8.4 (Schleimer) — Detecting reducible Heegaard splittings
v1.3 research notesIs there an algorithm to detect whether a Heegaard splitting is reducible and, if so, find a reducing curve?...
8.5 (Schleimer) — Classification of strongly irreducible Heegaard splittings
v1.3 research notesIs there a classification of the strongly irreducible Heegaard splittings of a given $3$-manifold?...
8.7 (Tillmann) — Higher-dimensional multisections and stabilization
v1.3 research notesDo higher-dimensional smooth manifolds always admit multisections? What is the correct generalization of uniqueness up to stabilization for multisecti...
Problem 2.1 — Determine the finiteness properties of Ig.
v1.3 research notesDetermine the finiteness properties of Ig. For which k is Hk(Ig) finitely gen- erated? For which k is there a K(Ig, 1) with finite k-skeleton (one say...
Question 2.9 — Is it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free?
v1.3 research notesIs it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free? Gromov discovered the analogous phe...
Conjecture 3.15 — (Density of pseudo-Anosovs).
v1.3 research notes(Density of pseudo-Anosovs). LetP denote the set of pseudo-Anosov ele- ments of Modg. Then d(P) = 1. J. Maher [ Mah] has recently proven that a random...
Problem 3.2 — Construct any representations of Mg, finite or infinite, which do not factor through Sp(2g, Z).
v1.3 research notesConstruct any representations of Mg, finite or infinite, which do not factor through Sp(2g, Z). In a very different direction, every mathematician woul...
Problem 1: — Determine the metric completion of the Gromov boundary of C(S) and relate this metric completion to the geometry of C…
v1.3 research notesDetermine the metric completion of the Gromov boundary of C(S) and relate this metric completion to the geometry of C(S). There is yet another way to ...
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 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...
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 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, ...
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 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 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 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...
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...