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For simplicial complexes, there is a combinatorial approach to defining the fundamental group $\pi_1$, involving maximal trees and generators with relations (see for instance Armstrong's book pg 133-135). This definition does not use the idea of "loops" explicitly.

How does induced homomorphism work in this context?

For instance, given a simplicial map $f:K\to L$, can we combinatorially define the induced homomorphism $f_*: \pi_1(K)\to\pi_1(L)$, without explicitly using "loops"?

What I mean is given a set of generators and relations in $\pi_1(K)$, can we determine what $f_*$ does to them? Ideally, without explicitly using loops, i.e. avoid defining $f_*$ by $f_*(\gamma)=[f\circ\gamma]$ for a loop $\gamma$.

I have been searching for this approach in books, but did not quite find it. Is there such a thing?

Thanks a lot.

yoyostein
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    I do not believe that such a thing exists. For each maximal tree $T$ in $K$ you obtain a presentation of the fundamental group $\pi_1(K)$. What you would need is an induced mapping between presentations, But in general $f(T)$ is not contained in any maximal tree for $L$ so that you cannot assign generators to generators. – Paul Frost Apr 27 '18 at 10:39

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