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please where i can found the prove of this:

If $X$ is a topological space and $(X_{\alpha})_{\alpha\in I}$ is the family of it's path connected components.

Prove that for each $n\in \mathbb{N}$, $$H_n(X;\mathbb{A})=\bigoplus_{\alpha\in I} H_n(X_{\alpha};\mathbb{A})$$

Please help me

thank you.

Vrouvrou
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  • Try to do it for $n=0$ first. It is clear that 0-dim homology splits over path-components right? If two 0-simplices are in the same path-component, then by definition they are equivalent in homology. – M.B. Oct 28 '13 at 16:37
  • for n=0 , i have $H_0(X;\mathbb{A})=\mathbb{A}$ ! is this theorem have a name ? – Vrouvrou Oct 28 '13 at 16:49
  • It is even true at the level of chain complexes. Hint: The standard simplex is path-connected. – Carsten S Oct 28 '13 at 16:54

1 Answers1

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Let $n\in\mathbb N$ and $s$ be a singular $n$-simplex of $X$, i.e. $s:\Delta^n\to X$, where $\Delta^n$ is the standard $n$-simplex. Since $\Delta^n$ is path-connected, so is its image $s(\Delta^n)$, hence there is an $\alpha\in I$ such that $s(\Delta^n)\subseteq X_\alpha$.

If $c$ is a singular $n$-chain then we can write $\displaystyle c=\sum_{\alpha\in I} c_\alpha$, where each $c_\alpha$ is a chain with support in $X_\alpha$.

Let $\displaystyle z=\sum_{\alpha\in I}z_\alpha$ be a $n$-cycle. Since the $X_\alpha$'s are disjoint sets, so are the supports of each $z_\alpha$ and the boundaries $\partial z_\alpha$ as well, hence

$$0=\partial z=\partial\left(\sum_{\alpha\in I}z_\alpha\right)=\sum_{\alpha\in I}\partial z_\alpha\Rightarrow \forall\alpha\in I,\ \partial z_\alpha=0$$

i.e. each $z_\alpha$ is a $n$-cycle (with support) in $X_\alpha$.

If you identify each $n$-simplex $s:\Delta^n\to X_\alpha$ with the map $s:\Delta^n\to X$, then you can write $z_\alpha\in Z_n(X_\alpha,\mathbb A)$ and $[z_\alpha]\in H_n(X_\alpha,\mathbb A)$.

Using similar arguments with the singular $(n+1)$-boundaries, you can prove that the application below is well defined and is an isomorphism.

$$\Phi_n:\left\{\begin{array}{rl}H_n(X,\mathbb A)&\longrightarrow \bigoplus_{\alpha\in I} H_n(X_\alpha,\mathbb A) \\ [z]&\longmapsto \oplus_\alpha[z_\alpha]\end{array}\right.$$

Also, I invite you to watch this video (and other videos of the same author as well).

Philippe Malot
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  • Thank you, i just want to ask you a question: is $C_p(X_{\alpha},\mathbb{A})$ and $H_n(X_{\alpha},\mathbb{A})$ have a structure of vectoriel space, and why ? because if they don't have it we can't define $\bigoplus_{\alpha\in I} H_n(X_\alpha,\mathbb A)$ please, thank you. – Vrouvrou Nov 02 '13 at 16:04
  • They are $\mathbb A$-modules! – Philippe Malot Nov 02 '13 at 17:24
  • because they are $\mathbb{A}-$modules they have a structure of a vectoriel space ? – Vrouvrou Nov 02 '13 at 17:35
  • $\mathbb A$-modules share many properties with vector spaces. When $\mathbb A$ is a field, a $\mathbb A$-module is just a $\mathbb A$-vector space. Check http://en.wikipedia.org/wiki/Module_(mathematics) – Philippe Malot Nov 02 '13 at 20:18
  • thank you, i missed that the module is a generalisation of a vector space – Vrouvrou Nov 02 '13 at 20:30