Braided surfaces and their characteristic maps...Characteristic maps of braided surfaces Elementary...

Characteristic maps of braided surfacesElementary quotients of surface groups

Lifting one stepProfinite separability

Braided surfaces and theircharacteristic maps

Louis Funar (joint work with Pablo Pagotto)

K-OSOctober 22, 2020

1 L.Funar Braided surfaces



Abstract

We show that branched coverings of surfaces of large enoughgenus arise as characteristic maps of braided surfaces, thusbeing 2-prems. In the reverse direction we show that anynonabelian surface group has infinitely many finite simplenonabelian groups quotients with characteristic kernels whichdo not contain any simple loops and hence the quotient maps donot factor through free groups. By a pullback construction,finite dimensional Hermitian representations of braid groupsprovide invariants for the braided surfaces. We show that thestrong equivalence classes of braided surfaces are separated bysuch invariants if and only if they are profinitely separated.




Plan

I. Characteristic maps for Braided surfaces

II. Elementary quotients of surface groups

III. Lifting one step

IV. Profinite separability




Let Σ denote a closed orientable surface. A braided surfaceover Σ is an embedding of a surface j : S → Σ× R2, suchthat the composition with the first factor projection

f : Sj→ Σ× R2 p→ Σ

is a branched covering. The composition p j is called thecharacteristic map of the braided surface S.Two braided surfaces ji : S → Σ× R2, i = 0, 1 over Σ areequivalent if there exists some ambient isotopyht : Σ× R2 → Σ× R2, h0 = id such that ht isfiber-preserving (i.e. there exists a homeomorphismϕ : Σ→ Σ such that p ht = ϕ p) and h1 j0 = j1.When ϕ can be taken to be isotopic to the identity rel thebranch locus, we say that the braided surfaces are stronglyequivalent.Viro, Rudolph ’83, Kamada ’94, Carter-Kamada,Nakamura ’11Edmonds ’99: f unramified covering, S contained in anorientable plane bundle, then its Euler class is torsion.




Geometric Lifting Problem:When a ramified covering f : S → Σ lifts to a braided surfaceembedding ϕ : S → Σ× R2?

We could instead ask ϕ be an immersion and the embeddingmight be smooth, PL topologically flat, topological, etc.

One might take f be a generic smooth/PL map and ask ifit lifts to an embedding. Melikhov ’15.




Two branched coverings f0, f1 : S → Σ are equivalent ifthere exist homeomorphisms Φ : S → S and φ : Σ→ Σ suchthat

f1 Φ = φ f0

When φ is isotopic to the identity rel the branch locus,then the branched coverings are strongly equivalent.

A degree n branched covering f : S → Σ of surfacesdetermines a holonomy homomorphism

f∗ : π1(Σ \B)→ Sn

where B is the set of branch points.

Hurwitz branched coverings Classification: Twobranched coverings of surfaces are strongly equivalent if andonly if their holonomy homomorphisms are conjugate.Moreover, they are equivalent if and only if the conjugacyclasses of their holonomy homomorphisms are equivalentunder the left action of the pure mapping class groupΓ(Σ \B).




A braided surfaces ϕ : S → Σ has degree n if itscharacteristic homomorphism f : S → Σ has degree n.

A degree n braided surface ϕ : S → Σ× R2 of surfacesdetermines a holonomy homomorphism

ϕ∗ : π1(Σ \B)→ Bn

where B is the set of branch points of its characteristic mapand Bn is the braid group on n strands.

Braided surfaces Classification: Two branchedcoverings of surfaces are strongly equivalent if and only iftheir holonomy homomorphisms are conjugate. Moreover,they are equivalent if and only if the conjugacy classes oftheir holonomy homomorphisms are equivalent under theleft action of the pure mapping class group Γ(Σ \B).




Our first result gives a positive answer to the geometriclifting problem, for large enough genus:

Theorem

There exists some hn,m such that every degree n branchedcovering S → Σ of a closed orientable surface Σ of genusg ≥ hn,m with at most n branch points occurs as thecharacteristic map of some braided surface.

Petersen ’90 have proved that solvable unramified coveringscan be lifted.




Algebraic Lifting ProblemGiven a surjective group homomorphism p : G→ G, when doesa homomorphism f : π1(Σ)→ G lift to ϕ : π1(Σ)→ G?

We will restrict to surjective homomorphisms f and Σ willbe a closed orientable surface.

