Homology of S^n x R: Hatcher's Theorem 2B.1

[quasar987]

Hi people.

In the proof of theorem 2B.1 page 169-170 of Hatcher (generalized Jordan curve theorem), item (b) is proved by induction on k and the case k=0 is handled by noticing that S^n - h(S^0) is homeomorphic to S^(n-1) x R. How does he know that \widetilde{H}_i(\mathbb{S}^{n-1} \times \mathbb{R}) is Z if i=n-1 and 0 otherwise?

 

[wofsy]

Hi people.

In the proof of theorem 2B.1 page 169-170 of Hatcher (generalized Jordan curve theorem), item (b) is proved by induction on k and the case k=0 is handled by noticing that S^n - h(S^0) is homeomorphic to S^(n-1) x R. How does he know that \widetilde{H}_*(\mathbb{S}^{n-1} \times \mathbb{R})=0?

I'm not sure what homology you are using but H(S^n-1xR) is not zero if you mean Z homology. It is zero in dimension,n.

 

[quasar987]

Excuse me, I meant

"How does he know that \widetilde{H}_i(\mathbb{S}^{n-1} \times \mathbb{R}) is Z if i=n-1 and 0 otherwise?"
(as in the statement of the theorem).

And yes, we are talking about homology with coefficient in Z.

 

[matt grime]

What is k? You're inducting on k, but there is no mention of k in the statement of the problem. Actually, have you stated the problem?

A decent homology theory will turn products into graded tensor products of groups. I.e. the homology of (UxV) in degree n will be the direct sum (over all i) of H_i(U)xH_{n-i}(V) and presumably you've worked out the homology for S^n and R.

 

[yyat]

S^{n-1}\times\mathbb{R} is homotopy equivalent to S^{n-1} by a deformation retraction, and the reduced homology of the sphere is already known.

 

[matt grime]

That's a lot better an explanation than mine.

 

[wofsy]

Excuse me, I meant

"How does he know that \widetilde{H}_i(\mathbb{S}^{n-1} \times \mathbb{R}) is Z if i=n-1 and 0 otherwise?"
(as in the statement of the theorem).

And yes, we are talking about homology with coefficient in Z.

S^n-1 x R has the same homology as S^n-1. the deformation retract argument given in this thread is correct.

You could also use the Kunneth formula and the knowledge that the homology of R is zero except in dimension zero.