Hey, I've been reading Hatcher and doing the exercises, couldn't answer this one:
(p.132 ex. 17)
Tried various directions but I'm pretty much lost. Any help will be appreciated!
Edit: I did get that since is the wedge of two tori, that
Hey, I've been reading Hatcher and doing the exercises, couldn't answer this one:
(p.132 ex. 17)
Tried various directions but I'm pretty much lost. Any help will be appreciated!
Edit: I did get that since is the wedge of two tori, that
I am very, very novice at homology theory so take what I am about to say as probably wrong (and possibly gobbeldy-gook). That said, since no one has answered your question I'll take a stab at my interpretation (see below) of it.
So, what is the in ? Is it the cellular decomposition (I have Hatcher, but not with me. And, I haven't read it yet)? If so, have you discussed the fact yet that the homology is independent of the decomposition? If you have then I think I can help with a) (maybe). Why can't you decompose into a zero-cell, two one-cells, and one two-cell in the natural way so that our chain complex becomes . I think (giving the outward normal orientation of course) it's easy to show that and are all trivial. From there it's pretty easy to calculate that
Haven't gone through cellular homology yet, so only used singular homology, exact sequences and excision for these:
for a), note that is a good pair, and so . The quotient's 1-skeleton has only one path-component, and therefore .
For n=1, use the long exact sequence of the pair X,A:
and and . The first two imply that the map is an isomorphism and from the latter we have the result for n=1.
For n=2, and therefore .
For any n>2, the result is obviously 0.
Similar arguments give us the same for . For the first homology group, the long exact sequence we used to calculate is
and so we get
To finish (b), we note that is homotopy equivalent to a torus with 2 points identified, and since we already know that homotopic spaces have the same homology groups, and since is a good pair, we get:
But we already calculated this one in (a).