On the Nonexistence of Certain Branched Covers

G GG T T T G T G TT G T G T G T G T G T G T G T G T GG G G T T T eometry & G opology T Volume 16 (2012)

On the nonexistence of certain branched covers

PEKKA PANKKA JUAN SOUTO

msp eometry & opology 16 (2012) 1321–1344 msp G T

On the nonexistence of certain branched covers

PEKKA PANKKA JUAN SOUTO

We prove that while there are maps T 4 #3.S2 S2/ of arbitrarily large degree, ! there is no branched cover from the 4–torus to #3.S2 S2/. More generally, we obtain that, as long as a closed manifold N satisﬁes a suitable cohomological condition, any n 1 –surjective branched cover T N is a homeomorphism. ! This article is incorrect. Text of retraction received 14 February 2019

57M12; 30C65, 57R19 1 Introduction

Starting with the classical result of Alexander[1], asserting that every closed, orientable PL–manifold of dimension n admits a branched cover onto the n–dimensional sphere Sn , a series of authors such as Berstein and Edmonds[3], Edmonds[11], Hilden[15], Hirsch[16], Iori and Piergallini[18], Montesinos[19] and Piergallini[20] have proved the existence of branched covers satisfying certain conditions between certain manifolds. In this note, the starting point is a general existence theorem due to Edmonds[11]: Every 1 –surjective map f M N between closed, orientable W ! 3–manifolds with deg.f / 3 is homotopic to a branched cover. We show that, already in 4 dimensions, there are subtle obstructions for the existence of branched covering mappings.

Theorem 1.1 There is no branched cover from the 4–dimensional torus T 4 to #3.S2 S2/, the connected sum of three copies of S2 S2 . On the other hand, 4 3 2 2 there are, a fortiori 1 –surjective, maps T # .S S / of arbitrarily large degree. ! Recall that a branched cover between closed PL–manifolds is a discrete and open PL–map; in the sequel we work always in the PL–category and consider only oriented manifolds and orientation preserving maps; this is no reduction of generality as the torus admits an orientation reversing involution. One of the reasons why we ﬁnd Edmonds’s theorem to be rather surprising is because it asserts that, in 3 dimensions, there are many more branched covers than we would

Published: 10 July 2012 DOI: 10.2140/gt.2012.16.1321 1322 Pekka Pankka and Juan Souto naively have expected. A different interpretation of the same theorem is that it is not clear how to distinguish between branched covers and maps of positive degree. When trying to rule out the existence of branched coverings between two closed manifolds M and N , one can perhaps observe that any branched cover f M N induces, via W ! pullback, an injective homomorphism (1-1) f H .N R/ H .M R/ W I ! I between the associated cohomology rings. Although this poses a huge restriction to the existence of branched covers, the same restriction applies for the existence of mere maps of nonzero degree. In fact, it is due to Duan and Wang[10] that, at least when the target N is a simply connected 4–dimensional manifold, the existence of maps M N ! of positive degree is equivalent to the existence of maps H 2.N Z/ H 2.M Z/ I ! I scaling the intersection form. In the light of Theorem 1.1, the existence of branched covers is a more subtle issue which does not seem to be detected by any standard cohomological property. Cohomological considerations play a central role in the proof of Theorem 1.1, but they come into our argument in a perhaps unexpected way.

Quasiregularly elliptic manifolds: the original motivation We would like to mention that our interest to investigate the existence of branched covers from T 4 onto #3.S2 S2/ stems from a question of Gromov[12; 13] and Rickman[21], who asked if the manifolds #k .S2 S2/ are K–quasiregularly elliptic for some K. Recall that a continuous mapping f M N between oriented Riemannian n– W ! manifolds is said to be K–quasiregular, with (outer) distortion K 1, if f is a 1;n Sobolev mapping in W .M N / and satisﬁes the quasiconformality condition loc I Df n KJ a:e:; j j Ä f where Df is the operator norm of the differential Df and J is the Jacobian determi- j j f nant. A Riemannian manifold of dimension n is K–quasiregularly elliptic if it admits a nonconstant K–quasiregular mapping from Rn . We refer to Rickman[22] and Bonk– Heinonen[4] for a detailed discussion on quasiregular mappings and quasiregularly elliptic manifolds respectively. Returning to the quasiregular ellipticity of manifolds #k .S2 S2/, recall that Bonk and Heinonen showed in[4] that K–quasiregularly elliptic manifolds have quantitatively bounded cohomology. In particular, there exists a bound k0 k0.K/ depending only k 2 2 D on K, so that # .S S / is not K–quasiregularly elliptic for k > k0 . Without a priori bounds on the dilatation K, the question on the quasiregular ellipticity of manifolds #k .S2 S2/ is only answered for k 1; 2. We sketch the argument D

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1323 used in these two cases. Observing that branched covers between closed manifolds are quasiregular, it follows that every manifold N for which there is a branched cover f T 4 N is quasiregularly elliptic. Considering T 4 as the product of two 2– W ! dimensional tori and taking on each factor a branched covering T 2 S2 , one obtains ! a branched cover T 4 S2 S2 and deduces that the latter manifold is quasiregularly ! elliptic. What is really much more surprising, is that there is also a branched cover T 4 #2.S2 S2/; see[23] for this construction. In particular #2.S2 S2/ is also ! quasiregularly elliptic. The unfortunate offshoot of Theorem 1.1 is that this argument cannot be applied to #k .S2 S2/ for k 3. Notice that for k > 3, this observation follows directly from D the injectivity of the homomorphism (1-1).

Sketch of the proof of Theorem 1.1 We will deduce the existence of maps T 4 ! #3.S2 S2/ with arbitrarily large degree from the work of Duan and Wang[10]. The nonexistence of branched covers in Theorem 1.1 is a special case of the following more general result:

Theorem 1.2 Let N be a closed, connected, and oriented n–manifold, n 2, so that dim H r .N R/ dim H r .T n R/ for some 1 r < n. Then every branched cover n I D I Ä n T N is a cover. In particular, every 1 –surjective branched cover T N is a ! ! homeomorphism.

