A simplicial complex with vanishing first homology but nonzero fundamental group

Question

I'm interested in the simplicial complex and I do not know much of algebraic topology to use the answers that question:

https://mathoverflow.net/questions/110158/what-if-i-want-to-look-for-a-space-with-vanishing-first-homology-but-nonzero-fun#

Thanks!!

Answer 1

Consider the alternating group $A_5$ which is perfect. Take a wedge sum of 60 oriented circles indexed by elements of $A_n$ and to each relation in $A_5$---for example, $(1,2,3)(1,3,2)=\mathrm{id}$---glue a $2$-cell in such a way that its boundary will correspond to the relation, in this case, to the circle $(1,2,3)(1,3,2)$. This will trivialize the loop $(1,2,3)(1,3,2)$. In this way, you obtain a $2$-complex with fundamental group $A_5$ and the first homology group, being its abelianization, will be zero.