Definition

The Schur class sc(f) ∈ H2(G) is the image f∗([Σ]) of thefundamental class [Σ] of Σ.

The action of Aut(π1(Σ) on Hom(π1(Σ), G) preserves theSchur classes.

Moreover, the G-conjugacy acts trivially on H2(G). Thusthe Schur class descends to a function:

sc : Γ(Σ)\Hom(π1(Σ), G)/G→ H2(G)




A homomorphism π1(Σ′)→ π1(Σ) is a pinch map if its isinduced by the quotient (degree one) map Σ′ → Σ whichcrushes several 1-handles to points.

Definition

A stabilization of f : π1(Σ)→ G is the composition with a pinchmap.

Two homomorphisms are stably equivalent if they havestabilization equivalent under the Aut+(π1(Σ) action. Thestable equivalence descends also to G-conjugacy classes ofhomomorphisms.

Observe that the image of a homomorphism is an invariantof its (stable) equivalence class. For this reason we shallrestrict to surjective homomorphisms.

Note that the Schur class of a homomorphism does notchange under stabilization.




Theorem (Livingston ’85, Zimmermann ’87)

Surjective homomomorphisms are stably equivalent if and only iftheir Schur classes agree.

If Ωn(X) is the dimension n orientable bordism group of X,then Thom proved that the natural map Ωn(X)→ Hn(X)is an isomorphism if n ≤ 3 and an epimorphism, if n ≤ 6.Two maps f : Σ→ X, f ′ : Σ′ → X representing the sameclass in H2(X) are therefore bordant and thus there existsa 3-manifold M3 whose boundary is Σ t Σ′ and a commonextension F : M3 → X.Consider a Heegaard surface Σ′′ in M3, decomposing it intothe union of two compression bodies C ∪ C ′, glued togetheralong their common boundary Σ′′ by means of ahomeomorphism ψ. A compression body is obtained fromΣ′′ × [0, 1] by attaching 2-handles along disjoint nontrivialsimple closed curves on Σ′′ × 1.Then F |Σ′′ is a stabilization of f and f ′, up to equivalence,throughout the restrictions F |C and F |C′ .




The stable algebraic lifting problem has a solution:

Corollary

Given a surjective p : G→ G, then a surjective homomorphismf : π1(Σ)→ G lifts stably to G if and only if there exists some

class a ∈ H2(G) such that p∗(a) = sc(f).

The Livingston-Zimmermann result was improved in the casewhen the target G is a finite group, as follows:

Theorem (Dunfield-Thurston ’06)

If G is finite, then there exists some g(G) such that every twosurjective homomorphisms π1(Σ)→ G with the same Schurclass, for a closed orientable surface Σ of genus g ≥ g(G), areequivalent. In particular, every such surjective homomorphismf : π1(Σ)→ G lifts to G, if there exists some a ∈ H2(G) withp∗(a) = sc(f).

A key ingredient is that for large enough genus every surjectivef should be a stabilization.




The last step, in the unramified case, is the following:

Lemma

If G ⊂ Sn and p : Bn → Sn is the projection, then we have asurjective homomorphism in 2-homology:

H2(p−1(G))→ H2(G)→ 1

Proof: Let Pn denote the pure braid group on n strands. Thefive terms exact sequence in homology reads:

H2(p−1(G))→ H2(G)→ H1(Pn)G → H1(p−1(G))→ H1(G)

observe that H1(Pn)G ∼= ZS(n)G ∼= Z[S(n)/G], whereS(n) = (i, j); 1 ≤ i < j ≤ n. In particular, H1(Pn)G is atorsion-free group, while H2(G) is torsion. Therefore everyhomomorphism H2(G)→ H1(Pn)G must be trivial.




In the ramified case (B 6= ∅) we need to adapt the proofabove to surjective homomorphisms π1(Σ \B)→ G, havingprescribed value of the peripheral loops, i.e. encircling onceevery branch point.The characteristic homomorphism of a branched coveringmaps peripheral loops into nontrivial elements of Sn. Whatabout braided surfaces?A link L ⊂ S1 ×D2 is completely split if there existpairwise disjoint disks D2

i ⊂ D2 such that each componentLi of the link L is contained in one solid torus S1 ×D2

i .A braid b ∈ Bn is completely splittable if its closure Lwithin the solid torus is completely split, while L is atrivial link in the sphere S3.Kamada ’96: Local monodromy of PL topologically flatembeddings ϕ around branch points completely splittablebraids An ⊂ Bn.Schur classes for homomorphisms with fixed peripheralholonomy, Catanese-Lonne-Perroni ’16 and extension ofLivingston-Zimmermann and Dunfield-Thurston results.