Suppose that f T n N is a branched cover as in the statement of Theorem 1.2. W ! By a result of Berstein and Edmonds[2], there are a compact polyhedron X , a finite group G of automorphisms of X and a subgroup H of G such that N X=G , D T n X=H and such that the map f is just the orbit map X=H X=G . Our first D ! goal is to prove that the group G acts on the H –invariant cohomology H .X R/H I of X ; this is the content of Section 3 below. In Section 4, we use the action of G on H 1.X R/H to construct, by averaging, 1 I H cocycles ‚1;:::;‚n which form a basis of H .X R/ and behave well with respect I to the action of G on C 1.X R/, the vector space of 1–cochains on X . I These cocycles give rise to a map X Rn=Zn which semiconjugates the action of G ! on X to a certain action on Rn=Zn . This map is defined in the same way as the classical Abel–Jacobi map and this is the way we will refer to it below. The reader can find the construction and some properties of the Abel–Jacobi map in Section 5 and Section 6. The Abel–Jacobi map and its G –equivariance properties are the key tools to show that the map f T n N factors as T n N N where the first W ! ! 0 !

eometry & opology, Volume 16 (2012) G T 1324 Pekka Pankka and Juan Souto

arrow is a cover and the second arrow has at most degree 2; moreover, N 0 is homotopy equivalent to a torus. In Section 7 we will conclude the proof of Theorem 1.2 showing that the branched cover N N is a homeomorphism. In order to do so, we have to rule out that it has 0 ! degree 2. Every degree 2 branched cover is regular, meaning that there is an involution N N such that N N = and that the map N N N = is the W 0 ! 0 D 0 h i 0 ! D 0 h i orbit map. At this point we will already know that the involution acts as id on n H1.N R/ R . The Lefschetz fixed-point theorem implies that the fixed-point set 0I ' Fix./ is not empty. The final contradiction is obtained when we show, using Smith theory, that Fix./ is in fact empty. Once Theorem 1.2 is proved, we discuss Theorem 1.1 in Section 8.

Acknowledgements The ﬁrst author would like to thank Seppo Rickman for frequent discussions on these topics and encouragement. The second author would like to express his gratitude to the Department of Mathematics and Statistics of the University of Helsinki for its hospitality. Both authors would like to thank the referee for kind remarks and suggestions improving the manuscript. The ﬁrst author has been supported by Academy of Finland project number 1126836 and by NSF grant DMS-0757732. The second author has been partially supported by NSF grant DMS-0706878, NSF Career award 0952106 and the Alfred P Sloan Foundation. The second author also wishes to dedicate this paper to Paul el pulpo.

2 Preliminaries

In this section we remind the reader of a few well-known facts and deﬁnitions. Through- out this note we will work in the PL–category. We refer the reader to Hudson[17] and Rourke and Sanderson[24] for basic facts on PL–theory.

2.1 Chains, cochains and homology

Suppose that X is a compact polyhedron, ie the geometric realization of a ﬁnite simplicial complex and let R be either R or Z. Denote by C .X R/, Z .X R/, C k .X R/ k I k I I and Zk .X R/ the R–modules of singular k –chains, k –cycles, k –cochains and k – I cocycles with coefﬁcients in R. The associated homology and cohomology groups are denoted, as always, by H .X R/ and H k .X R/. The homology (resp. cohomology) k I I class represented by a chain (resp. cochain) ˛ will be denoted by Œ˛.

If X Y is a continuous map to another polyhedron Y , denote by # C .X R/ W ! W k I ! C .Y R/ and by # C k .Y R/ C k .X R/ the pushforward and pullback of chains k I W I ! I

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1325 and cochains, respectively. On homological level, we denote the pushforward and pullback by H .X R/ H .Y R/ and H .Y R/ H .X R/ respectively. W I ! I W I ! I Finally, denote by ; C k .X R/ C .X R/ R h iW I k I ! the canonical evaluation form. Notice that, with the same notation as above, we have for ! C k .Y R/ and c C .X R/ 2 I 2 k I # (2-1) .!/; c !; #.c/ : h i D h i The induced bilinear form on H k .X R/ H .X R/ is also denoted by ; ; again I k I h i (2-1) is satisﬁed

(2-2) .Œ!/; Œc Œ!; .Œc/ : h i D h i Recall that if c and ! are a chain and a cochain representing the homology and cohomology classes Œc and Œ! we have !; c Œ!; Œc . h i D h i Continuing with the same notation, suppose that G is a ﬁnite group acting on the polyhedron X . We denote by X=G the quotient space and by

G X X=G W ! the orbit map. The G –invariant real cohomology H .X R/G ! H .X R/ g .!/ ! for all g G I D f 2 I j D 2 g of X is isomorphic to the real cohomology H .X=G R/ of the orbit space X=G . I Given that this fact will be used over and over again, we state it as a proposition:

Proposition 2.1 Let X be a polyhedron and G a ﬁnite group acting on X . The pull- G back of the orbit map G X X=G induces an isomorphism between H .X R/ W ! I and H .X=G R/. I Proposition 2.1 is a consequence of the natural transfer homomorphism H .X=G R/ I ! H .X R/; see eg Bredon[6, III.2]. Below we will need the following closely related I fact, whose proof we leave to the reader:

Lemma 2.2 Let X be a polyhedron, G a ﬁnite group acting on X and suppose that c is a k –chain whose pushforward .G/#.c/ under the orbit map G X X=G is a P W ! k –cycle in X=G . Then g G g#.c/ is a k –cycle in X . 2 Moreover, if Œ! is a k –cohomology class in X=G then X .Œ!/; g#.c/ G Œ!; Œ.G/#.c/ : G D j jh i g G 2

eometry & opology, Volume 16 (2012) G T 1326 Pekka Pankka and Juan Souto

We refer the reader to[6;7; 14] for basic facts of algebraic topology.

2.2 Branched covers

A branched cover between oriented PL–manifolds is an orientation preserving discrete and open PL–map. Given any such branched cover f M N we say that x M W ! 2 is a singular point of f if f is not a local homeomorphism in a neighborhood of x. The image f .x/ of a singular point is a branching point of f . We denote by S M f and B f .S / N the sets of singular points and branch points of f respectively. f D f We also denote by Z f 1.B / the preimage of the set of branching points of f ; f D f notice that S Z . f f

Before moving on, observe that since f is PL, discrete and open, the sets Sf ; Bf and Zf are polyhedra of codimension at least 2. Remark also that the restriction of f to M Z is a covering map onto N B . n f n f

Remark It is worth noticing that the above statement does not require the mapping to be PL. Indeed, given a discrete and open mapping between n–manifolds the singular set has topological codimension at least 2 by a result of Cernavskiˇ ˘ı[8] and Väisälä[25]. Moreover, Bf and Sf have the same (topological) dimension.

