Let XX be a CWCW-Complex such that in the cell decomposition of XX there is no 11 cell.Is there any ‘easy’ way to show that π1(X)\pi_1(X) is zero?

I am looking for a easy proof without using some big results like ‘simplicial approximation theorem’

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try to use seifert van kampen.

– Thomas Rot

2 days ago

@ThomasRot Since i am new to fundamental groups,so i dont even have van Kampen theorem with me yet.

– Victor Barg

2 days ago

By “topological space” you probably mean a CW complex, otherwise the claim is trivially false. On the matter of substance, try to show that the n-sphere of dimension >1 is simply connected without using any “big results” (which are not really that “big”), like Seifert – Van Kampen theorem, or simplicial approximation theorem or Sard’s theorem. To the best of my knowledge, such proofs do not exist.

– Moishe Cohen

2 days ago

@MoisheCohen: I know SnS^n is simply connected for n>1n>1 but how does this help?

– Victor Barg

2 days ago

@VictorBarg: Of course, but how do you know this? Imagine yoy have a “Peano circle” which is a surjective continuous map from S1S^1 to S2S^2. How do you know that this map is null-homotopic?

– Moishe Cohen

2 days ago

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