Definition

We say that a homomorphism f : π1(Σ)→ G is elementary, if itfactors through a free group.

By a well-known result of Stallings and Jaco ’69, this isequivalent to the fact that f factors through the mapπ1(Σ)→ π1(H), where H is the handlebody bounded by Σ.

Corollary

If G is finite, then there exists g(G) such that anynull-homologous surjective homomorphism f : π1(Σ)→ G,where Σ is a surface of genus g ≥ g(G), is elementary. Inparticular, f lifts to any G.




The thickness t(f) of a nullhomologous f is the smallest gfor which there exists some 3-manifold M3 with boundaryΣ and Heegaard genus g, and an extension F : π1(M3)→ GLiechti-Marche ’19 considered the torus case.

Proposition

t(f) is the smallest genus of an elementary stabilization of f .

Let π1(Σ) = 〈ai, bi;∏gi=1[ai, bi]〉 and G = F/R be a

presentation of G. Set ocl(f) to be the minimal n such thatwe can write

g∏i=1

[f(ai), f(bi)] =

n∏i=1

[rj , fj ]

where f(ai), f(bi) denote lifts to F and rj ∈ R, fj ∈ F.

Proposition (Hopf type formula)

t(f) = ocl(f).




Conjecture (Wiegold)

For any finite simple nonabelian group G and n ≥ 3 we have

|Out(Fn)\Epi(Fn, G)/Aut(G)| = 1

McCullough-Wanderley ’03: Epimorphisms becomeequivalent after µ(G) (= the minimal number of generatorsof G) stabilizations, for all finite G.

McCullough-Wanderley ’03: For large enoughn ≥ 1 + |G| log2 |G| any two epimorphisms into G areequivalent.

There exist nonequivalent epimorphisms onto infinitegroups G.




Conjecture (virtual solvability – Lubotzky)

For any finite dimensional representation of Aut(Fn), the imageof the inner automorphisms subgroup Fn is virtually solvable.

Formanek-Procesi ’92 proved that the image of the freesubgroup of Fn on two standard generators is virtually solvable.

Conjecture (free factors - Lubotzky)

For finite simple nonabelian G and surjective homomorphismf : Fn → G, n ≥ 3, there exist a proper free factor H ⊂ Fn withf(H) = G, i.e. for any system of generators g1, g2, . . . , gn of Fn,we can drop one such that their images by f still generate G.

Conjecture (characteristic quotients - Lubotzky)

There is no finite simple characteristic quotient of Fn, n ≥ 3.




Gilman ’77: There exists a large orbit of Out(Fn) onEpi(Fn, G)/Aut(G), whose size N goes to infinity with |G|and on which the action is by the alternating group AN orthe symmetric group SN .

Therefore, if Wiegold’s Conjecture holds, then there are nofinite simple characteristic quotients of Fn, n ≥ 3.

Similar questions were asked by Lubotzky ’11 about surfacegroups π1(Σ).

Theorem (F-Lochak ’18)

For surface groups of genus g ≥ 2 the virtual solvabilityconjecture and Wiegold-type conjecture do not hold. Inparticular, surface groups have infinitely many finite simplecharacteristic quotients.

It is unknown whether a single stabilization is enough to makeequivalent nullhomologous epimorphisms of a surface group ontoa finite simple nonabelian group.




Theorem

If g ≥ 2, the there exist infinitely many epimorphismsπ1(Σ)→ G onto finite simple nonabelian groups G, whosekernels do not contain any simple loop and hence arenonelementary.

Livingston ’00, Gabai: finite quotients without simple loopsin the kernel

Pikaart ’01: finite caracteristic quotients without simpleloops in the kernel

Proof sketch: If elementary, it has simple loops in the kernel,corresponding to (nonseparating) meridians of the handlebody.Since the kernel is characteristic, if it contains onenonseparating simple loop, it should contain all nonseparatingsimple loops and hence it would be trivial. The order ofseparating loops in the quotients are explicitly computed byusing their TQFT description.




