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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.

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    @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

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