A branched cover f M N is said to be regular if there is a group G of PL– W ! automorphisms of M such that there is an identiﬁcation of N with M=G in such a way that the map f becomes the orbit map G M M=G N . The following W ! D result, due to Berstein and Edmonds[2, Proposition 2.2], asserts that every branched cover is a, nonregular, orbit map:

Proposition 2.3 Given a branched cover f M N between closed orientable W ! manifolds of dimension n, there exists a connected polyhedron Xf of dimension n, a group G of automorphisms of X , a subgroup H of G and identiﬁcations M X =H f D f and N X =G so that the following diagram commutes: D f

H G y % Xf =H M / N Xf =G D f D

Moreover, H n.X ; R/ R and G acts trivially on H n.X ; R/. f ' f

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1327

We observe that in general the polyhedron Xf is not a manifold. However, the comple- 1 1 ment of .B / .Z / in X is a manifold and the restriction of G and H G f D H f f to that set is a covering map. Just to clarify terminology, we would like to mention that under a cover or covering map we understand, as usual, a branched cover whose singular set is empty. Before going any further we state, with the same notation as in Proposition 2.3, the following simple consequence of the fact that H n.X R/ R and that G acts triv- f I ' ially on H n.X R/. If K is namely a subgroup of G then H n.X =K R/ R by f I f I ' Proposition 2.1. Hence, if V1; V2 are oriented n–manifolds then any two continuous maps f1 V1 X =K; f2 X =K V2 W ! f W f ! have a well-deﬁned degree deg.fi/ which satisﬁes the usual rule

deg.f2 f2/ deg.f2/ deg.f1/: ı D This fact will be of some importance below.

2.3 Branched covers of degree 1 or 2

Below, we will show that any branched cover f T n N as in the statement of W ! Theorem 1.2 is the composition of a covering map and a branched cover of degree one or two. It is well-known that such branched covers are very particular:

Proposition 2.4 (1) A branched cover of degree 1 is a homeomorphism. (2) Let f M N be a degree 2 branched cover between closed, connected, W ! orientable PL–manifolds. Then there is an orientation preserving involution M M such that N M= and such that the map f is just the natural W ! D h i orbit map M M= N . ! h i D In the statement of Proposition 2.4, as well as in the future, we denote by the group h i generated by . We conclude this section with the following observations on the ﬁxed point sets of involutions on n–manifolds that have the homotopy type of an n–torus.

Proposition 2.5 Let M be a closed orientable n–manifold with nontrivial orientation- preserving involution such that M= is an n–manifold. Then each component of h i the ﬁxed point set of is a Z2 –cohomology .n 2/–manifold. In particular, if n 3 then the components of Fix./ are not Z2 –acyclic.

eometry & opology, Volume 16 (2012) G T 1328 Pekka Pankka and Juan Souto

By classical Smith theory, the ﬁxed point set of a Zp –action, for p prime, on an orientable manifold is a Zp –cohomology r –manifold for some r < n; see eg[5, Theorem 7.4].

Recall that a Zp –cohomology r –manifold, for p prime, is a topological space satisfying a few properties, the most important of which is that it admits a cover such that B the one-point compactification of every element in has the same Zp –cohomology B relative to the point at infinity, as the r –dimensional sphere Sr relative to one of its points. See[5, Definition 1.1] for details and notice that it follows directly from the definition that a connected polyhedron which is a Zp –cohomology 0–manifold is a singleton.

Proof of Proposition 2.5 By the comment above, it sufﬁces to show that r n 2. D Let M M= be the quotient map. W ! h i Let † be a component of Fix./ and A a top dimensional simplex in †. Then A is an r –simplex. Let x A be an interior point. By passing to a subdivision 2 on M and M= if necessary, we may assume that the link Lk..x/; M= / is an h i h i .n 1/–sphere and that Lk.x; M / Lk.x; M / Lk..x/; M= / is a simplicial j W ! h i map.

Let S1 A Lk.x; M / and S2 Lk.A; M /. Then S1 and S2 are r – and .n r 1/– D \ D spheres in Lk.x; M /, respectively. Moreover, acts trivially on S1 and freely on S2 n r 1 by top dimensionality of A. Thus .S2/ has the homotopy type of RP .

Let † Lk..x/; M= /. Since † is a join of .S1/ and .S2/ and the pair D h i n 1 r 1 .†; .S1// is the standard sphere pair .S ; S /, we have that † .S1/ has the n r 1 n r 1 n homotopy type of both S and RP . The claim now follows.

Proposition 2.6 Let M be an n–manifold with the homotopy type of the n–torus and let be an involution of M that acts by multiplication by . 1/ on H 1.M; Z/. Then the ﬁxed point set Fix./ consists of 2n points.

Proof Since M is homotopy equivalent to an n–torus, we may identify 1.M / n n D H1.M Z/ Z and H .M R/ H .T R/, where R is either Z or R. I D I D I 1 V n Since acts by . 1/ to H .M; Z/ and H .M; Z/ is isomorphic to the ring Z , we know acts on H s.M Z/ by multiplication by . 1/s . In particular, M M I W ! has Lefschetz number L./ 2n . It follows from the Lefschetz ﬁxed point theorem D that

(2-3) Fix./ ∅: ¤

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1329

We show now that components of Fix./ are points. The claim then follows from the Lefschetz–Hopf theorem. Let N M= . We observe that H 1.N R/ 0. Indeed, D h i I D (2-4) H 1.N R/ H 1.M= R/ H 1.M R/ 0: I D h iI D I h i D We prove ﬁrst that the inclusion of Fix./ into M is trivial on homology.

Claim 1 The map H1.Fix./ Z/ H1.M Z/ is trivial. I ! I

Seeking a contradiction suppose that Œ˛ H1.Fix./ Z/ is not trivial in H1.M Z/ 2 I I and let Œˇ H n 1.M R/ be the unique cohomology class with 2 I Œ; Œ˛ Œ Œˇ h i D ^ for all Œ H 1.M R/. Since Œ˛ is ﬁxed by we deduce that Œˇ is also ﬁxed by . 2 I This implies that H n 1.M= R/ H n 1.M R/ 0; h iI D I h i ¤ contradicting (2-4). This completes the proof of Claim 1.

Claim 2 Every connected component † of Fix./ lifts homeomorphically to the universal cover M of M .

Since the homomorphism 1.M / H1.M R/ is injective, given a connected com- ! I ponent † of Fix./ we have the following commutative diagram:

1.†/ / 1.M /

H1.† R/ / H1.M R/ I I As we just observed the right vertical arrow is injective. On the other hand, the lower horizontal arrow is trivial by Claim 1. This implies that the homomorphism 1.†/ 1.M / is trivial as well. In other words, † lifts homeomorphically to the ! universal cover M of M . Thus Claim 2 holds. Let now † Fix./ be a connected component and lift it to † M . Since M is z homotopy equivalent to T n , M is homotopy equivalent to Rn and hence contractible. Choosing a point x † there is a unique lift of to M with .x/ x. Observe z 2 z z z z D z that is an involution and that z † Fix./: z z

eometry & opology, Volume 16 (2012) G T 1330 Pekka Pankka and Juan Souto

The involution is obviously a transformation of the contractible space M of prime z period 2. It is a standard consequence of Smith theory[6, III. Theorem 5.2] that Fix./ z is Z2 –acyclic and hence connected. Since Fix./ is Z2 –cohomology manifold, we z observe that † Fix./ is a point by Z2 –acyclicity and connectedness. Thus † is a z D z point. This concludes the proof of Proposition 2.6.

