Mathematics Problem Archive

Prove (or disprove) that the even Mumford-Morita-Miller classes e2i∈H 4i(Ig; Q) are non-trivial, in a suitable stable range, as cohomology classes of ...

Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be finitely generated by Johnson [42])....

Let ug denote the graded Lie algebra associated to the prounipotent radical of the relative Malcev completion of Ig defined by Hain [29] and let ug→hQ...

Prove that all the secondary classes d2,d 3,··· are non-trivial. Here is a problem concerning the first class d1. Let C be a separating simple closed ...

Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg. In particular, determine whether the Magnus representation Ig,1→GL(2g; Z[H]) of t...

The classes µi are non-trivial for all i = 1, 2,···. More generally we have the following....

Produce non-trivial rational (co)homology classes of OutFn. Next we consider the group IOut n. In [ 38] Igusa defined higher Franz-Reidemeister torsio...

(Igusa). Prove that the higher Franz-Reidemeister torsion classes τ2i∈H 4i(IOutn; R) are non-trivial in a suitable stable range. 22. Cohomological str...

Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the right hand side is generated by (cert...

Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorphisms in (9)....

Define unstable (co)homology classes of GL(n, Z). In particular, what can be said about the image of µi ∈ H4i(OutF2i+2; Q) in H4i(GL(2i + 2, Z); Q) un...

The stable rational cohomology of OutFn is trivial. Namely lim n→∞ ˜H ∗(OutFn; Q) = 0. We can aslo ask how the cohomology of Out Fn with twisted coeffic...

Compute the cohomology of AutFn and OutFn with coefficients in various GL(n, Q)-modules. 360 S. Morita For example, we could ask how Looijenga’s result ...

Determine whether the natural homomorphisms ˜H ∗(AutF2g; Q)−→˜H ∗(Mg,1; Q) ˜H ∗(OutF2g; Q)−→˜H ∗(Mg,∗; Q) induced by the inclusions Mg,1→AutF2g, Mg,∗→...

The classes e1,t 3,t 5,··· are all non-trivial. Furthermore they are linearly independent and form a basis of H 2(hQ g,1)Sp....

Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial....

Determine the image as well as the cokernel of the homomorphism (15) explic- itly. Note that Hain [ 29] proved that the image of (15), after tensored ...

Describe the Galois images in hg,1⊗ Zℓ. The above result was proved by analyzing the number theoretical enhancement of the Johnson homomorphism where ...

Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as possible. Also compare these clas...

Study the central extension (20) from the point of view of group cohomology as well as geometric topology. In particular determine the Euler class of ...

1. ¯σ∗(˜t2k+1) is non-trivial in H 2(Hg,1) for any k 2. σ∗(˜t2k+1) is trivial in H 2(Hg,1) for any k. The first part of the above conjecture is the “g...

Determine the abelianization of the group Hg,1. Is it trivial? Also determine the second homology group H2(Hg,1; Z). Is the rank of it equal to 1 give...

Generalize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of homology cobordism classes of homo...

Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g. The cohomology classes in (22) are stable w...

Study whether the homology groups of BDiffδ +Σg stabilize with respect to g or not. The same problem for the group SympδΣg. 374 S. Morita Acknowledgmen...

Which properties of the braid groups can be extended to the mapping class groups?...

The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory should go, analogous to the Weil conjectures in alge...

The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category of finite spectra. That is, if f is a map of fin...

Find some geometric meaning for elliptic cohomology. I believe this problem may be solvable--we keep learning new things about it. One thing I will sa...

On the same theme, find some way of doing index theory related to elliptic cohomology. This is not really algebraic topology, but would have a major i...

The chromatic splitting conjecture, which is considerably more complicated to state. Basically nothing is known about this, and so this one may be mor...

Classify all finite loop spaces. This is the long term project of Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the Lie groups are...

Say something general about the stable or unstable homotopy groups of spheres. For example, Ravenel has suggested that the size of the nth homotopy gr...

Does the possible Kervaire-invariant-one element $\theta_6\in\pi_{126}^{S}$ exist; equivalently, is $h_6^2$ a permanent cycle in the mod-2 Adams spect...

Once again, I am not sure whether this problem deserves to be called major, but it is annoying that the the R. Cohen - Goerss result proving that h_0 ...

Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree. This would follow from the chromatic splitting conjecture, I think. (...

Show that the Picard group is finitely generated over the p-adics. I don't think this is known even for the algebraic Picard group, which is obtained ...

Elucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard an...

As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local category. Consider the spectral sequence, for exam...

Find the shadow of the thick subcategory theorem in the K(n)-local category. There is only one thick subcategory of small spectra in the K(n)-local ca...

We now know that Morava E-theory admits an action of the stabillizer group S. This is the famous Hopkins-Miller result, which one day I hope will see ...

Presumably one should be able to form a category of E-S module spectra; spectra with an action of the ring spectrum E and a compatible action of the g...

Understand the relationship between the K(n)-local category and some sort of (algebraic) derived category of E_*-S-modules. Jens Franke has claimed th...

One of the corollaries of the Hopkins-Miller theorem, together with the Devinatz-Hopkins fixed point business, is that the famous class zeta in contin...

Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-theory. Franke claims to have generalized all this...

Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do with 2-categories. If you think about vector bu...

Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines. They wanted to use gerbes, or 2-gerbes maybe, instead. I could never understa...

Yet another idea is to go back to a decription of cobordism I once heard. I think this description is in print somewhere, but I don't know where or wh...

Introduce stable homotopy theory into the world of C^*-algebras, like Voevodsky has done in algebraic geometry. More specifically, find a model struct...

Neil Strickland points out that several different moduli spaces are used in differential geometry and physics. For example, there is the moduli space ...