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...
Morava K- and E-theory 3 — Elucidate the connection between the Morava stabilizer groups and the K(n)-local category.
v1.3 research notesElucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard an...
Morava K- and E-theory 4 — As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local ca…
v1.3 research notesAs 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...
Morava K- and E-theory 5 — Find the shadow of the thick subcategory theorem in the K(n)-local category.
v1.3 research notesFind 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...
Morava K- and E-theory 7 — Presumably one should be able to form a category of E-S module spectra; spectra with an action of the…
v1.3 research notesPresumably 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...
Morava K- and E-theory 10 — Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-th…
v1.3 research notesBousfield 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...
Elliptic cohomology 2 — Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do…
v1.3 research notesAlmost 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...
Elliptic cohomology 3 — Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines.
v1.3 research notesDennis McLaughlin and Jean-Luc Brylinski also thought along these lines. They wanted to use gerbes, or 2-gerbes maybe, instead. I could never understa...
Elliptic cohomology 4 — Yet another idea is to go back to a decription of cobordism I once heard.
v1.3 research notesYet 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...
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 2 — Characterize the stable homotopy category up to equivalence.
v1.3 research notesCharacterize the stable homotopy category up to equivalence. This has been done for categories that are homotopy categories of model categories by Sch...
Axiomatic stable homotopy 3 — Show that there is only a set of localizing subcategories.
v1.3 research notesShow that there is only a set of localizing subcategories. It is known that there is only a set of Bousfield classes (Ohkawa; Strickland simplified hi...
Axiomatic stable homotopy 4 — In one of Bob Thomason's last papers, he determined the thick subcategories of finite objects in the d…
v1.3 research notesIn one of Bob Thomason's last papers, he determined the thick subcategories of finite objects in the derived category of a scheme. For the derived cat...
Axiomatic stable homotopy 5 — John Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he…
v1.3 research notesJohn Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he has found a computable ring which is f-isomorph...
Axiomatic stable homotopy 6 — My general feeling about stable homotopy categories is that they are like commutative rings.
v1.3 research notesMy general feeling about stable homotopy categories is that they are like commutative rings. Follow this up; define Spec C for example, for a stable h...
Axiomatic stable homotopy 7 — The equivariant stable homotopy category is not treated very well in our memoir.
v1.3 research notesThe equivariant stable homotopy category is not treated very well in our memoir. That is, we assume that the generators have to be dualizable. This is...
Axiomatic stable homotopy 8 — From an axiomatic point of view, I don't understand Grojnowski's equivariant elliptic cohomology.
v1.3 research notesFrom an axiomatic point of view, I don't understand Grojnowski's equivariant elliptic cohomology. This theory takes values in an abelian category that...
Axiomatic stable homotopy 9 — Suppose G is a self-equivalence of the stable homotopy category.
v1.3 research notesSuppose G is a self-equivalence of the stable homotopy category. Must G be some iterate of the suspension functor? If G commutes with the suspension, ...
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 ...
Equivariant homotopy 4 — As a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey…
v1.3 research notesAs a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey functors over a Green functor, and analyze its ...
Model categories 1 — The safest sort of problem to work on with model categories is building one of interest in applications.
v1.3 research notesThe safest sort of problem to work on with model categories is building one of interest in applications. The essential idea is: whenever someone uses ...
Model categories 2 — A scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n.
v1.3 research notesA scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n. So maybe there is some kind of model structure on s...
Model categories 3 — Every stable homotopy category I know of comes from a model category.
v1.3 research notesEvery stable homotopy category I know of comes from a model category. Well, that used to be true, but it is no longer. Given a flat Hopf algebroid, St...
Model categories 4 — Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the…
v1.3 research notesGiven a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model categ...
Model categories 5 — The second step: show that the category of algebras over a cofibrant operad admits a model structure,…
v1.3 research notesThe second step: show that the category of algebras over a cofibrant operad admits a model structure, where the fibrations and weak equivalences are t...
Model categories 6 — Find conditions under which algebras over a noncofibrant operad admit a model structure that generaliz…
v1.3 research notesFind conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley. This woul...
Model categories 7 — Let A be a cofibrant operad as above.
v1.3 research notesLet A be a cofibrant operad as above. Use the above results to construct spectral sequences that converge to the homotopy groups of the space of A-alg...
Model categories 8 — My general theory is that the category of model categories is not itself a model category, but a 2-mod…
v1.3 research notesMy general theory is that the category of model categories is not itself a model category, but a 2-model category. Weak equivalences of model categori...
Model categories 9 — The 2-category of simplicial model categories is supposed to be (according to me) 2-Quillen equivalent…
v1.3 research notesThe 2-category of simplicial model categories is supposed to be (according to me) 2-Quillen equivalent to the 2-category of model categories. Even wit...
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 11 — Charles Rezk has a homotopy theory of homotopy theories.
v1.3 research notesCharles Rezk has a homotopy theory of homotopy theories. This is just a category, though it is large. The objects are generalizations of categories wh...
Model categories 12 — In the appendix to my book on model categories, I said maybe what we are doing in associating to a mod…
v1.3 research notesIn the appendix to my book on model categories, I said maybe what we are doing in associating to a model category its homotopy category is the wrong t...
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 3 — Suppose X is a simply connected finite complex.
v1.3 research notesSuppose X is a simply connected finite complex. Do the Steenrod reduced powers P^t act trivially on the mod p cohomology of the loop space of X when p...
Miscellaneous problems 3 — This one is due to Mike Hopkins.
v1.3 research notesThis one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-in...