Question 3.8 — Can Q(n,m ) have infinitely many finite quotients?
v1.3 research notesCan Q(n,m ) have infinitely many finite quotients? Is it residually finite?...
Question 4.1 — Can one detect the growth of a surface or free-group homomorphism by its action on the homology of a characteristic s…
v1.3 research notesCan one detect the growth of a surface or free-group homomorphism by its action on the homology of a characteristic subgroup of finite index? Notice t...
Question 4.2 — Classify those φ∈ Aut(Fn) for which Fn ⋊φ Z is automatic and those for which it is CAT(0).
v1.3 research notesClassify those φ∈ Aut(Fn) for which Fn ⋊φ Z is automatic and those for which it is CAT(0). Of central importance in trying to understand mapping tori ...
Question 4.3 — Is there an alogrithm to decide isomorphism among groups of the form F ⋊ Z.
v1.3 research notesIs there an alogrithm to decide isomorphism among groups of the form F ⋊ Z. In the purest form of this question one is given the groups as finite pres...
Question 4.4 — Is the conjugacy problem solvable in Out(Fn)?
v1.3 research notesIs the conjugacy problem solvable in Out(Fn)? Martin Lustig posted a detailed outline of a solution to this problem on his web page some years ago [ 6...
Question 5.1 — Where precisely does the rational homology of Aut(Fn) stabilize?
v1.3 research notesWhere precisely does the rational homology of Aut(Fn) stabilize? And for Out(Fn)? There are only two known non-trivial classes in the (unstable) ratio...
Question 5.2 — Are Morita’s original cycles non-trivial in homology?
v1.3 research notesAre Morita’s original cycles non-trivial in homology? Are the generalizations due to Morita and to Conant and Vogtmann non-trivial in homology? No oth...
Question 5.3 — Do the Morita classes generate all of the rational homology of Out(Fn)?
v1.3 research notesDo the Morita classes generate all of the rational homology of Out(Fn)? The maximum dimension of a Morita class is about 4 n/3. Morita’s cycles lift n...
Question 5.4 — Is the image of the second Morita class in H8(GL(6, Z); Q)) non-trivial?
v1.3 research notesIs the image of the second Morita class in H8(GL(6, Z); Q)) non-trivial? For further discussion of the cohomology of Aut( Fn) and Out( Fn) we refer to...
Question 6.1 — Is there a set of simple Steinberg-type relations for the mapping class group?
v1.3 research notesIs there a set of simple Steinberg-type relations for the mapping class group? There is also a presentation of Aut( Fn) coming from the action of Aut(...
Question 6.2 — Can Out(Fn) and Mod±(Sg) be obtained as a pushout of a finite subsystem of their finite subgroups, i.e.
v1.3 research notesCan Out(Fn) and Mod±(Sg) be obtained as a pushout of a finite subsystem of their finite subgroups, i.e. is either the fundamental group of a developab...
Question 6.3 — Establish finiteness properties of the kernel IA(n) of the map from Out(Fn) to GL(n, Z).
v1.3 research notesEstablish finiteness properties of the kernel IA(n) of the map from Out(Fn) to GL(n, Z). In particular, determine whether IA(n) is finitely presentabl...
Question 7.2 — What are the higher-dimensional isoperimetric functions of GL(n, Z), Aut(Fn)and Out(Fn)?
v1.3 research notesWhat are the higher-dimensional isoperimetric functions of GL(n, Z), Aut(Fn)and Out(Fn)?...
Question 7.3 — Is Aut(Fn) automatic for n> 3?
v1.3 research notesIs Aut(Fn) automatic for n> 3?...
Conjecture 2.1 — The natural homomorphisms ( Λ∗Λ3HQ )Sp →H ∗(Mg,∗; Q), (Λ∗UQ)Sp→H ∗(Mg; Q) induce isomorphisms ( Λ∗Λ3H ∗ Q/ ( [12]tore…
v1.3 research notesThe natural homomorphisms ( Λ∗Λ3HQ )Sp →H ∗(Mg,∗; Q), (Λ∗UQ)Sp→H ∗(Mg; Q) induce isomorphisms ( Λ∗Λ3H ∗ Q/ ( [12]torelli⊕ [22] ))Sp ∼=R∗(Mg,∗) ( Λ∗U ∗...
Problem 3.1 — Prove (or disprove) that the even Mumford-Morita-Miller classes e2i∈H 4i(Ig; Q) are non-trivial, in a suitable stable…
v1.3 research notesProve (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 ...
Problem 3.2 — Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be fini…
v1.3 research notesDetermine 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])....
Problem 3.3 — Let ug denote the graded Lie algebra associated to the prounipotent radical of the relative Malcev completion of Ig d…
v1.3 research notesLet 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...
Problem 3.4 — Prove that all the secondary classes d2,d 3,··· are non-trivial.
v1.3 research notesProve 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 ...
Problem 3.5 — Find explicit way of calculating d1(ϕ) for any given element ϕ ∈ Kg.
v1.3 research notesFind 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...
Conjecture 4.2 — The classes µi are non-trivial for all i = 1, 2,···.
v1.3 research notesThe classes µi are non-trivial for all i = 1, 2,···. More generally we have the following....
Problem 4.4 — (Igusa).
v1.3 research notes(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...
Problem 4.5 — Prove (or disprove) that the natural homomorphism H 4(OutF4; Q)∼= Q−→H 4(IOut4; Q)GL is an isomorphism where the righ…
v1.3 research notesProve (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...
Problem 4.6 — Determine the homomorphisms H 8(M3,∗; Q) i∗ ←−H 8(OutF6; Q) p∗ ←−H 8(GL(6, Z); Q) (10) induced by the above homomorph…
v1.3 research notesDetermine 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)....
Problem 4.8 — Define unstable (co)homology classes of GL(n, Z).
v1.3 research notesDefine 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...
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...