We combine these two propositions to the following corollary.

Corollary 2.7 Let M be an n–manifold with the homotopy type of the n–torus. For n 3 there is no involution on M that acts by multiplication by . 1/ on H 1.M Z/ I and M= is a manifold. h i Remark An equivalent formulation for Corollary 2.7 is that if M and N are PL– manifolds, with M homotopy equivalent to the n–dimensional torus with n 3 and with H1.N R/ 0, then there is no 2–fold branched cover M N . In the I D ! topological category, this result is due to Connolly, Davis and Kahn[9].

At this point we ﬁx some notation that will be used throughout the whole paper:

Notation 2.8 Suppose that f T n N is a branched cover as in the statement W ! of Theorem 1.2; in other words, we may ﬁx 1 r < n so that dim H r .N R/ Ä I D dim H r .T n R/. Let also X X be the polyhedron provided by Proposition 2.3 and I D f H G the groups of automorphisms of X with T n X=H and N X=G . Also, D D let n H X X=H T D W ! D be the orbit map onto X=H .

3 The key observation

The key tool in the proof of Theorem 1.2 is the analysis of the action of G on the singular (real) cohomology ring H .X R/ of X . I Recall that if G denotes a subgroup of G , then H .X R/ is the –invariant I cohomology of X and that the orbit map X X= induces, via the pullback, an ! isomorphism between H .X= R/ and H .X R/ . I I Since H is a subgroup of G , it is immediate that H .X R/G is H –invariant. The I main result of this section is a partial converse. Our notation is as in Notation 2.8.

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1331

Proposition 3.1 The subspace H 1.X R/H is G –invariant. Moreover, for any g G I 2 we have that the restriction of g to H 1.X R/H is equal to id. I ˙ We begin with two observations.

Lemma 3.2 Elements of G act trivially on H r .X R/H . I Proof Observe that

dim H r .X R/G dim H r .X=G R/ dim H r .N R/ I D I D I and, similarly,

dim H r .X R/H dim H r .X=H R/ dim H r .T n R/: I D I D I Since dim H r .N R/ dim H r .T n R/ and H r .X R/G H r .X R/H , it follows I D I I I that H r .X R/G H r .X R/H . Thus G acts trivially on H r .X R/H by deﬁnition. I D I I

Lemma 3.3 Let r 1, V be a vector space, W a subspace of V with dim W r 1, r r r C and ' V V a linear mapping so that V ' id V W V W . Then W is W ! D W ! ' –invariant and ' W id. j D ˙ Proof Since Vr ' id on Vr W , all r –dimensional subspaces of W are invariant D under ' ; hence W is ' –invariant. Let v W , v 0, and let U1;:::; Us be r – 2 ¤ T dimensional subspaces of W whose intersection L Ui is the line containing D i v. Since subspaces Ui , i 1;:::; s, are ' –invariant, the line L is also invariant. It D follows that the restriction of ' to W ﬁxes every line in L and hence is a homothety of ratio R: '.v/ v for all v W . 2 D 2 We claim that 1. In order to see that this is the case let v1; : : : ; vr W with D ˙ 2 v1 vr 0 and compute ^ ^ ¤ Vr v1 vr . '/.v1 vr / '.v1/ '.vr / ^ ^ D ^ ^ D ^ ^ r .v1/ .vr / .v1 vr /: D ^ ^ D ^ ^ The claim follows.

Proof of Proposition 3.1 We are going to apply Lemma 3.3 to V H 1.X R/, W D I D H 1.X R/H and ' g for g G . Notice that dim H 1.X R/H dim H 1.T n R/ I D 2 I D I D n r 1. C

eometry & opology, Volume 16 (2012) G T 1332 Pekka Pankka and Juan Souto

Consider the commutative diagram

r r V .H 1.X=H R// ^ / H .X=H R/ I I Vr (3-1) Vr .H 1.X R/H / / H r .X R/H I I where the vertical arrows are the pullback of the orbit map X X=H , while W ! the horizontal maps are the cup product in cohomology. By Proposition 2.1, the two vertical arrows are isomorphisms. On the other hand, the upper horizontal arrow is also an isomorphism because X=H T n is an n–dimensional torus. This proves that the D homomorphism Vr H r H .H1.X R/ / H .X R/ I ! I is an isomorphism.

r H Vr H Lemma 3.2 says g acts trivially on H .X R/ and hence on .H1.X R/ /. In I I other words, Lemma 3.3 applies and the claim follows.

Let G Hom.H 1.X R/H ; H 1.X R/H / be the map g g . By Proposition 3.1, ! I I 7! the image of this homomorphism is id . Identifying id with the multiplicative f˙ g f˙ g group 1 of two elements, we obtain a homomorphism f˙ g (3-2) ı G 1 : W ! f˙ g Notice that by deﬁnition H Ker.ı/. With the same notation as above we can now show that H .X R/H is in fact G – I invariant and describe the action

G Õ H .X R/H : I Corollary 3.4 For s 0; 1;:::; n and Œ! H s.X R/H we have D 2 I g Œ! ı.g/sŒ! D for all g G . 2 Proof Using the same argument as in the proof of Proposition 3.1, more concretely using diagram (3-1) and the discussion following that diagram, we deduce that the standard homomorphism

Vs H s H H1.X R/ H .X R/ I ! I

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1333 is an isomorphism. By Proposition 3.1, and by deﬁnition of ı. /, we have that g acts on H 1.X R/H by multiplication by ı.g/. Hence, g acts on H s.X R/H by I I multiplication by ı.g/s , as we claimed.

From here we obtain the ﬁrst nontrivial restrictions on any manifold N as in Notation 2.8.

Corollary 3.5 Using Notation 2.8 the following holds: (1) The pullback f H 2 .N R/ H 2 .T n R/ is an isomorphism. W I ! I (2) If n or r is odd, then f H .N R/ H .T n R/ is an isomorphism. W I ! I Proof Since H s.N R/ H s.X R/G H s.X R/H we deduce from Corollary 3.4 I D I I that H s.N R/ .H s.X R/H /G I D I for s even. Also by Corollary 3.4 we have that the action G Õ H s.X R/H is trivial I for even s; hence H s.N R/ .H s.X R/H /G H s.X R/H H s.T n R/ I D I D I D I for even s. This proves (1). Suppose now, for the sake of concreteness, that n is odd. Notice that on the one hand g is trivial on H n.X R/ by Proposition 2.3 while on the other hand it acts by I multiplication by ı.g/n by Corollary 3.4. Hence ı.g/ 1 for all g G . Observe that D 2 the same argument applies if r is odd. Once we know that ı.g/ 1 for all g G , the D 2 second claim follows as (1).

