Fundamental groups of noncompact surfaces

Question

I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology is that if $S$ is a noncompact connected surface, then $\pi_1(S)$ is a free group (possibly trivial or $\mathbb{Z}$). I've had a lot of people ask me for references for this fact. I know of two such references:

1) In section 44A of Ahlfors's book on Riemann surfaces, he gives a very complicated combinatorial proof of this fact.

2) This isn't a reference, but a high-powered 2-line proof. Introducing a conformal structure, the uniformization theorem shows that the universal cover of $S$ is contractible. In other words, $S$ is a $K(\pi,1)$ for $\pi=\pi_1(S)$. Next, since $S$ is a noncompact $2$-manifold, its integral homology groups vanish in dimensions greater than or equal to $2$. We conclude that $\pi_1(S)$ is a group of cohomological dimension $1$, so a deep theorem of Stallings and Swan says that $\pi_1(S)$ is free.

There should be a proof of this that you can present in a first course in topology! Does anyone know a reference for one?

Answer 1

I'm reluctant to advertise, but since no one else has answered yet, I'll mention the proof on pp. 142--144 of my book Classical Topology and Combinatorial Group Theory.

Answer 2

If you assume the existence of smooth structure on a noncompact surface then it is easy to show the existence of a proper morse function with no local maximum.This shows that the surface is homotopic to a one dim CW complex. This is the smooth version of Igor's answer.

EDIT BY ANDY PUTMAN: Mohan isn't registered and thus isn't able to comment, but he sent me an email with more details. The result is true in all dimensions : any noncompact smooth n-manifold is homotopy equivalent to an n-1 complex. The key is to construct a strictly subharmonic morse exhaustion function. The subharmonicity prevents the function from having local maxima. Details of this can be found in his paper "Elementary Construction of Exhausting Subsolutions of Ellitpic Operators", which was joint with Napier and was published in L’Enseignement Math´ematique, t. 50 (2004), p. 1–24.

Answer 3

I just ran across this question, and thought I would give a precise version of the proof Ilya suggested. I believe I learned this proof in Richie Miller's topology course, Michigan State University, 1977 or so.

Choose a triangulation of the surface $S$, equipped with the simplicial metric. Choose a maximal one-ended subtree $T$ of the dual 1-skeleton $S^{(1)}$. The subtree $T$ contains every dual $0$-cell, that is, the barycenter of every 2-simplex. Also, $T$ contains dual 1-cells crossing certain $1$-simplices. Let $U$ be the union of the open 2-simplices and open 1-simplices that contain a point of $T$. The metric completion of $U$, denoted $\bar U$, is a closed disc with one boundary point removed, and so there is a deformation retraction from $\bar U$ onto its boundary $\partial \bar U$. Attaching $\bar U$ to $S - U$ in the obvious way to form the surface $S$, the deformation retraction $\bar U \to \partial\bar U$ induces a deformation retraction of $S$ onto $S-U$, wnich is a subcomplex of the 1-skeleton.

By the way, the subtree $T \subset S^{(1)}$ can be constructed by an explicit process. Enumerate the dual $0$-cells $v_1,v_2,\ldots \in S^{(1)}$. Construct one-ended subtrees $T_1,T_2,\ldots \subset S^{(1)}$ as follows. $T_1$ is any proper ray based at $v_1$. If $v_n \in T_{n-1}$ then $T_n = T_{n-1}$. If $v_n \not\in T_{n-1}$, let $T_n$ be the union of $T_{n-1}$ with any arc $\alpha \subset S^{(1)}$ having one endpoint at $v_n$ and intersecting $T_{n-1}$ in its opposite endpoint. Each $T_n$ is a one-ended tree by induction, and since the radius $r$ neighborhood of $v_1$ in $T_n$ stabilizes as $n \to \infty$, it follows that $T = \cup_n T_n$ is a one-ended subtree of $S^{(1)}$, and it is maximal because it contains each $v_i$.

I think this proof generalizes to any dimension, to give the theorem that Igor Belegradek refers to.

--- Edited to simplify and clarify the argument ---

Answer 4

Not to take anything away from the other answers, but I believe that the result (the more general n-dimensional version mentioned by @Igor Belegradek) is actually due to J. H. C. Whitehead: The immersion of an open 3-manifold in euclidean space, Proc. London Math. Soc 11 1961, 81-90., lemma 2.1 (JHC was a little modest calling this a lemma).

Answer 5

In case anyone is interested, I wrote up a detailed account synthesizing the various answers here and correcting some issues I ran into. It is entitled "Spines of manifolds and the freeness of fundamental groups of noncompact surfaces" and can be downloaded from my page of notes here.

Answer 6

I don't have a reference but here is one way to prove what you want. It is a basic result of PL-topology that any open PL-manifold deformations retracts to a subcomplex of lower dimension. Thus you are reduced to showing that the fundamental group of a graph is free, so collapse a maximal subtree to a point, get a wedge of circles and apply Van Kampen.

Answer 7

Related to Mohan's answer, one can give an overkill proof using the fact that non-compact Riemann surfaces are Stein, and every complex $n$-dimensional Stein manifold is homotopy equivalent to an n-dimensional CW complex. This is Theorem 7.2 in Milnor's book on Morse theory.

Answer 8

Let $M$ be a non-compact connected surface. One can assume that $\partial M = \varnothing$. It follows from Lemma 2.2 this paper

• Epstein, D. B. A. Curves on 2-manifolds and isotopies. Acta Math. 115 1966 83–107.

that

1) $\pi_1 M$ is locally free, i.e. every finitely generated subgroup $G$ of $\pi_1 M$ is free;

2) $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups.

However there are locally free groups that are not free.

