Book recommendation: Homology and Cohomology

Question

I need to learn some concepts about Homology and Cohomology theory to apply in riemannian geometry basically, but really I have not time to read about that.

I know just two books of W. S. Massey, Algebraic Topology: An introduction and A basic course of Algebraic Topology, about this (they're big references here), but I don't know what's the difference between them. Is there really much difference for a quick reading? What's the better?

I would like to know if someone has a quick book or lectures notes about Homology and Cohomology theory to help me.

Thanks so much.

Answer 1

Although Amazon says that Massey's Singular Homology Theory is a sequel to Massey's Algebraic Topology: An Introduction, the earlier book is not a logical prerequisite for Singular Homology Theory.

There are plenty of online lecture notes that might be a right fit for you, for example: http://www.math.harvard.edu/~ctm/home/text/class/harvard/213b/01/html/course/course.pdf.

Start by googling, pick a set of introductory lecture notes, and try to do the exercises in them. If you have any questions, ask them here.

Answer 2

I think for differential topology, and geometry it is probably best to learn de Rham cohomology. Bott and Tu's book is the canonical reference.