4 Averaging

In this section we construct n cocycles in C 1.X R/ which represent a basis of I H 1.X R/H and have some highly desirable properties; notation is as in Notation 2.8. I n 1 n To begin, choose bases Œc1; : : : ; Œcn of H1.T Z/ and Œ1; : : : ; Œn of H .T R/ I I with Œi; Œcj ıij ; h i D where ıij is the Kronecker symbol.

1 Proposition 4.1 There are 1–cocycles ‚1;:::;‚n C .X R/ with the following 2 I properties:

(1) .Œi/ Œ‚i for i 1;:::; n. D D

eometry & opology, Volume 16 (2012) G T 1334 Pekka Pankka and Juan Souto

# (2) g .‚i/ ı.g/‚i for any g G and i 1;:::; n. Here ı. / is the homomor- D 2 D phism (3-2).

(3) ‚i; cj ıij for any 1–chain cj C1.X R/ in X such that #.cj / is 1–cycle h z i D z 2 I z representing Œcj .

(4) ‚i; c Z for every integral 1–chain c C1.X Z/ whose pushforward #.c/ h i 2 2 I is a 1–cycle in X=H T n . D 1 n Proof We begin choosing 1–cocycles 1; : : : ; n C .T R/ representing the co- 2 I homology classes Œ1; : : : ; Œn and define ‚i as a twisted G –average of the pull- # back .i/: 1 X # # ‚i ı.g/g . .i//: D G g G j j 2 # # Observe that ‚i is a cocycle because it is a weighted sum of the cocycles g . .i//. We claim that the cocycles ‚1;:::;‚n fulfill the claims of Proposition 4.1. # To show (1), notice that, since .i/ represents the cohomology class .Œi/ 2 .H 1.X=H R// H 1.X R/H , we have by Proposition 3.1 that I D I # # # Œg . .i// g Œ .i/ ı.g/ .Œi/: D D Since ı.g/2 1 for all g G we have D 2 1 X # # 1 X 2 Œ‚i ı.g/g . .i// ı.g/ .Œi/ Œi; D G D G D g G g G j j 2 j j 2 as claimed. The validity of (2) follows from a simple computation and the fact that ı is a homomorphism. To show (3) observe that the homomorphism ı from (3-2) induces a well-defined map ı G=H 1 W ! f˙ g because H Ker.ı/. From now on we choose a representative g for every class gH G=H and hence identify G=H with a subset of G . We start rewriting ‚i; cj : 2 h z i 1 X # # ‚i; cj ı.g/g . .i//; cj h z i D G z j j g G 2 1 # X .i/; ı.g/g#.cj / D G z g G j j 2

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1335

1 # X X .i/; ı.gh/.gh/#.cj / D G z j j g G=H h H 2 2 1 # X X .i/; ı.g/g#.h#.cj // D G z j j g G=H h H 2 2 1 # X X .i/; ı.g/g# h#.cj / D G z j j g G=H h H 2 2 1 X # # X (4-1) ı.g/g . .i//; h#.cj / : D G z j j g G=H h H 2 2 P Notice that, by Lemma 2.2, h H h#.cj / is a 1–cycle satisfying 2 z X (4-2) h#.cj / H Œcj : z D j j h H 2 Observing that in the last equation in (4-1) we are evaluating a 1–cocycle and a 1–cycle we obtain the same result if we evaluate the corresponding cohomology and homology classes: 1 X # # X ‚i; cj ı.g/g . .i//; h#.cj / h z i D G z j j g G=H h H 2 2 1 X # # X ı.g/g . .i// ; h#.cj / (4-3) D G z j j g G=H h H 2 2 1 X X ı.g/g . .Œi//; h#.cj / : D G z j j g G=H h H 2 2 By Proposition 3.1, and ı.g/2 1, we have D (4-4) ı.g/g . .Œi// Œi D for all g G . Therefore, we obtain from (4-3) and (4-4) 2 1 X X ‚i; cj ı.g/g . .Œi//; h#.cj / h z i D G z j j g G=H h H 2 2 G=H X (4-5) j j .Œi/; h#.cj / D G z j j h H 2 G=H X G=H ˝ ˛ j j Œi; h#.cj / j j Œi; H Œcj ; D G z D G j j j j h H j j 2

eometry & opology, Volume 16 (2012) G T 1336 Pekka Pankka and Juan Souto

where the last equality holds by (4-2). It follows that

‚i; cj Œi; Œcj ıij ; h z i D h i D as claimed. To prove (4), we observe that a similar computation as in the proof of (3) shows that

‚i; c Œi; Œ#.c/ : h i D h i The claim (4) follows because Œ#.c/ H1.X=H; Z/ can be written as a linear combi- 2 nation of the basis Œc1; : : : ; Œcn with integer coefﬁcients and because Œi; Œcj ıij h i D by the choice of the bases. This concludes the proof of Proposition 4.1.

5 The Abel–Jacobi map

Still using Notation 2.8, we construct in this section a map ‰ X Rn=Zn and study W ! its equivariance properties with respect to the action of G on X . The construction of the map ‰ is motivated by the construction of the classical Abel–Jacobi map of a Riemann surface into its Jacobian. To begin with we consider the linear map

n ‰ C1.X R/ R ; ‰.c/ . ‚i; c /i 1;:::;n; z W I ! z D h i D given by evaluating the 1–cocycles ‚1;:::;‚n in Proposition 4.1 on singular chains. n By (4) in Proposition 4.1, the image of Z1.X Z/ under ‰ is contained in Z . In I z particular, ‰ induces a map z n n ‰ C1.X R/=Z1.X Z/ R =Z : 0W I I ! In order to obtain a map deﬁned on X we choose a base point x0 . Suppose that x X is another point and that I and I are two paths starting at x0 and ending 2 0 in x. Considering both paths as 1–chains we have ‰ .I/ ‰ .I / because I I is 0 D 0 0 0 a 1–cycle. It follows that the value ‰0.I/ does not depend on the chosen path I but only on the point x. We obtain hence a well-deﬁned map

‰ X Rn=Zn: W ! We will refer to ‰ as the Abel–Jacobi map.

Proposition 5.1 The Abel–Jacobi map ‰ X Rn=Zn descends to a homotopy W ! equivalence ‰ T n X=H Rn=Zn . y W D !

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1337

We start by investigating how the action of G on X is reﬂected by the Abel–Jacobi map. In order to do so we consider Rn=Zn as an abelian group.

n n Lemma 5.2 For all g G there is Ag R =Z with 2 2 ‰.g.x// ı.g/‰.x/ Ag D C for all x X . 2

Proof For x X denote by Ix a path from x0 to x. In order to compute ‰.g.x// 2 we consider the juxtaposition I g.Ix/ of I and g.Ix/. We have thus g.x0/ C g.x0/ ‰.g.x// ‰ .I g.Ix// ‰ .I / ‰ .g.Ix// ‰.g.x0// ‰ .g.Ix//: D 0 g.x0/ C D 0 g.x0/ C 0 D C 0 We set Ag ‰.g.x0// and compute D ‰0.g.Ix// . ‚i; gIx /i 1;:::;n: D h i D We have for i 1;:::; n D # ‚i; g.Ix/ ‚i; g#.Ix/ g ‚i; Ix ı.g/ ‚i; Ix ; h i D h i D h i D h i where the last equation holds by Proposition 4.1. The claim follows.

