname{Hom}\left(\frac{A}{[A,A]},B\right)\xrightarrow{\ \cong \ }\operatorname{Hom}(A,B)$$ Thus, we have a chain of isomorphisms: $$H^1(X;G)\xrightarrow{\ \tau \ }\operatorname{Hom}(H_1(X),G)\xrightarrow{\ -\circ\bar{h} \ }\operatorname{Hom}(\pi_1(X,x_0)^{ab},G)\xrightarrow{\ -\circ\pi \ }\operatorname{Hom}(\pi_1(X,x_0),G).$$ The map $\tau$ sends a cocycle $[c:C_1(X)\to G]$ to the induced map of the restriction: $$\tau[c:C_1(X)\to G] = \overline{c|_{Z_1(X)}}:H_1(X)\to G,$$ and the Hurewicz map $\bar{h}\circ\pi = h:\pi_1(X,x_0)\to H_1(X)$ sends a homotopy class $\langle\omega\rangle$ to its corresponding homology class $[\omega]$. Thus, the above chain of isomorphisms takes a cocycle $[c:C_1(X)\to G]$ and sends it to the map $$\overline{c|_{Z_1(X)}}\circ h:\pi_1(X,x_0)\to G.$$