Open Questions

Research questions I am currently thinking about:

• Under what operations is the class closed, which is given by those groups G for which the norm map (in the derived category) of Q[G]-Modules is an equivalence. Both finite groups as well as groups with finitely dominated BG are contained in this class.

• Which properties of groups are stable under ultraproducts?

• Is the class of groups that satisfy the Farrell-Jones conjecture closed under infinite products?

• The Whitehead group of the cyclic group C_n of order n is particularly simple: It is given by a free abelian group of finite rank r, where r is the difference of the number of irreducible real representations and the number of irreducible rational representations. Concrete elements called Bass-Cyclic Units can be given via a formula, however these may fail to generate the entire group. For example for n=5, the element 1-t^2-t^3 in Z[C_5] is the generator of the Whitehead group, but not given by a Bass-Cyclic Unit. Provide concrete formulas of the generators as Elements in the group ring for all n.

• Related to this: Can the kernel of the trace map to THH as a subgroup of the Whitehead Group always be completely described?

• This question was told to me by Jim Davis: Given an amalgamated product (= pushout) of groups G = K_1 *_H K_2, Waldhausen proved that there is a Mayer-Vietoris-type long exact sequence in K-theory ... -> K_1(Z[K_1])+K_1(Z[K_2]) -> K_1(Z[G])/Nil -> K_0(Z[H]) ->... Is it possible to find an example of G such that the resulting sequence 0 -> cok -> K_1(Z[G])/Nil -> ker -> 0 does not split?

• Given a map of spaces X -> Y with finitely dominated fibers, there are wrong way maps in stable homotopy and K-Theory - One can consider the trace map from K-theory to stable homotopy. Does the resulting square commute? (Currently working on this with Marlkiewich, Naef)

• Given an E_infty-ring k and an E_1-group G, there should exist a generalization of the Cartan map Perf_k[G] -> Fun(BG, Perf_k) which exists for finite groups. The norm map involving the dualizing spectrum of G of the K-theory of Perf_k should factor through the K-theory of this generalized Cartan map. Similarly, this should hold for all localizing invariants. This will potentially only work in the infinity category of dualizable stable infinity-categories, and potentially needs a finiteness assumption on BG.

• By recent work of Ramzi-Sosnilo-Winges, the K-theory functor Cat^perf -> Sp has a section. This implies in particular that Segal K-theory, which group completes the core of a symmetric monoidal category, factors as a functor SymmMonCat -> Cat^Perf. Can this functor be improved in a nice way? Maybe also modeling the classical comparison map K( Proj(R) ) -> K( Perf_R ).

• Carlsson-Pederson '95 gave an explicit model for the assembly map in K and L-theory using controlled algebra. Barthels-Efimov-Nikolaus do the same using the stable category of sheaves. There should be a comparison of these two approaches given by a functor "Controlled modules" -> Calk( hatted CoSh( X, R-Mod ) ), which induces an equivalence in K and L-theory.

• There should exist a 6-functor formalism on condensed Anima, that encompasses both the 6-functor formalism of locally compact Hausdorff spaces, which sends such a space X to CoSh(X, Sp), as well as the 6-functor formalism of anima, which sends an anima Y to Fun(Y,Sp). The natural functor CoSh(X, Sp) -> Fun(X,Sp) should be incorporated by the assignment of a locally compact Hausdorff space X to its homotopy type (viewed as a condensed anima).

• Given an E_1-group G and an analytic ring R, there should be a notion of a "condensed Von-Neumann E_1-algebra" that generalizes the known Von Neumann Algebra in the case when R is the complex numbers. There should be a notion of generalized dimension theory for some notion of module over such algebras. These should give better control over L_2-Betti numbers and related invariants.

• The Rosenberg Conjectures. Given a Banach Algebra A over the reals R the two statement should  be true:

  • The algebraic K-theory of A and the topological K-theory agree with mod l coefficients in positive degrees.

  • The non-positive algebraic K-theory of A is homotopy invariant.

• Given a functor F : C -> D, one can define an endofunctor D -> D as the Right Kan Extension of F along itself (if it exists). This is called a codensity monad. One can define "condensed objects" for any codensity-monad. How much of the theory of condensed sets does still work in this context?