Clearly, the map

n n n n (5-1) G R =Z R =Z ;.g; x/ ı.g/x Ag; ! 7! C is an action of G on Rn=Zn .

Lemma 5.3 The restriction of the G –action (5-1) to H is trivial.

Proof Notice that for any h H the path Ih.x0/ projects to a closed loop in X=H . 2 n In other words, #.I / is a 1–cycle in X=H T . It follows from (4) in h.x0/ D Proposition 4.1 that ‰.I / Zn , meaning that z h.x0/ 2 A ‰.I / mod Zn 0: h D z h.x0/ D Since H Ker.ı/ we deduce from Lemma 5.2 that the action of H on Rn=Zn is trivial.

Proof of Proposition 5.1 Lemma 5.3 implies directly that the Abel–Jacobi map ‰ descends to a map ‰ X=H Rn=Zn: y W !

eometry & opology, Volume 16 (2012) G T 1338 Pekka Pankka and Juan Souto

We claim now that ‰ is a homotopy equivalence. In order to see that this is the case let y n c1 be a loop in X=H T representing the homology class Œc1 H1.X=H Z/ D 2 I D 1.X=H / chosen in the previous section. Suppose also that c1 is based at .x0/ and avoids Zf , the preimage of the branching locus of f . This last condition implies that there is a unique lift c1 of c1 starting at x0 . Applying Proposition 4.1(3) to c1 we z z obtain that ‰.c1/ .1; 0;:::; 0/: z z D n n n n Identifying 1.R =Z / with the deck-transformation group Z of the cover R n n ! R =Z we have that the image ‰.c1/ of c1 is homotopic to the ﬁrst standard generator n n n y of 1.R =Z / Z . Arguing in the same way for c2;:::; cn we have proved that the ' homomorphism n n 1.‰/ 1.X=H; .x0// 1.R =Z ; ˆ..x0/// y W ! y n n n induced by ‰ maps the basis Œc1; : : : ; Œcn to the standard basis of Z 1.R =Z /. y D In other words, .‰/ is an isomorphism. Since both X=H and Rn=Zn are tori, and 1 y hence have contractible universal coverings, this implies that ‰ X=H Rn=Zn is a W ! homotopy equivalence, as we wanted to show.

We can now start drawing a diagram to which we will continue adding arrows and objects in the next section:

X ‰ G x ‰ f ' Rn=Zn o y X=H T n / X=G N D D 6 The subgroup K G Still using Notation 2.8 deﬁne K Ker.ı/ D to be the kernel of the homomorphism ı; see (3-2). By construction K is a normal subgroup of G with at most index 2. Notice that H K Ker.ı/ and that it follows D from Corollary 3.4 that K G if the dimension n of the involved manifolds is odd. D Remark In the next section, we will also prove that K G if n is even. D By Lemma 5.2, the action (5-1) of G on Rn=Zn restricts to an action of K by translations. Observe that since H K, this action is not effective by Lemma 5.3. In spite of this, we will denote by .Rn=Zn/=K the quotient of Rn=Zn under the action by translations of K. In this section we prove:

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1339

Proposition 6.1 The homotopy equivalence ‰ X=H Rn=Zn descends to a map y W ! ‰ X=K .Rn=Zn/=K. In particular, the following holds: x W ! n n n n (1) The orbit map p1 R =Z .R =Z /=K has degree K=H . W ! j j (2) H is a normal subgroup of K. (3) The action of K=H on Rn=Zn is effective and .Rn=Zn/=K is a torus. (4) The action of K=H on X=H is free. In particular, the space X=K is in fact a manifold and the orbit map p2 X=H X=K is a covering map. W ! (5) The map ‰ X=K .Rn=Zn/=K is a homotopy equivalence. x W ! Proof The fact that ‰ X=H Rn=Zn descends to a map ‰ X=K .Rn=Zn/=K y W ! x W ! follows just from the deﬁnitions and is left to the reader. Notice that since K acts on the n–dimensional torus Rn=Zn by translations, the quotient is an n–dimensional a torus as well. This proves the second part of (3). At this point we can enlarge the diagram above as follows: X ‰

v ‰ G (6-1) Rn=Zn o y X=H T n D f p1 p2

‰ p3 ' .Rn=Zn/=K o x X=K / X=G N D where p1 , p2 and p3 are the obvious orbit maps. Since the square in the left bottom corner of the diagram commutes we have that

deg.p1/ deg.‰/ deg.p1 ‰/ deg.‰ p2/ deg.‰/ deg.p2/: y D ı y D x ı D x Taking into account that ‰ is a homotopy equivalence and that p2 has degree K=H y j j we obtain deg.p1/ K=H deg.‰/: D j j x On the other hand, p has positive degree. This implies that ‰ has to have positive 1 x degree as well; hence deg.p1/ K=H : j j On the other hand, deg.p1/ K=H because the subgroup H of K acts trivially on Ä j j Rn=Zn . We have proved (1). Recall that H is contained in the kernel of the homomorphism K Aut.Rn=Zn/ !

eometry & opology, Volume 16 (2012) G T 1340 Pekka Pankka and Juan Souto

by Lemma 5.3. The image of this homomorphism has deg.p1/ K=H elements. D j j It follows that H is in fact precisely the kernel of this homomorphism and that it is therefore normal in K. We have proved (2) and the first part of (3). Now H is normal in K, so K=H acts on X=H and that X=K .X=H /=.K=H /. D We claim that the action of K=H on X=H is free. In order to see this, suppose that we have x X=H and k K=H with kx x. Then we have 2 2 D ‰.x/ ‰.kx/ k‰.x/: y D y D y Since the action of K=H on Rn=Zn is effective and by translations, an element k K=H can only fix the point ‰.x/ Rn=Zn if k is the neutral element in K=H . 2 2 We have proved (4). It remains to show that the map ‰ X=K .Rn=Zn/=K is a homotopy equivalence. x W ! As in the proof of Proposition 5.1, it suffices to show that .1/ .‰/ 1.X=K/ n n x W ! 1..R =Z /=K/ is an isomorphism. Notice at this point that we have the following commutative diagram:

1 / 1.X=H / / 1.X=K/ / K=H / 1

.1/ .‰/ .1/ .‰/ id y x n n n n 1 / 1.R =Z / / 1..R =Z /=K/ / K=H / 1 Since the ﬁrst and third vertical arrows are isomorphisms it follows that the middle arrow is an isomorphism as well. This concludes the proof of Proposition 6.1.