Recall that a connected subsurface $N \subset M$ is incompressible if the homomorphism $\pi_1 N \to \pi_1 M$ is injective, so one can regard $\pi_1 N$ as a subgroup of $\pi_1 M$.

Lemma 2.2 of Epstein's paper. Let $X \subset M$ be a compact subset and $G$ be a finitely generated subgroup of $\pi_1 M$. Then there is a compact incompressible subsurface $N \subset M$ such that

• $X \subset int(N)$

• $G \subset \pi_1 N \subset \pi_1 M$.

The proof is elementary and is based on Jordan curve theorem and properties of covering spaces.

Proof that $\pi_1 M$ is locally free. Indeed, since $N$ is compact and has non-empty boundary, $N$ can be deformed onto a finite graph, and therefore $\pi_1 N$ is free and contains $G$. Hence, by Nielsen–Schreier theorem, $G$ is free as well.

Proof that $\pi_1 M$ is a union of an increasing countable sequence of finitely generated free subgroups. Represent $M$ as a countable union of compact subsets $X_1 \subset X_2 \subset \cdots $ such that $M = \cup_i X_i$. Let also $G_0 = 1$ be the unit subgroup of $\pi_1 M$, and $N_0 \subset M$ be an incompressible subsurface such that

• $X_0 \subset int(N_0)$ and $G_0 \subset \pi_1 N_0 \subset \pi_1 M$.

Denote $G_1 = \pi_1 N_0$, and let $N_1 \subset M$ be an incompressible subsurface such that

• $X_1 \subset int(N_1)$ and $G_1 \subset \pi_1 N_1 \subset \pi_1 M$,

repeating this process we will obtain that an increasing sequence of incompressible compact subsurfaces $N_0 \subset N_1 \cdots $ such that $M = \cup_i N_i$.

Since every loop in $M$ is contained in some compact subset and therefore in some $N_i$, it follows that $\pi_1 M $ is a union of its finitely generated free subgroups $\pi_1 N_0 \subset \pi_1 N_1 \subset \cdots $

Answer 9

A new approach to spines is available via mass transport theory and Kantorovich duality. This is developed in my PhD thesis.

The idea is elementary: consider the retract $x\mapsto x/|x|$ from the closed unit ball $B:=\{x \in \mathbb{R}^{N} | ~~ |x| \leq 1\}$ to its boundary sphere $\partial B$. The retract has locus-of-discontinuity $Z=\{pt\}$ equal to a point, namely $x=0$. Observe that the inclusion $\{pt\} \hookrightarrow B$ is a homotopy-equivalence. Claim: this is general principle which follows from Kantorovich duality and the optimal transport theory.

For example, let $S$ be closed hyperbolic surface with metric $d$, and $C\hookrightarrow S$ an embedded Cantor set. Let $X:=S-C$ be the Cantor-punctured surface, and let $\sigma$ be the Hausdorff measure on $X$. Similarly let $\tau$ be the Hausdorff measure of $C$ viewed as a subset of $(S,d)$. Now consider the function $c: X \times C \to (0,\infty)$ defined by the rule $$c(x,y_0):= [\int_C d(x,y)^{-2} d\tau(y) ] - \frac{1}{2} d(x,y_0)^{-2}.$$ We view $c(x,y_0)$ as the cost of transporting a unit mass from the source $x\in X$ to target $y_0\in C$. If $\int_X \sigma > \int_C \tau$, then there exist semicoupling measures $\pi$ on $X\times C$ with the property $$proj_X \# \pi \leq \sigma, ~~~\text{and}~~~~proj_C \# \pi = \tau.$$ In otherwords, $\pi$ is a transference plan from the abundant source $\sigma$ to the prescribed target $tau$. (Such measures are called "Semicouplings"). It is standard result of optimal transport that there exists a unique $c$-optimal semicoupling $\pi_*$ which minimizes the total cost $$c[\pi]:=\int_{X\times C} c(x,y) d\pi(x,y).$$

Now imagine we rescale the target measure $\tau\mapsto \lambda \tau$ for scalar $\lambda>0$. If $\lambda \int_C \tau$ is sufficiently close to $\int_X \sigma$, then the $c$-optimal semicoupling $\pi_*$ will have a "locus-of-discontinuity" $Z \hookrightarrow X$ such that $Z$ is a strong-deformation retract of $X$ and $Z$ will be codimension-one (i.e., the "singularity" is the spine).

More specifically, the $c$-optimal semicouplings $\pi_*$ are characterized by the existence of a $c$-concave potentials $\psi: C \to \mathbb{R}$ satisfying $(\psi^c)^c=\psi$. This is Kantorovich's duality theory. The $c$-optimal transport has the form $x\mapsto \partial^c \psi^c (x)$ for every $x\in X$. Here $\partial^c \psi^c(x)$ is a subset of $C$, namely the $c$-subdifferential of $\psi^c$ at $x\in X$. The "locus-of-discontinuity" is more precisely described as the set of $x\in X$ where $\# \partial^c \psi^c (x) \geq 2$, i.e. where the $c$-convex potential $\psi^c$ is not uniquely differentiable. The locus-of-discontinuity $Z$, where $\psi^c$ is finite and not uniquely differentiable, is a closed lipschitz subvariety of $X$. And Kantorovich duality shows $Z \hookrightarrow X$ is a deformation-retract. The existence of this retract is probably not obvious, unless you are well-studied in the mass-transport theory...

But all the details are in my thesis, including applications to spines for the Teichmueller spaces and symmetric spaces of arithmetic-groups. I'd be happy to share the details, since my supervisor has absolutely zero interest in topological applications, and is quite plainly indifferent to algebraic topology.