Sources for some algebraic topology

manpreet Best Answer 2 years ago

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 MM is the boundary of a compact manifold, then the Euler characteristic of MM is even.

2. Show that RP2nRP2n and CP2nCP2n cannot be boundaries.

3. Show that CP2#CP2CP2#CP2 cannot be the boundary of an orientable 55-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.