Is it true that fundamental group of CPn\mathbb CP^n is zero ? If yes,then how to prove this?

I know that the fundamental group of RPn\mathbb RP^n is C2C_2 the cyclic group of order 22 but i cant see why it is zero for CPn\mathbb CP^n Any ideas/hints?

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The answer can be found on wikipedia and the proofs in any textbook on algebraic topology; did you check any of that?

– Peter Franek

2 days ago

@PeterFranek: I am reading from Munkres topology book and its not given there.Could you give some idea? Is it true that if the cell decomposition of a space does not have any 11 cell then fundamental group of space is zero?

– Victor Barg

2 days ago

There is a simple CW model of the projective space that only has cells in even dimension; no cell in dimension 1. The simplicial approximation theorem then provides a nullhomotopy of loops.

– Peter Franek

2 days ago

@PeterFranek: So if cell decomposition does not have any 11 cell then fundamental group is zero? Is there any ‘easy’ way to see this?

– Victor Barg

2 days ago

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@PeterFranek: Thanks since i have just started learning about fundamental groups so i am looking for an ‘elementary’ answer.

– Victor Barg

2 days ago

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