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I am about halfway through Hatcher's Algebraic topology book, and have learnt something about homology and cohomology. However, there are a few things that I am looking forward to which doesn't appear yet.

  1. The relation between Euler characteristic and genus of a surface. The word genus has been used several times without a formal definition.
  2. The concept of orientablity. Again, this is used informally without definition.

Both of the above concepts are pretty famous, and they doesn't seem to be very difficult (high school students know it, although not rigorously).

I know that orientability is not far away in the book. But is there a reference that proves results about the relation between homology groups and genus?

Ma Joad
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  • Wikipedia: "The genus of a connected, orientable surface (possibly with boundary) is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected." And the euler characterisitc of such a surface is $\chi = 2 − 2g − b$ where b is the number of boundary components. A manifold M is orientable if there is a set of charts that cover M such that all of the transition maps have determinant one. Poincaré duality explains the link of orientability to (co-)homology. Everything here is covered in Munkres. – Noel Lundström Jan 29 '20 at 04:31
  • @NoelLundström The thing is there is no proofs on wikipedia. – Ma Joad Jan 29 '20 at 04:46
  • Everything I just said is covered in Munkres. – Noel Lundström Jan 29 '20 at 05:01
  • The first chapter or two of Armstrong: Topology got you covered I think – Duncan W Jan 29 '20 at 05:25
  • The first part of this seems to be closely related to your earlier question – almagest Jan 29 '20 at 08:50
  • @NoelLundström Sorry if this is a stupid question but I was having a similar problem and I couldn't find any mentioning of the word "genus" in Munkres' book. I'm using the one published by Pearson limited edition (2014). May I ask where can you find the discussion of genus in your book? – BigbearZzz Jul 17 '20 at 14:00

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