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Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his 'Algebraic Topology' who uses a very intuitive and natural analogy with the definition of the boundary operator in simplicial homology. (this is the simple one, page 108).

On the other hand we have Rotman in his 'An introduction to homological Algebra' who states that the definition used by Hatcher is wrong because the images under this operator aren't singular simplexes (page 29). I think this state must have something to do with the baricentric coordinates, but I'm not sure. In fact I don't understand his alternative definition that uses face maps where he puts down σε. Is that some kind of product? If anyone could give me an example for n=2 would be awesome.

However the main question is about the differences between these two definitions. Any help is welcome.

Thanks.

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    $\begingroup$ Both authors are defining exactly the same thing, just with slight differences of notation. $\endgroup$
    – Reid Barton
    Commented Dec 23, 2009 at 6:23

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The point Rotman is trying to make is the following: if you have a singular $q$-simplex $\sigma:\Delta^q\to X$, then for example the restriction $\sigma|\_{[e_0,e_2,\dots,e_q]}$ is not a singular $(q-1)$-simplex, simply because its domain $[e_0,e_2,\dots,e_q]$ is not the standard simplex $\Delta^{q-1}$, which is instead $[e_0,e_1,\dots,e_{q-1}]$.

He fixes this by composing with the face maps $\varepsilon$, so as to get the domains right.

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    $\begingroup$ Hatcher also mentions this, immediately after the definition on page 108, but chooses not to reflect it in the notation. $\endgroup$
    – Reid Barton
    Commented Dec 23, 2009 at 6:21

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