User joshua - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:26:33Z http://mathoverflow.net/feeds/user/23831 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97449/computing-the-euler-characteristic-of-the-complex-projective-plane-using-differen Computing the Euler characteristic of the complex projective plane using differential topology Joshua 2012-05-20T01:56:22Z 2012-10-14T15:59:15Z <p>I am trying to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for this result, which uses the cellular decomposition of $\mathbb{C}\mathrm{P}^2$ to get $\chi(\mathbb{C}\mathrm{P}^2) = 3$, but I would like to find a proof of this result that relies on concepts like indices of isolated zeros on a vector field. So for the purposes of this question, I would like to utilize the following definition of the Euler characteristic: For a closed orientable manifold $M$ we define $\chi(M) = \sum_i \mathrm{Ind}_{d_i} \mathrm{v}$ where $\mathrm{v}$ is a vector field on $M$ with isolated zeros.</p> <p>In my first attempt at this problem I thought about finding a vector field on $\tilde{\mathrm{v}}$ on $S^5$, and then using the identification $\mathbb{C}\mathrm{P}^2 \cong S^5/\mathrm{U}(1)$, seeing if $\tilde{\mathrm{v}}$ descended to a vector field on $\mathbb{C}\mathrm{P}^2$ with isolated zeros that lent itself to computing the Euler characteristic. I had difficulty making this work out, so I am unsure if this is a good approach to tackling the problem. Any insights?</p> http://mathoverflow.net/questions/98535/tensor-product-of-acyclic-complexes-of-free-abelian-groups Tensor product of acyclic complexes of free abelian groups Joshua 2012-06-01T03:22:46Z 2012-06-01T07:57:57Z <p>I am trying to find a relatively elementary proof of the following result: If $(K,d^K)$ and $(L,d^L)$ are acyclic chain complexes of free abelian groups, then their tensor product $(K \otimes L,d^{K \otimes L})$ is acyclic. I am familiar with the usual proof of this result in terms of the Kunneth tensor product formula, but since that result is considerably more general than this one, I was wondering if there is a simpler proof that works in this case. Anyone have ideas?</p> http://mathoverflow.net/questions/98535/tensor-product-of-acyclic-complexes-of-free-abelian-groups Comment by Joshua Joshua 2012-06-01T04:03:30Z 2012-06-01T04:03:30Z Solving the problem via homotopy equivalences was actually my first idea for a solution, but I could not figure out an obvious equivalence.