Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?I know, fundamental group of GL(n,C) is isomorphic to Z. Actually, I've succeed in proving this, but my proof is two pages long and very technical. I want1. to find some better proofs of this fact (in order to compare to mine);2. to find some book or article, using which I can learn, how to calculate fundamental groups and, more generally, connectedness components of maps from one space to another;3. to find something for the reference, which I can use in order to learn, how to write this proofs nicely, using standard terminology.

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Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?I know, fundamental group of GL(n,C) is isomorphic to Z. Actually, I&#039;ve succeed in proving this, but my proof is two pages long and very technical. I want1. to find some better proofs of this fact (in order to compare to mine);2. to find some book or article, using which I can learn, how to calculate fundamental groups and, more generally, connectedness components of maps from one space to another;3. to find something for the reference, which I can use in order to learn, how to write this proofs nicely, using standard terminology.

Santiago Collier · Accepted Answer

Step 1The first thing you have to do is to note that the inclusion   U  (  n  )  →      G    L    (  n  ,      C    ) induces an isomorphism on the fundamental groups. This can be done by noting that a loop in GL(n,C) can be deformed to one in U(n) by performing the Gram-Schmidt procedure at each point of the loop, and checking that this can be done continuously and so on.Step 2Next, considering the beginning of the long exact sequence for the homotopy groups of the spaces appearing in the fibration  U  (  n  −  1  )  →  U  (  n  )  →      S          2      n      −      1      which arises from the transitive linear action of U(n) on       S          2      n      −      1        ⊆            C              n       you can prove, by induction, that the inclusion   U  (  1  )  →  U  (  n  ) induces an isomorphism on fundamental groups. Then you can explicitly describe U(1) as a space homeomorphic to       S    1  .

Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?

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