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This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space?

We disregard the case of $H_0$ as it has the very satisfactory explanation that it is the direction sum of $\mathbb Z$ over the path-connected components of the space.

Now, handwaving aside, we consider the most important example of this "detecting hole" phenomenon, viz,, the fact that for $i \geq 1$ $H_i(S^n) = \mathbb(Z)$ if and only if $i = n$. For this we use Mayer-Vietoris and a decomposition of $S_n$ into a union of two open sets which are the complements of the north pole and south pole. And the intersection deformation retracts to $S^{n -1}$ and from the long exact sequence we get the isomorphisms $H_i \cong H_{i -1}$.

Now, by the above computation, it seems that the "hole detection" is achieved via Mayer-Vietoris and going up from the dimension below, using the long exact sequence. Mayer-Vietoris on the other hand depends on the snake lemma, which is very un-geometric and difficult to visualize.

So I would be most grateful for a more intuitive explanation of this hole capturing phenomenon. I can see that it is very natural that boundaries should be cancelled out as the solid simplices can be contracted to the central point. I can also "feel" that a hollow $n$-simplex, there should be a nontrivial $n$-chain which is not a boundary of an $n+1$-chain. But I am still left with a feeling of partial understanding. I hope this fundamental vagueness of understanding of mine can be cleared here.

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Have you seen the (sub)section "The Idea of Homology" of Hatcher's book? It helped me a lot. –  Zbigniew Błaszczyk May 13 '10 at 9:49
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Mayer-Vietoris predates the snake lemma, if I have my history correct, and is quite geometric: if $X = U \cup V$ and $\tau$ is a cycle in $X$, then you cut it along the interface between $U$ and $V$ to obtain a chain $\tau'$ lying in $U$, which may no longer be a cycle; so you take its boundary, which will lie in $U\cap V$. (Draw a picture.) This is the map from $H_i(X)$ to $H_{i-1}(U\cap V)$. The other maps in the long exact sequence are similarly geometric. If this is not clear from the text you are using, than I suggest you find another text. –  Emerton May 15 '10 at 5:43
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2 Answers

up vote 3 down vote accepted

The "hole detection" is rather in the very definition of homology. Consider, for example, $H_2$: it is morally the set of closed surfaces in you space modulo those that bound a $3$-dimensional body, and if a surface is not the boundary of any $3$-dimensional body then surely there must be a hole entrapped in it, no?

(Morally because when you want to actually implement this, you get a slightly different thing... Although I'd be thrilled to be informed that, in the case of a manifold at least, say, one can somehow construct a free abelian group on the set of maps $\Sigma\to M$ from $k$-manifolds $\Sigma$ to $M$ which, when one mods out the subgroup of those maps that extend to a manifold-with-boundary $N$ such that $\partial N=\Sigma$, gets you $H_k(M)$ or something close)

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It seems that you are describing the cobordism functor $X \mapsto \Omega_n(X)$ that factors the Hurewicz map: $\pi_n \to \Omega_n \to H_n$. These maps are neither injective nor surjective in general. In particular, not all homology classes are represented by maps from manifolds. –  Sammy Black May 13 '10 at 4:22
    
If you allow something slightly more general than manifold, a trust-worthy source has assures me that the statement becomes true. If that's any consolation. –  David Steinberg May 13 '10 at 8:15
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I prefer true statements to manifolds :) Any references and/or details, David? –  Mariano Suárez-Alvarez May 13 '10 at 8:34
    
Yes, from the definition it is easy to get an intuitive feeling. However making it precise was hard. If it were so easy, there would be an "intuitive" derivation of the homology of S^n, rather than going through Mayer-Vietories. –  Akela May 13 '10 at 8:49
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Homology is bordism of pseudo-manifolds, ie, simplicial complexes such that every codim 1 face meets two top faces (or only 1 on the boundary). a related reference: ams.org/mathscinet-getitem?mr=413113 James McClure complains that the book does not state the theorem and he was going to write something, but I don't know that he did. –  Ben Wieland May 13 '10 at 17:31
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Rather than thinking directly about "holes", I suggest you think about how a circle (in the guise of the boundary of a triangle) is obtained by gluing three intervals at their endpoints, or how a 2-sphere (in the guise of the surface of a tetrahedron) is obtained by gluing together four triangles along their edges. In general, the boundary of an $n+1$ simplex, which is topologically a sphere, is the sum of $n+1$ different $n$-simplices, and this sum is the non-trivial $n$-cycle giving the top homology of the sphere.

If you have trouble connecting this picture with the definitions of singular homology, then try learning simplicial homology first. The computations are then much more explicit, and you can compute the homology of a sphere directly from the preceding triangulations, rather than from a diagram-chasing interpretation of Mayer--Vietoris.

Once you are comfortable with computations in that context, return to the singular theory.

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