2
$\begingroup$

I've been looking at some quals problems for algebraic topology that I found online. The problem is that I don't know if I can solve them with the amount of algebraic topology that I know, but nevertheless, they seem interesting. Also I know my committee tends to ask questions about topics not on the syllabus... The problems are as follows:

  1. Show that if a connected manifold $M$ is the boundary of a compact manifold, then the Euler characteristic of $M$ is even.

  2. Show that $\mathbb{R}P^{2n}$ and $\mathbb{C}P^{2n}$ cannot be boundaries.

  3. Show that $\mathbb{C}P^2\# \mathbb{C}P^2$ cannot be the boundary of an orientable $5$-manifold.

  4. Show that the Euler characteristic of a closed manifold of odd dimension is zero.

I haven't found anything in Hatcher that would link manifolds, their dimensions etc. to the Euler characteristic. In particular, I don't know what information in the definition of a manifold would help with computing the Euler characteristic. If someone could provide me with some book, lecture note or anything like that or provide some basic hints, so that I could try to construct enough of the theory myself in order to do the problems above, I'd appreciate it.

  • 0
    For (1), suppose that the Euler characteristic of $M$ is odd. Then come up with a contradiction.2011-03-21
  • 0
    For (2), I guess one can show that the Euler Characteristic is odd.2011-03-21
  • 0
    Well, I assume that is the obvious thing. However, I know of no results that connect the Euler characteristic of a manifold to its boundary. My only idea is that this has something to do with the connected sum, but that would require "filling" the hole with something.2011-03-21
  • 0
    PEV: Regarding your comment to two, it's easy to compute the euler characteristic of both projective spaces, but you can't just blindly apply the first problem, because there's nothing in (2) to rule out boundaries of non-compact manifolds. I don't even know if these are from the same source as I just compiled a long list, so (1) and (2) might not be related...2011-03-21
  • 0
    @dstt: Every manifold $M$ is the boundary of a non-compact manifold: just look at $M\times[0,1)$. The problem has compact ones in mind.2011-03-21
  • 0
    For (4), you need Poincaré duality and the universal coefficient theorem.2011-03-21
  • 0
    For (1) you form the *double* of the bounding manifold, which is a closed manifold. You then apply Poincare duality to the double, to relate its Euler characteristic to the euler characteristics of the bounding manifold and the intersection, which is $M$.2011-03-21
  • 0
    Have you studied Milnor and Stasheff's "Characteristic Classes" book? It gives you tools for (3). These tools are also in Hatcher's Vector Bundles notes. FYI, $\mathbb CP^2 \# \mathbb CP^2$ is the boundary of a non-orientable manifold -- think of a $1$-handle attachment to $[0,1] \times \mathbb CP^2$. Similarly, $\mathbb CP^2 \# -\mathbb CP^2$ bounds an orientable manifold.2011-03-21
  • 0
    @dstt: Apparently (1) and (2) are part of the same problem here: http://www.math.wisc.edu/~maxim/top5S10.pdf2011-03-21
  • 0
    Here is a nice problem that you definitely should be able to solve: http://math.stackexchange.com/questions/24784/why-isnt-mathbbcp2-a-covering-space-for-any-other-manifold . This comes from a nice big list on the Berkeley math grad student association website (google Berkeley MGSA). I think other schools also have similar sites, or at least you can glean problems from people's transcripts.2011-03-21
  • 0
    @Ryan: I haven't studied characteristic classes. I should probably look into them. I figured out the rest of the problems, so thanks for the help. A friend of mine also told me to look at Bredon, which is supposed to cover material like this.2011-03-22

0 Answers 0