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I was wondering if there's any way to fit in homotopy into the definition of singular homology. Assume that two singular $n$-simplices are homotopic as maps, does this relate them in any way as elements of the homology group $H_n$?

I guess we should at least get some information for $H_1$ as homotopic loops would generate the same thing in the fundamental group, so when taking the abelianization, they would definitely represent the same element in $H_1$ as singular $1$-simplices.

How much can we say in general? Are there any interesting cases where this provides us with interesting information?

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up vote 5 down vote accepted

$\Delta^n$ doesn't really carry any interesting homological information, since simplices are contractible.

If we add the restriction that the boundary of the simplex maps to a single point, the analog of a loop (indeed, a singular cycle), we in fact see that $\Delta^n / \partial \Delta^n \cong S^n$, so maybe we should be looking at maps out of spheres. Alternately, you could just be thinking about $\partial \Delta^n = S^{n-1}$ to begin with.

In fact, maps from spheres are much more fruitful. A map $f : S^n \to X$ gives an induced map $f_* : H_n(S^n) \to H_n(X)$. Homotopic maps $f,g : S^n \to X$ give equal maps $f_*,g_*$. Notice that $H_n(S^n) = \mathbb{Z}$, and so $f_*$ picks out an element of $H_n(X)$. So in this sense, homotopic maps of spheres pick out equal elements of homology.

If you know about higher homotopy groups, this defines a natural homomorphism $\pi_n(X) \to H_n(X)$ called the Hurewicz map.

This map has a very useful property: if $\pi_1$ is nonvanishing, then the map $\pi_1(X) \to H_1(X)$ is the abelianization homomorphism. If $\pi_k$ vanishes for $k < n$, and $n \ge 2$, then $\pi_n(X) \to H_n(X)$ is in fact an isomorphism! This is a very useful tool for computing the first nonvanishing homotopy group of a space, since homology is typically much easier to compute than the homotopy groups are.

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Thanks. Yes, I should have thought about the obvious fact that you can contract anything to a point on the simplex side, so we need to fix the boundary to prevent the contractibility. That essentially gives higher homotopy groups. – pki Nov 30 '11 at 3:51

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