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 $X^n=\lor_{\alpha} S_n^{\alpha}$, we have $H_n(X^n)=F$. Now author constructed maps $\{f_{\beta}:S^n\rightarrow X^n\}_{\beta}$ such that $deg(p_{\alpha}\circ f_{\beta})=d_{\beta \alpha}$ where, $p_{\alpha}:X^n\rightarrow S_{\alpha}^n$ 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 $\tilde H_i(X)=0,\forall i\not =n$. Here $n\geq 1$.

I haven't understood the next lines of this problem where $f_{\beta}$ has been constructed. Also I have a dobut in computing the homology group of this space. Any help will be appreciated. Actually I need some rigorous argument for this problem.

  • $\begingroup$ So the construction is like this in order to apply the cellural boundary formula, right? Together with the fact that homology of wedge sum is a direct product of homologies. $\endgroup$ – freakish Feb 12 at 12:28
  • $\begingroup$ Can you explain the construction of $f_{\beta}$? $\endgroup$ – Mathlover Feb 12 at 12:43

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 $X$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.