Let γ0G = G, γk+1G = [γkG,G] denote the lower centralseries of the group G. It is well-known that Pn is residuallytorsion-free nilpotent, namely

⋂∞k=0 γkPn = 1 and

Ak = γk−1Pn

γkPnare finitely generated torsion-free abelian

groups.

We then have a series of abelian extensions

1→ Ak+1 →Bn

γk+1Pn→ Bn

γkPn→ 1

Whether a homomorphism fk : π1(Σ)→ Bn

γkPnadmits a lift

to fk+1 : π1(Σ)→ Bn

γk+1Pncan be reformulated in purely

cohomological terms.

For every k ≥ 1 there exist examples of homomorphisms fkwhich admit no lift.




Proposition

Every homomorphism f0 : π1(Σ)→ Bn

γ0Pn= Sn admits a lift

f1 : π1(Σ)→ Bn

γ1Pn.

Every f : π1(Σ)→ Sn induces a π1(Σ)-module on A1 = H1(Pn).The key ingredient of the proof is the following

Lemma

Let P : π(Σ′)→ π1(Σ) be a pinch map. Then

P ∗ : H2(π1(Σ), A1)→ H2(π1(Σ′), A1)

is injective.

Then f0 admits a lift f1 if and only if the pull-back of theextension Bn

γ1Pnof Sn by A1 by f0 is a split extension. Since

every homomorphism lifts stably to Bn and hence to Bn

γ1Pn, the

cohomological obstruction stably vanishes. Lemma abovecompletes the proof.




Equivalence classes of degree n braided surfaces with B 6= ∅correspond to double cosets Bn\Bm+1

n /Bn.Let K ⊆ H be a pair of groups and ρ : H → U(V ) be afinite dimensional representation of H preserving aHermitian form 〈, 〉. A K-spherical function on H is amatrix coefficient

φ : H → C, φ(x) = 〈ρ(x)v, w〉,where v, w belong to the space of K-invariants vectors V K .Then φ is bi-K-invariant, namely it descends to K\H/K.

Proposition

For K ⊂ H finite groups or compact connected Lie groups, theunitary K-spherical functions separate points of K\H/K.

Unitary representations R : Bn → U induce topologicalinvariants of braided surfaces, pulling-back U -sphericalfunctions under the map:

R∗ : Bn\Bm+1n /Bn → U\Um+1/U




We can assemble all U -spherical functions in a single formalseries Φ. Let i index the finite dimensional irreduciblerepresentations Vi of SU(2). The space of invariant vectorsH0(SU(2), Vi1 ⊗ Vi2 ⊗ · · ·Vik) has a basis BI indexed bythe set of partitions α = (αst)s,t=1,...,k with

∑t αst = is.

Φ =∑I,(αst)

1

α!β!

∏s,t

xαyβ(φI,α,β)

Here we set xα =∏s,t x

αstst , α! =

∏s,t αst!.

Neretin ’10 proved that:

Φ(A) = det(1−AXA⊥Y )−1/2

for A ∈ SU(2)k, where X = (Xij), Y = (Yij) are matrices

of blocks Xij =

(0 xij−xij 0

), Yij =

(0 yij−yij 0

),

Xji = −Xij , Yji = −Yij for i < j, Xii = Yii = 0 and xij , yijare variables.This provides a polynomial invariant Φ(A)−2 of degree 3braided coverings via the unitary Burau representation.




Consider the profinite completion of the group H:

H = lim← G/H;|H/G|<∞

H/G

This is a totally disconnected compact group. If K ⊂ H isa subgroup, then K denotes the closure of K in thetopological group H.

Elements of K\H/K are profinitely separated if their

images in K\H/K are distinct.

Theorem

If H is of finite type, then Hermitian K-spherical functions onK\H/K separate precisely those elements which are profinitelyseparated.

Note that the conjugacy separability of Bn is unknown (forn ≥ 4).




Problem

What can be said about braided surfaces (in particular theirfundamental groups) which cannot be distinguished by unitaryspherical functions?

Unitary spherical functions are not invariants of strongequivalence classes, since mapping class group actergodically on the moduli spaces of repersentations (seeGoldman ’99, Pickrell-Xia ’03).

However, we can construct strong equivalence invariants byusing instead regular functions on moduli spaces ofG-bundles

Γ(Σ \B)\Homstable(π1(Σ \B), G)/G

for suitable noncompact Lie groups, e.g. SL(n,C).


Braided surfaces and their characteristic maps...Characteristic maps of braided surfaces Elementary...

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