Re: Some questions about higher homotopy groups

I am the said friend studying with Avital. We also do not understand
the statement at the top of the page, regarding simply connected cell
complexes to be finitely generated. The author proved only that higher
homotopy groups are abelian, and immediately went on to place this
statement. Perhaps someone could guide us as to how this follows?


On Jun 30, 9:58 pm, avital <avitaloli...@xxxxxxxxx> wrote:
Hi.

A friend of mine are self-studying Algebraic Topology using the book
"Algebraic Topology: An intuitive approach" by Hajime Sato. We are
reading about higher homotopy groups and do not understand a certain
set of explanations (the book does not go into complete proofs).

A scan of the page we have a problem with is athttp://thewe.net/algtop/problem.pdf

Our question regards examples 3.13-3.15:

(a) How would you prove thatis isomorphic to 0? We
understand it intuitively (if you try to "wrap" around a circle using
the x-axis you must lost the fixed point on the top and bottom lines
of the square)
(b) What allows the author to say so simply "if we regard ... as the
quotient space ... hence our claim" (3.13 and 3.14)?
(c) What is this Hopf map (3.15) and wouldn't a representative of 1 be
a function from [;D^3;] to [;S^2;]?

Thanks all,
Avital.

[To see formulas:http://thewe.net/tex]

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