|WikiProject Mathematics||(Rated Start-class, Low-importance)|
This page is very nice and clean. Can someone add details about the boundary map in the homology sequence?
Also, a particularly nice example is given (), yet I believe even knowning the cellular decomposition of many higher dimensional CW complexes does not allow one necessarily to compute the homology. Anyone know more about this? MotherFunctor 04:01, 22 May 2006 (UTC)
- It depends on what you mean be knowing. As long as you know all the relevant degrees associated to the cellular attaching maps, you can compute the homology. This article is missing the relevant details but they're in the standard textbooks like Hatcher and Spanier. Rybu (talk) 20:23, 22 November 2009 (UTC)
- I've added some detail about the boundary map, it's Hatcher's definition.
When it says,
doesn't that depend (in homology) on a chosen orientation of the source? So, a choice is required and that choice affects the degree entering the boundary map. So, the resulting homology would be affected. Is the definition only for oriented CW-complexes? If not, why does a set of choices of these isomorphisms with spheres not affect the resulting homology of the complex? Or does it? It would be great if someone could clarify this. I couldn't find answer in Hatcher's book either. Thanks.
- The cells of a CW-complex come equipped with more than an orientation, they have characteristic maps which give an explicit parametrization of them. Under the homotopy-equivalence to the wedge of spheres you mention, the characteristic maps induce canonical orientations on the spheres (or equivalently one takes the homotopy-equivalences to preserve orientation with the standard orientation of the sphere). Rybu (talk) 15:27, 5 September 2013 (UTC)
- I see, thanks. So, the cells are naturally oriented, and that leaves no room for ambiguity, and the characteristic map dictates already canonically such maps to the spheres on the level of sets, not only homotopy. My confusion came from considering CW-complexes with a group action: When one says, G-CW-complex for a group G, does one automatically assume that the group action intertwines the characteristic maps? Given your answer, probably yes. (However, I see that this point is not related to the page anymore, since it does not speak about the equivariant version.)