Mathematics Problem Archive

f f is pseudo-Anosov element of a mapping class group ModS with sufficiently big dilatation coefficient, then the subgroup of ModS normally generated by f...

Are there any other relations between N -th powers Tγ = tN γ of Dehn twists for sufficiently high N? In other words, do the above relations provide a pr...

s the subgroup of ModS generated by the N -th powers of all elements of ModS of infinite index in ModS for sufficiently big N? Notice that such a subgro...

LetS and R be closed surfaces. Let Γ be a subgroup of finite index in ModS. If the genus of R is less than the genus of S, then there is no homomorphi...

For every finitely generated subgroups G of ModS, the group Φf (G) is nilpotent. For a a little bit more detailed discussion, see [ I4], Section 10.10...

Is there an endomorphism of the mapping class group of a (closed) orientable surface onto an infinite index infinite subgroup? If there is one such en...

Let Γ be a subgroup of finite index in the mapping class group Mod1,2 and let φ: Γ → Γ be an automorphism. Is φ the restriction of an automorphism of ...

Let g > hand let Γ be a finite index subgroup of Modg. Assume that g≥ 3 and φ: Γ → Modh is a homomorphism. (a) Is the image of φ necessarily finite? (...

Let Γ be a finite index subgroup of Modg. Is it true that H 1(Γ; Z) = 0? There are some partial answers to Problem 2.6. If g≥ 3 and if Γ contains the ...

Suppose that ta1ta2··· tan = 1 in Modg, where n≥ 1. Let G denote the quotient H1(S)/⟨[a1], [a2],..., [an]⟩, where [ai] denotes the homology class of t...

Given a factorization ta1ta2··· tan = 1 of the identity into a product of right Dehn twists in Modg, is it always possible to lift this factorization ...

Suppose that a mapping class f ∈ Modb g, b≥ 1, is a product of right Dehn twists. Does there exist a constant Cf, depending on f, such that whenever f...

Compute ϕ(g,n ). Is it constant? If not, for g < h, compare ϕ(g,n ) and ϕ(h,n ). 90 M. Korkmaz...

Let g≥ 3 and b≥ 1. Let a1,a 2,... be an infinite sequence of nonseparating simple closed curves on an oriented surface S of genus g with b boundary co...

Let r be a positive integer and let Γr be the (normal) subgroup of Modg generated by the rth powers of all Dehn twists. Is Γr of infinite index? If we...

(a) It is known from [33] that H2(Mod3; Z) and H2(Mod1 3; Z) are either Z or Z⊕ Z2. What are they? Also compute H2(PModb 3,p; Z) for all p and b. (b) ...

(a) Is it possible to generate the mapping class group of a closed nonori- entable surface by two elements? (b) Is it possible to generate the mapping...

Compute the (outer) automorphism group of Mod(N )....

Letg >h, and let N andN ′ denote the closed nonorientable surfaces of genera g andh respectively. Is it true that any homomorphism φ: Mod(N )→ Mod(N ′...

Let φ: Mod( N )→ Mod(N ) be a homomorphism such that the image of φ is of finite index. Is φ necessarily an automorphism? How about if we take the dom...

Study homomorphisms Modg→ Mod(N ) and Mod(N )→ Modg. 92 M. Korkmaz It is known by the work of Birman and Chillingworth [ 3] that the mapping class gro...

Give a homotopy theoretic construction of a map ρh: Ω ∞CP ∞ −1→K Sp(Z) with ρ≃ ρh◦α, at least after localization at a regular prime. (The 2-local case...

. Is κ2i = 0 in H ∗(BT∞; Q)? Very little is known about the cohomology of the Torelli group past dimension 1, and as far as I know it might be possibl...

. Find a direct description of some particular simple torsion classes in H ∗(BΓ∞; Z). There is an interesting connection between the higher Reidemeist...

. Is Z×BAut(F∞)+ homotopy equivalent to Ω ∞S∞? 4. The interplay between homotopy theory and the mapping class group is inspired by the study of confor...

. Find an analogue of Theorem 4.1 for topological manifolds, and relate it to surgery and pseudo-isotopy theory. Waldhausen’s functor A(X) is the alge...

. Is there a geometric map from BC2 into topological cyclic homology of a point? A recent manuscript by Kevin Costello, [ 3], relates conformal field ...

. Can one generalize Theorem 2.1 to the Deligne-Mumford compactification of the moduli space of Riemann surfaces?...

Understand either classically or as quantum geometric objects the non-Hausdorff quotients of PL 0(F ) or PL(F ) by MC (F ) or PMC (F ). T (F ) has been...

Given a tuple ×N i=1(mi,ti) ∈ ZN, give a tractable expression in terms of Dehn- Thurston or other coordinates for the number of components of the corr...

Give a useful (piecewise) tropical description of the two elementary transformations. One thus immediately derives a (piecewise) tropical polynomial r...

Does the recipe in Theorem 4 give virtually all pA maps? That is, given a pA map f, is there some iterate fn, for n≥ 1, so that fn arises from the rec...

For a given surface F, calculate Σ( F ). More modestly, calculate the least element of Σ( F ) or the least gap among elements of Σ( F ). Characterize ...

Calculate the topological type (PL-homeomorphism, homotopy, homology... type) of the Arc-complexes. The first non-trivial case is the calculation of t...

Devise a matrix model (cf. [28]) for the calculation of the Euler characteristics of Arc-complexes....

[Contributed by the referee] Does Theorem 5 say anything about the structure of the end of Riemann’s moduli space? For instance, what is the homology ...

[Bounded Distortion Conjecture] Given a hyperbolic structure on F, associate its combinatorial invariant, namely, an ideal cell decomposition of F tog...

Though the (virtual) Euler characteristics are already known [28,56], devise a matrix model using screens to calculate these invariants for ¯M (F )....

[LevelN Torelli Franchetta Problem] What is the second cohomology group of the levelN Torelli group? The cell decomposition of ¯M(F ) described before...

Calculate various group-theoretic boundaries of mapping class and Torelli groups, for instance, Tits boundaries....

What are the kernels of the Magnus representations? Finally in [48] by taking contractions of powers of our canonical one cocycle, new combinatorially...

Which Artin groups admit non-geometric embeddings into M (S)? If we do not require the homomorphism to be geometric we do not need to restrict the que...

Do there exist any other examples of non commuting homeomorphisms g andh which are not both Dehn twists and satisfy a braid relation ghg...   mi,j...

Does there exist a set of at least three pseudo-Anosov homeomorpisms such that every pair satisfies a braid relation. The referee to this paper observ...

What is the maximal length of such a product? Can it be arbitrarily long?...

Are the relations (1) - (7) the only relations which express the twist along the boundary as a product of positive twists, up to the positive equivale...

Let αi, i = 1,...,n be a configuration of curves on a surface S of genus g with one boundary component δ such that every pair of curves intersect in 0...

Consider the Artin group A5 (the braid group on six strings) divided by the relation (a1a2a3a4)5 = a5a4a3a2a2 1a2a3a4a5. Is it true that every positiv...

Is the Hurwitz problem for mapping class group factorizations decidable? Are there interesting criteria which can be used to conclude that two given f...

(Donaldson). Is it possible to enumerate all matching paths in a Lefschetz fibration with given monodromy factorization?...