Surfaces of infinite-type are non-Hopfian Sumanta Das and Siddhartha Gadgil Abstract

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Surfaces of inﬁnite-type are non-Hopﬁan

Sumanta Das and Siddhartha Gadgil

Abstract

We show that ﬁnite-type surfaces are characterized by a topological analog of the Hopf property.

Namely, an oriented surface Σis of ﬁnite-type if and only if every proper map f:Σ→Σof degree

one is homotopic to a homeomorphism.

1 Introduction

All surfaces will be assumed to be connected and orientable throughout this note. We will say a surface is

of ﬁnite-type if its fundamental group is ﬁnitely generated; otherwise, we will say it is of inﬁnite-type.

Recall that a group Gis said to be Hopﬁan if every surjective homomorphism φ:G↠Gis an isomorphism.

It is well known that a ﬁnitely generated free group is Hopﬁan, for instance, as a consequence of Grushko’s

theorem. On the other hand, a free group generated by an inﬁnite set Sis not Hopﬁan as a surjective

function f:S→Sthat is not injective extends to a surjective homomorphism on the free group generated

by Swhich is not injective.

In this note, we show that there is an analogous characterization for orientable surfaces of ﬁnite-type. The

natural topological analog of a surjective group homomorphism is a proper map of degree one, and that of

an isomorphism is a homotopy equivalence.

One-half of this characterization is classical, namely that any proper map of degree one from a surface

of ﬁnite-type to itself is a homotopy equivalence. For instance, a theorem of Olum (see [2, Corollary

3.4]) says that every proper map of degree one between two oriented manifolds of the same dimension is

π1-surjective. Now, the fundamental group of any surface is residually ﬁnite (see [4]). Also, any ﬁnitely

generated residually ﬁnite group is Hopﬁan. Thus, every degree one self map of a ﬁnite-type surface is a

weak homotopy equivalence, hence a homotopy equivalence by Whitehead’s theorem.

Our main result is that inﬁnite-type surfaces are not Hopﬁan.

Theorem 1.1 Let Σbe any inﬁnite-type surface. Then there exists a proper map f:Σ→Σof degree one

such that π1(f) : π1(Σ)→π1(Σ) is not injective. In particular, fis not a homotopy equivalence.

2 Background

Asurface is a connected, orientable two-dimensional manifold without boundary, and a bordered surface

is a connected, orientable two-dimensional manifold with a non-empty boundary. A (possibly bordered)

subsurface Σ′of a surface Σis an embedded submanifold of codimension zero.

Let Σbe a non-compact surface. A boundary component of Σis a nested sequence P1⊇P2⊇ · · · of

open, connected subsets of Σsuch that the followings hold:

arXiv:2210.03395v2 [math.GT] 6 Jun 2023

• the closure (in Σ) of each Pnis non-compact,

• the boundary of each Pnis compact, and

• for any subset Awith compact closure (in Σ), we have Pn∩A=∅for all large n.

We say that two boundary components P1⊇P2⊇ · · · and P′

1⊇P′

2⊇ · · · of Σare equivalent if for

any positive integer nthere are positive integers kn, ℓnsuch that Pkn⊆P′

nand P′

ℓn⊆Pn. For a boundary

component P=P1⊇P2⊇ · · · , we let [P] to denote the equivalence class of P.

The space of ends Ends(Σ) of Σis the topological space having equivalence classes of boundary components

of Σas elements, i.e., as a set Ends(Σ):={[P]|Pis a boundary component }; with the following

topology: For any set Xwith compact boundary, at ﬁrst, deﬁne

X†:={[P=P1⊇P2⊇ · · · ]|X⊇Pn⊇Pn+1⊇ · · · for some large n}.

Now, take the set of all such X†as a basis for the topology of Ends(Σ). The topological space Ends(Σ) is

compact, separable, totally disconnected, and metrizable, i.e., homeomorphic to a non-empty closed subset

of the Cantor set.

For a boundary component [P] with P=P1⊇P2⊇ · · · , we say [P] is planar if Pnare homeomorphic

to open subsets R2for all large n. Deﬁne Endsnp(Σ):={[P]:[P] is not planar}. Thus, Endsnp(Σ) is a

closed subset of Ends(Σ). Also, deﬁne the genus of Σas g(Σ):=sup g(S), where Sis a compact bordered

subsurface of Σ.

Theorem 2.1 Ker´

ekj´

art´

o’s classiﬁcation theorem [7, Theorem 1]Let Σ1,Σ2be two non-compact

surfaces. Then Σ1is homeomorphic to Σ2if and only if g(Σ1)=g(Σ2), and there is a homeomorphism

Φ: Ends(Σ1)→Ends(Σ2) with ΦEndsnp(Σ1)=Endsnp(Σ2).

Let Σbe a non-compact surface, and let Enp(Σ)⊆E(Σ) be two closed, totally-disconnected subsets of S2

such that the pair Endsnp(Σ)⊆Ends(Σ) is homeomorphic to the pair Enp(Σ)⊆E(Σ). Consider a pairwise

disjoint collection {Di⊆S2\E(Σ) : i∈A}of closed disks, where |A|=g(Σ), such that the following

holds: For p∈S2, any open neighborhood (in S2) of pcontains inﬁnitely many Diif and only if p∈Enp(Σ).

The proof of [7, Theorem 2] describes constructing such a collection of disks.

Now, let M:=(S2\E(Σ)) \⊔i∈Aint(Di) and N:=⊔i∈AS1,1, where S1,1is the genus one compact bordered

surface with one boundary component. Deﬁne a non-compact surface Σhandle as follows: Σhandle :=

M⊔∂M≡∂NN. Then we have the following theorem.

Theorem 2.2 Richards’ representation theorem [7, Theorems 2 and 3]The surface Σhandle is homeo-

morphic to Σ.

3 Proof of Theorem 1.1

Let Mand Nbe two non-compact, oriented, connected, boundaryless smooth n-manifolds. Then the

singular cohomology groups with compact support Hn

c(M;Z) and Hn

c(N;Z) are inﬁnite cyclic with preferred

generators [M] and [N]. If f:M→Nis a proper map then the degree of fis the unique integer deg(f) deﬁned

as follows: Hn

c(f)([N]) =deg(f)·[M]. Note that deg is proper-homotopy invariant and multiplicative. See

[2, Section 1] for more details.

We will use the following well-known characterization of degree.

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Surfacesofinfinite-typearenon-HopfianSumantaDasandSiddharthaGadgilAbstractWeshowthatfinite-typesurfacesarecharacterizedbyatopologicalanalogoftheHopfproperty.Namely,anorientedsurfaceΣisoffinite-typeifandonlyifeverypropermapf:Σ→Σofdegreeoneishomotopictoahomeomorphism.1IntroductionAllsurfaceswillbeassu...

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Surfaces of infinite-type are non-Hopfian Sumanta Das and Siddhartha Gadgil Abstract

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