We conclude this section with some remarks needed to prove Theorem 1.2 below. The following is just a direct consequence of parts (3) and (5) of Proposition 6.1:

Corollary 6.2 The manifold X=K is homotopy equivalent to an n–torus.

Recall now that K G is normal of at most index 2 because it is the kernel of the homomorphism (3-2). In particular, the group G=K acts on X=K with X=G D .X=K/=.G=K/ and the map p3 X=K X=G N in (6-1) is just the orbit map. W ! D We prove now:

Lemma 6.3 If K G denote by the nontrivial element in G=K. Then acts ¤ as id on H 1.X=K R/. I We suggest the reader to compare with Corollary 3.5.

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1341

Proof Denote by G a representative of in G . Notice that, by the very deﬁnition z 2 of the action (5-1), acts on H 1.Rn=Zn R/ by id. Since K acts on Rn=Zn by z I translations, it follows that also acts on H 1..Rn=Zn/=K R/ by id. Since the I homotopy equivalence ‰ satisﬁes ‰..x// .‰.x//, we deduce that acts on x x D x H 1.X=K R/ as id as well. I

7 The ﬁnal step of the proof of Theorem 1.2

In this section we conclude the proof of Theorem 1.2; as always, we use Notation 2.8. We copy (6-1) here for the convenience of the reader:

X ‰

v ‰ G Rn=Zn o y X=H T n D f p1 p2

‰ p3 ' .Rn=Zn/=K o x X=K / X=G N D Having this diagram in mind, we conclude the proof of Theorem 1.2:

Theorem 1.2 Let N be a closed, connected, and oriented n–manifold, n 2 so that dim H r .N R/ dim H r .T n R/ for some 1 r < n. Then every branched cover n I D I Ä n f T N is a cover. In particular, every 1 –surjective branched cover T N is W ! ! a homeomorphism.

Proof If G=K is the trivial group then p3 id and, in particular, we have D

f p3 p2 p2: D ı D

By Proposition 6.1, the map p2 , and hence f , is a covering map. Thus it sufﬁces to show that K G . D Seeking a contradiction assume that K G . Hence G=K is the group of order ¤ two. As in Lemma 6.3, denote by the nontrivial element in G=K. It follows from Corollary 6.2 that X=K is homotopy equivalent to an n–torus. Furthermore, it follows from Lemma 6.3 that acts on H s.X=K R/ by multiplication by . 1/s . Since I .X=K/=.G=K/ X=G N is an orientable manifold, this contradicts Corollary 2.7. D D The proof is complete.

eometry & opology, Volume 16 (2012) G T 1342 Pekka Pankka and Juan Souto

8 Brief discussion of Theorem 1.1

Theorem 1.1 There is no branched cover from the 4–dimensional torus T 4 to #3.S2 S2/, the connected sum of three copies of S2 S2 . On the other hand, 4 3 2 2 there are, a fortiori 1 –surjective, maps T # .S S / of arbitrarily large degree. ! The claim that there is no branched covering T 4 #3.S2 S2/ follows directly from ! Theorem 1.2. Maps of arbitrarily large degree f T 4 #3.S2 S2/ W ! can either be constructed directly, or be shown to exist using the work of Duan and Wang[10]. These authors prove namely the following remarkable theorem: Theorem (Duan–Wang[10, Theorem 3]) Suppose M and L are closed oriented 4–manifolds and denote by H 2.M Z/ and H 2.L Z/ the torsion free part of the x I x I corresponding second cohomology group. Let also ˛ and ˇ be respectively bases of H 2.M Z/ and H 2.L Z/ and let A and B be the intersection matrices on M and L x I x I with respect to the bases ˛ and ˇ . Finally, let P be an m–by–l –matrix, where m dim H 2.M / and l dim H 2.L/. D x D x If L is simply connected, then there is a map f M L of degree k such that W ! f .ˇ / ˛ P if and only if P t AP kB . Moreover, there is a map f M L of D D W ! degree 1 if and only if the intersection form of L is isomorphic to a direct summand of the intersection form of M . In[10, Example 4], this is applied in the particular case of maps T 4 #3.S2 S2/ to ob- ! tain mappings of every degree. We ﬁnish by recalling this example. Let Œ1; : : : ; Œ4 be 1 4 2 4 generators of H .T Z/ as in Section 4. The basis .Œi Œj /1 i

30 1 L : 1 0 Thus the linear mapping given by the matrix

30 k P L D 1 0 with respect to these bases of H 2.#3.S2 S2/ R/ and H 2.T 4 R/ yields the result. I I

eometry & opology, Volume 16 (2012) G T On the nonexistence of certain branched covers 1343

References

[1] J W Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920) 370–372 MR1560318 [2] I Berstein, A L Edmonds, The degree and branch set of a branched covering, Invent. Math. 45 (1978) 213–220 MR0474261 [3] I Berstein, A L Edmonds, On the construction of branched coverings of low- dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979) 87–124 MR517687 [4] M Bonk, J Heinonen, Quasiregular mappings and cohomology, Acta Math. 186 (2001) 219–238 MR1846030 [5] G E Bredon, Orientation in generalized manifolds and applications to the theory of transformation groups, Michigan Math. J. 7 (1960) 35–64 MR0116342 [6] G E Bredon, Introduction to compact transformation groups, Pure and Applied Math. 46, Academic Press, New York (1972) MR0413144 [7] G E Bredon, Topology and geometry, Graduate Texts in Math. 139, Springer, New York (1997) MR1700700 [8] AV Cernavskiˇ ˘ı, Finite-to-one open mappings of manifolds, Mat. Sb. 65 (107) (1964) 357–369 MR0172256 In Russian; translated in Amer. Math. Soc., Translat., II. Ser. 100 (1972) 253–267

