While reading the construction of Moore space from Hatcher's Algebraic topology on page 143, I faced the following problem:
Let $G$ be an abelian group and $0\rightarrow K\rightarrow F\rightarrow G\rightarrow 0$ be a free resolution of $G$ with $\{x_{\alpha}\}$ and $\{y_{\beta}\}$ as basis of $F$ and $K$ respectively such that $y_{\beta}=\sum_{\alpha} d_{\beta \alpha}\ x_{\alpha}$. Writing $X^n=\bigvee_{\alpha} \Bbb S_n^{\alpha}$, we have $H_n(X^n)=F$. Now author constructed maps $\left\{f_{\beta}:\Bbb S^n\rightarrow X^n\right\}_{\beta}$ such that $\deg(p_{\alpha}\circ f_{\beta})=d_{\beta \alpha}$ where, $p_{\alpha}:X^n\rightarrow \Bbb S_{\alpha}^n$ is the projection map. Now let $X$ be obtained from $X^n$ attaching $e_{\beta}^{n+1}$ via $f_{\beta}$. Then $H_n(X)=G$ and $\widetilde {H_i}(X)=0,\forall i\not =n$. Here $n\geq 1$.
I haven't understood the following lines of this problem where $f_{\beta}$ has been constructed. Also, I doubt computing the homology group of this space. Any help will be appreciated. I need some rigorous arguments for this problem.
Best Answer
Given your free resolution of the group $G$, we have $F=\displaystyle\bigoplus_{g\in G} \mathbb{Z}$ and $\{ y_\beta\}$ is a basis of $K\subseteq F$ the kernel of the projection map $F\rightarrow G$. Define, as you said, the CW complex: $$ X^n=\bigvee_{g\in G} S^n. $$ Since $\widetilde{H}_*$ is a reduced homology theory, it is in particular additive. Thus we get: $$ \widetilde{H}_n(X^n)\cong \bigoplus_{g\in G} \widetilde{H}_n(S^n)\cong \bigoplus_{g\in G} \mathbb{Z} = F. $$ For $n\geq 1$, the space $X^n$ is $(n-1)$-connected, so we can apply in particular the Hurewicz theorem which says there is an isomorphism: $$ h:\pi_n(X^n)\stackrel{\cong}\longrightarrow \widetilde{H}_n(X^n)\cong F. $$ Via the above isomorphism, for each basis element $y_\beta$ in $K$, there is a corresponding element $[f_\beta]_*$ in $\pi_n(X^n)$, i.e., a map $S^n\rightarrow X^n$ uniquely defined up to homotopy. This defines a map: $$ f:\bigvee_{\beta} S^n \stackrel{\bigvee f_\beta}\longrightarrow X^n. $$ Your desired CW complex is then obtained as the pushout: \begin{array} $\displaystyle\bigvee_{\beta}S^n &\stackrel{f}{\longrightarrow} & X^n \\ \downarrow & & \downarrow\\ \displaystyle \bigvee_{\beta} D^{n+1} & \longrightarrow & X. \end{array} Notice that the space $X$ is the mapping cone of the map $f$, this should help compute the homology of $X$.