Any text book or lecture notes regarding the algebraic part of geometry? - MathOverflow

Any text book or lecture notes regarding the algebraic part of geometry?

2012-08-21T21:16:29Z

I know there are text books of Algebraic topology. There are books of Differential geometry. But when I read papers, for example lots of papers talking about fundamental groups or higher homotopy groups of certain manifolds, sometimes lots of terminologies from abstract algebra pop out - nilpotent, solvable or amenable etc. I can understand those definitions, but I feel very uncomfortable that I don't have a geometric feeling of those languages. So I ask for a good reference, ideally written by geometer, that covers the material of this part?

Thanks in advance for any suggestions.

Answer by Igor Rivin for Any text book or lecture notes regarding the algebraic part of geometry?

2012-08-21T22:43:10Z

What about de la Harpe's topics in geometric group theory?

Answer by Mark Grant for Any text book or lecture notes regarding the algebraic part of geometry?

2012-08-22T02:17:23Z

As you mentioned higher homotopy groups, it's worth noting that these are always abelian, which is a stronger condition than each of nilpotent, solvable and amenable. So you probably won't find these adjectives used when describing higher homotopy groups.

Fundamental groups need not be abelian, but they may be almost abelian, and each of the three properties you mention give different ways of formalizing what this should mean. As Igor mentions in his answer, the study of group properties via their realization as fundamental groups of complexes is the topic of geometric group theory.

As for references, I do not know the geometry literature all that well, but Chapter 10 of G. W. Whitehead's book "Elements of Homotopy Theory" contains a nice introductions to nilpotency for topologists.