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.

  • $\begingroup$ For (1), suppose that the Euler characteristic of $M$ is odd. Then come up with a contradiction. $\endgroup$ – PrimeNumber Mar 21 '11 at 16:24
  • $\begingroup$ For (2), I guess one can show that the Euler Characteristic is odd. $\endgroup$ – PrimeNumber Mar 21 '11 at 16:25
  • $\begingroup$ 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. $\endgroup$ – dstt Mar 21 '11 at 16:28
  • $\begingroup$ 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... $\endgroup$ – dstt Mar 21 '11 at 16:47
  • $\begingroup$ @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. $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '11 at 16:55

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