Mathematics Problem Archive

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...

. 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?...

(Smith). Is there an a priori upper bound on the length of any factorization of the boundary twist δ as a product of positive Dehn twists in Mapg,n? E...

For which T∈ Map+ g,n is it possible to classify factorizations of T as a product of positive Dehn twists in Mapg,n? In particular, for which T is the...

Given two factorizations of the boundary twist δ as a product of positive Dehn twists along nonseparating curves in Mapg,n, such that the total spaces...

The ideas which were just described relate to the beginning of the lower central series of Ig. There is also the lower central series of Kg. The corre...

Assume, for this problem, that M is a 3-manifold with non-empty boundary. Then, on one side of the double coset HφH the handlebody subgroup needs to b...

How is the Nielsen-Thurston trichotomy related to the question of whether the distance is 0, 1, 2 or≥ 3? The next 3 problems concern the very non-cons...

Find an algorithm to compute the distance d(φ) of an arbitrary element φ∈M. We note that an algorithm to compute shortest paths between fixed vertices...

Knowing that d(φ)≤ 1, can we decide whether d(φ) = 0? Geometrically, if a Heegaard splitting is weakly reducible, can you decide if it’s reducible?...

Knowing that d(φ)≥ 1, can we decide whether it is ≥ 2? Knowing that it’s ≥ 2, can we decide whether it is ≥ 3?...

Schleimer has proved in [42] that each fixed 3-manifold M has a bound on the distances of its Heegaard splittings. Study this bound, with the goal of ...

Study the handlebody subgroup of Mg. A simplified presentation which would reveal new things about its structure, and/or anything new about its coset ...

Recall that we noted, earlier, that every genus g Heegaard splitting of every homology 3-sphere is obtained by allowing ϕ to range over Ig. We also no...

Uncover the structure in the mapping class group that relates to the classifica- tion theorem for the Heegaard splittings of graph manifolds in [44]....

Given a normal subgroup Gg ofMg, what basic properties are needed in a complex G(Sg) of curves on Sg so that Mg will turn out to be naturally isomorph...

Hempel’s distance function was chosen so that it would capture the geometry, and indeed it does that very well, yet in some ways it feels unnatural. T...

This one is a warm-up. Given α∈M g, say as a product of Dehn twists, expressβ∈M 2g+1 as a related product of Dehn twists. With that in hand, observe t...

In [43] it is proved that in the case of the trivial genus g surface bundle, i.e. Sg×S1 the bundle-related splitting is unique, up to equivalence. Are...

A 3-manifold is fibered if it admits a surface bundle structure. It is virtually fibered if it has a finite-sheeted cover that admits a surface bundle...

Is there a natural quotient complex of any one of the complexes discussed in §1 which might be useful for the construction of non-faithful representat...

Study the double coset HφH inM, using new finite or infinite quotients of M. In this regard we stress finite, because a principle difficulty when this p...

Are there quotients of Ig orKg in which the intersection of either Ig orKg with the handlebody group Hg is sufficiently tractible to allow one to study ...

Develop techniques to describe the sets Cχ,θ....