# Proof of Nielsen–Schreier Theorem using Covering Spaces

I have two questions about the proof of Theorem VI.34 in these lecture notes. I have typed all that is relevant, below.

Theorem. (Nielsen-Schreier) Any subgroup of a finitely generated free group is free.

Proof. Let $$F$$ be the free group on $$n$$ generators. Let $$X$$ be the wedge of $$n$$ circles, and let $$b$$ be the central vertex. Then, $$\pi_1(X,b) \cong F$$. Let $$H$$ be any subgroup of $$F$$. Then, by Theorem VI.32, there is a based covering $$p: (\tilde X, \tilde b) \to (X,b)$$ such that $$p_\ast \pi_1(\tilde X, \tilde b) = H$$. By Corollary VI.18, $$p_\ast$$ is injective, and so $$\pi_1(\tilde X,\tilde b) \cong H$$. Theorem VI.32 states that $$\tilde X$$ is a simplicial complex, and that $$p$$ is a simplicial map. Since $$p$$ is a local homeomorphism, $$\tilde X$$ can contain only zero and one-dimensional simplices. Therefore, $$\tilde X$$ is a graph, and so by Theorem IV.11, its fundamental group is free.

1. How does $$p$$ being a local homeomorphism ensure that $$\tilde X$$ contains only zero and one-dimensional simplices? A rigorous proof would be preferred, but intuition (which can be translated to a proof) is also welcome.

2. I believe the author meant to write connected graph instead of just graph. We know that the fundamental group of connected graphs is free, I don't think this holds for graphs in general. If I am right, why is this graph connected?

Thank you!

1. This is perhaps somewhat misguidingly put. The claim is true since the dimension of a simplex is a homeomorphism invariant, but this is non-trivial. Instead, I would refer to the proof of Theorem VI.32, where the simplicial structure explicitly constructed on $$\tilde{X}$$ has faces with at most as many vertices as a face in $$X$$ has vertices. Therefore, if $$X$$ has dimension $$\le1$$, so does $$\tilde{X}$$.
2. It is proven as part of Theorem VI.32 that $$\tilde{X}$$ is path-connected. However, this really does not matter. Fundamental groups only make sense relative to a basepoint and the fundamental group $$\pi_1(\tilde{X},\tilde{b})$$ only depends on the connected component of $$\tilde{X}$$ containing $$\tilde{b}$$, which is a connected graph even a priori. In particular, the fundamental group of any graph relative to any basepoint is free and this statement really has no more content than when you only state it for connected graphs.
• The dimension of a simplicial complex $X$ can be characterized as the largest value of $n$ for which there exists an $x\in X$ such that the local homology group $H_n(X,X\setminus\{x\})$ is non-trivial. This notion is invariant under homeomorphism. Commented Oct 27, 2021 at 21:16