Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm quite at a loss with this...I want to use Mayer-Vietoris with open covers $A=\Sigma_{2}\times (S^{1}\setminus \{p\})$ and $B=\Sigma_{2}\times (S^{1}\setminus \{q\})$ so that $A$ and $B$ both deformation retract to $\Sigma_{2}$ and $A\cap B$ deformation retracts to $\Sigma_{2}\times\{0,1\}$, but I don't understand how to think about the inclusion maps $H_{n}(A\cap B) \hookrightarrow H_{n}(A)\bigoplus H_{n}(B)$.

$\Sigma_2$ denotes the orientable surface of genus two.

share|improve this question
2  
What is $\Sigma_2$? –  Alex Becker Jul 18 '12 at 2:49
    
@AlexBecker Probably the orientable surface of genus $2$. It would be nice to have this explicit in the question, I agree. –  Dylan Moreland Jul 18 '12 at 2:50
    
@DylanMoreland That was my working hypothesis. I've drawn about 20 of those today though, so I want to be sure I'm not just seeing them everywhere. –  Alex Becker Jul 18 '12 at 2:52
    
Yes, $\Sigma_{2}$ is the orientable surface of genus 2. –  subfallen Jul 18 '12 at 17:37
    
@user36025, please add that information to the body of the question. –  Mariano Suárez-Alvarez Jul 18 '12 at 21:27

1 Answer 1

Note that $A$ and $B$ are both homotopic to $\Sigma_2$, and so $H_n(A \cap B) \cong H_n(A) + H_n(B)$ for all $n$, if you choose your embeddings $i$ and $j$ wisely.

share|improve this answer
    
So by cellular homology I have $H_{0}(\Sigma_{2}) \simeq H_{2}(\Sigma_{2}) \simeq \mathbb{Z}$ and $H_{1}(\Sigma_{2}) \simeq \mathbb{Z}^{4}$ where $H_{0}$ is generated by the vertex $v$, $H_{2}$ by the 2-cell $F$, and $H_{1}$ by the 4 loops in the 1-skeleton $e_{1},\dots ,e_{4}$. So the map $H_{1}(A\cap B) \hookrightarrow H_{1}(A)\bigoplus H_{1}(B)$ takes a generator $(e_{i},e_{j})$ to $\dots$what? :( –  subfallen Jul 18 '12 at 17:53
    
These embedding are what I'm confused about. :/ –  subfallen Jul 18 '12 at 17:59
    
Well, $H_1(A)$ should be isomorphic to $\mathbb{Z}^2$, so should have two generators $(e_1^A, e_2^A)$; similarly for $H_1(B)$. And $H_1(A \cap B) \cong H_1(A) \oplus H_1(B)$, so what's the most obvious map you can think of between $H_1(A) \oplus H_1(B)$ and $H_1(A) \oplus H_1(B)$? –  Kris Jul 18 '12 at 21:21
    
If $A$ is homotopic to $\Sigma_{2}$, then $H_{1}(A) \simeq \mathbb{Z}^{4}$ according to topospaces (topospaces.subwiki.org/wiki/… ). I'll try to think about this again tonight. –  subfallen Jul 20 '12 at 19:59

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.