[9] F Connolly, J F Davis, Q Khan, Topological rigidity and H1 –negative involutions on tori arXiv:1102.2660 [10] H B Duan, S C Wang, Non-zero degree maps between 2n–manifolds, Acta Math. Sin. .Engl. Ser./ 20 (2004) 1–14 MR2056551 [11] A L Edmonds, Deformation of maps to branched coverings in dimension three, Math. Ann. 245 (1979) 273–279 MR553345 [12] M Gromov, Hyperbolic manifolds, groups and actions, from “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, 1978)” (I Kra, B Maskit, editors), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 183–213 MR624814 [13] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Math. 152, Birkhäuser, Boston, MA (1999) MR1699320 Based on the 1981 French original [MR0682063], With appendices by M Katz, P Pansu and S Semmes, Translated from the French by S M Bates [14] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) MR1867354 [15] H M Hilden, Three-fold branched coverings of S 3 , Amer. J. Math. 98 (1976) 989–997 MR0425968 [16] U Hirsch, Über offene Abbildungen auf die 3–Sphäre, Math. Z. 140 (1974) 203–230 MR0362313

eometry & opology, Volume 16 (2012) G T 1344 Pekka Pankka and Juan Souto

[17] J F P Hudson, Piecewise linear topology, Univ. of Chicago Lecture Notes, Benjamin, New York-Amsterdam (1969) MR0248844 Prepared with the assistance of J L Shane- son and J Lees [18] M Iori, R Piergallini, 4–Manifolds as covers of the 4–sphere branched over non- singular surfaces, Geom. Topol. 6 (2002) 393–401 MR1914574 [19] J M Montesinos, Three-manifolds as 3–fold branched covers of S 3 , Quart. J. Math. Oxford Ser. 27 (1976) 85–94 MR0394630 [20] R Piergallini, Four-manifolds as 4–fold branched covers of S 4 , Topology 34 (1995) 497–508 MR1341805 [21] S Rickman, Existence of quasiregular mappings, from “Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986)” (D Drasin, C J Earle, F W Gehring, I Kra, A Marden, editors), Math. Sci. Res. Inst. Publ. 10, Springer, New York (1988) 179–185 MR955819 [22] S Rickman, Quasiregular mappings, Ergeb. Math. Grenzgeb. 26, Springer, Berlin (1993) MR1238941 [23] S Rickman, Simply connected quasiregularly elliptic 4–manifolds, Ann. Acad. Sci. Fenn. Math. 31 (2006) 97–110 MR2210111 [24] C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer, Berlin (1982) MR665919 [25] J Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I No. 392 (1966) 10 MR0200928

Department of Mathematics and Statistics, University of Helsinki PO Box 68 (Gustaf Hällströmin katu 2b), FI-00014 Helsinki, Finland Mathematics Department, University of British Columbia 1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada [email protected], [email protected] http://www.helsinki.fi/~pankka, http://www.math.ubc.ca/~jsouto

Proposed: David Gabai Received: 25 January 2011 Seconded: Benson Farb, Ronald Fintushel Revised: 27 January 2012

Geometry & Topology Publications, an imprint of mathematical sciences publishers msp GEOMETRY &TOPOLOGY www.msp.warwick.ac.uk/gt

EDITORS

MANAGING EDITORS Colin Rourke Brian Sanderson Walter Neumann [email protected] [email protected] [email protected] Mathematics Institute Mathematics Institute Department of Mathematics University of Warwick University of Warwick Barnard College, Columbia University Coventry CV4 7AL Coventry CV4 7AL New York, NY 10027-6902 UK UK USA

BOARDOF EDITORS Joan Birman Columbia University Cameron Gordon University of Texas [email protected] [email protected] Martin Bridson Imperial College, London Eleny Ionel Stanford University [email protected] [email protected] Jim Bryan University of British Columbia Vaughan Jones University of California, Berkeley [email protected] [email protected] Ben Chow University of California, San Diego Robion Kirby University of California, Berkeley [email protected] [email protected] Ralph Cohen Stanford University Frances Kirwan University of Oxford [email protected] [email protected] Tobias Colding Massachusetts Institute of Technology Wolfgang Lück Universität Münster [email protected] [email protected] Simon Donaldson Imperial College, London Haynes Miller Massachusetts Institute of Technology [email protected] [email protected] Bill Dwyer University of Notre Dame Shigeyuki Morita University of Tokyo [email protected] [email protected] Yasha Eliashberg Stanford University John Morgan Columbia University [email protected] [email protected] Benson Farb University of Chicago Tom Mrowka Massachusetts Institute of Technology [email protected] [email protected] Steve Ferry Rutgers University Walter Neumann Columbia University [email protected] [email protected] Ron Fintushel Michigan State University Jean-Pierre Otal Université d’Orleans ronﬁ[email protected] [email protected] Mike Freedman Microsoft Research Peter Ozsvath Columbia University [email protected] [email protected] David Gabai Princeton University Leonid Polterovich Tel Aviv University [email protected] [email protected] Paul Goerss Northwestern University Colin Rourke University of Warwick [email protected] [email protected] Lothar Göttsche Abdus Salam Int. Centre for Th. Physics Ron Stern University of California, Irvine [email protected] [email protected] Tom Goodwillie Brown University Peter Teichner University of California, Berkeley [email protected] [email protected] Gang Tian Massachusetts Institute of Technology [email protected] See inside back cover or www.msp.warwick.ac.uk/gt for submission instructions. The subscription price for 2012 is US $375/year for the electronic version, and $575/year ( $60 shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three yearsC and changes of subscribers address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berke- ley, CA 94704-3840, USA. Geometry & Topology is indexed by Mathematical Reviews, Zentralblatt MATH, Current Mathematical Publications and the Science Citation Index. Geometric & Topology (ISSN 1364-0380 electronic, 1465-3060 printed) at Mathematical Sciences Publisher, Department of Mathematics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing ofﬁces.

G&T peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2012 by Mathematical Sciences Publishers G EOMETRY GEOMETRY &TOPOLOGY Volume 16 Issue 3 (pages 1247–1880) 2012

1247Deformationspaces of T Kleinian& surface groups are not locally connected

AARON DMAGID OPOLOGY 1321Onthe nonexistence of certain branched covers PEKKA PANKKA and JUAN SOUTO .0/–groups 1345Geodesicﬂow for CAT ARTHUR BARTELS and WOLFGANG LÜCK 1393Allﬁnite groups are involved in the mapping class group

GREGOR MASBAUM and ALAN WREID 2012 1413Noncollapsingin mean-convex mean curvature ﬂow BEN ANDREWS 1419Whitney tower concordance of classical links

JAMES CONANT,ROB SCHNEIDERMAN and PETER 1247–1880) (pages 3 Issue 16, Vol. TEICHNER 1481Blobhomology SCOTT MORRISON and KEVIN WALKER 1609Prymvarieties of spectral covers TAMÁS HAUSEL and CHRISTIAN PAULY 1639Legendrian and transverse cables of positive torus knots JOHN BETNYRE,DOUGLAS JLAFOUNTAIN and BÜLENT TOSUN 1691Obstructionsto stably ﬁbering manifolds WOLFGANG STEIMLE 1725Towards representation stability for the second homology of the Torelli group SØREN KBOLDSEN and MIA HAUGE DOLLERUP .3/–Donaldson invariants of CP2 and mock theta functions 1767SO ANDREAS MALMENDIER and KEN ONO 1835Rigidityof certain solvable actions on the sphere MASAYUKI ASAOKA 1859TheNovikov conjecture and geometry of Banach spaces GENNADI KASPAROV and GUOLIANG YU