Problem 4.10 — Compute the cohomology of AutFn and OutFn with coefficients in various GL(n, Q)-modules.
v1.3 research notesCompute 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 ...
Problem 4.11 — Determine whether the natural homomorphisms ˜H ∗(AutF2g; Q)−→˜H ∗(Mg,1; Q) ˜H ∗(OutF2g; Q)−→˜H ∗(Mg,∗; Q) induced by…
v1.3 research notesDetermine 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,∗→...
Conjecture 6.1 — The classes e1,t 3,t 5,··· are all non-trivial.
v1.3 research notesThe 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....
Problem 7.2 — Find explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial.
v1.3 research notesFind explicit graphs Γ∈G odd such that the corresponding homology classes Φ(Γ) are non-trivial....
Problem 8.1 — Determine the image as well as the cokernel of the homomorphism (15) explic- itly.
v1.3 research notesDetermine 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 ...
Problem 8.2 — Describe the Galois images in hg,1⊗ Zℓ.
v1.3 research notesDescribe the Galois images in hg,1⊗ Zℓ. The above result was proved by analyzing the number theoretical enhancement of the Johnson homomorphism where ...
Problem 10.3 — Give examples of odd valent graphs Γ whose associated homology classes Φ(Γ)∈ H∗(OutFn; Q) are non-trivial as many as…
v1.3 research notesGive 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...
Problem 11.2 — Study the central extension (20) from the point of view of group cohomology as well as geometric topology.
v1.3 research notesStudy the central extension (20) from the point of view of group cohomology as well as geometric topology. In particular determine the Euler class of ...
Conjecture 11.3 — 1.
v1.3 research notes1. ¯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...
Problem 11.4 — Determine the abelianization of the group Hg,1.
v1.3 research notesDetermine 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...
Problem 11.5 — Generalize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of h…
v1.3 research notesGeneralize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of homology cobordism classes of homo...
Problem 12.1 — Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g.
v1.3 research notesProve that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g. The cohomology classes in (22) are stable w...
Problem 12.2 — Study whether the homology groups of BDiffδ +Σg stabilize with respect to g or not.
v1.3 research notesStudy 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...
Question — Which properties of the braid groups can be extended to the mapping class groups?
v1.3 research notesWhich properties of the braid groups can be extended to the mapping class groups?...
Major problems 1 — The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory shou…
v1.3 research notesThe 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...
Major problems 2 — The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category…
v1.3 research notesThe 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...
Applications 2 — Neil Strickland points out that several different moduli spaces are used in differential geometry and…
v1.3 research notesNeil Strickland points out that several different moduli spaces are used in differential geometry and physics. For example, there is the moduli space ...
Axiomatic stable homotopy 10 — What is the endomorphism ring of the identity functor on the stable homotopy category?
v1.3 research notesWhat is the endomorphism ring of the identity functor on the stable homotopy category? The ring Z splits off this ring, including by multiples of the ...
Model categories 10 — Is every monoidal model category Quillen equivalent to a simplicial monoidal model category?
v1.3 research notesIs every monoidal model category Quillen equivalent to a simplicial monoidal model category? This would remove the loose end in my book on model categ...
Model categories 13 — Find a model category you can prove is not cofibrantly generated.
v1.3 research notesFind a model category you can prove is not cofibrantly generated. This is just an annoyance, not a very significant problem, but it has been bugging m...
Unstable homotopy theory 1 — The Johnson question.
v1.3 research notesThe Johnson question. This says that if X is a space, and x is in BP_n (X), then x is not v_n torsion. My guess is that one should consider this quest...