Journal of Homotopy and Related Structures最新文献

This paper compiles and expands upon the author’s and his co-authors’ explicit calculations of the mod p Morava K-theory for various finite p-groups, a body of work currently scattered across different publications. The primary focus is on the author’s observations regarding the properties of formal group laws and the transfer in Morava K-theory. Using specific examples, this work aims to clarify the complex issues surrounding the multiplicative structure and the representation of Gröbner bases in terms of Chern classes and their transfers. A key computational question remains: is the mod 2 Morava K-theory of any finite 2-group completely generated by Chern classes and their transfers? While this conjecture by Hopkins et al. (J Am Math Soc 13:553–594, 2000), inspired by their generalized character theory of finite p-groups, was disproven for the mod (p>2) case by a counterexample in Kriz (Topology 36:1247–1273, 1997), the mod 2 case remains an open problem.

We reprove the generalized Nandakumar–Ramana Rao conjecture for the prime case using representation ring-graded Bredon cohomology. Our approach relies solely on the (RO(C_p))-graded cohomology of configuration spaces, viewed as a module over the (RO(C_p))-graded Bredon cohomology of a point.

In this note we prove that two seemingly different smooth 4-manifolds arising as quotients of (S^2times S^2) by free actions of (mathbb {Z}/4) are in fact diffeomorphic, answering a question of Hambleton and Hillman.

We compute the (E_2) page of the Adams spectral sequence converging to the connective KO-theory of the second mod 2 Eilenberg–MacLane space, (ko_*(K({mathbb Z}_2,2))), where ({mathbb Z}_2) is the cyclic group of order 2. This required a careful analysis of the structure of (H^*(K({mathbb Z}_2,2);{mathbb Z}_2)) as a module over the subalgebra of the Steenrod algebra generated by (operatorname {Sq}^1) and (operatorname {Sq}^2). Complete analysis of the spectral sequence is performed in [8].

We prove that for the action of a finite constant group scheme, equivariant algebraic K-theory is represented by a colimit of Grassmannians in the equivariant motivic homotopy category. Using this result we show that the set of endomorphisms of the equivariant motivic space defined by (K_0(G,-)) coincides with the set of endomorphisms of infinite Grassmannians in the equivariant motivic homotopy category by explicitly computing the equivariant K-theory of Grassmannians.

This paper builds on top of Positselski (J Homot Relat Struct 19(4):635–678, 2024). We consider a complete, separated topological ring ({mathfrak {R}}) with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left ({mathfrak {R}})-contramodules is equivalent to the derived category of the exact category of flat left ({mathfrak {R}})-contramodules, and also to the homotopy category of flat cotorsion left ({mathfrak {R}})-contramodules. In other words, a complex of flat ({mathfrak {R}})-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat ({mathfrak {R}})-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat ({mathfrak {R}})-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortés–Izurdiaga, and Estrada.

We prove a specific case of Rubin’s saturation conjecture about the realization of G-transfer systems, for G a finite cyclic group, by linear isometries (N_infty )-operads, namely the case of cyclic groups of order (p^nq^m) for p, q distinct primes and (n,min mathbb {N}).

Denote the virtual cohomological dimension of (textrm{SL}_n(mathbb {Z})) by (t=n(n-1)/2). Let St denote the Steinberg module of (textrm{SL}_n(mathbb {Q})) tensored with (mathbb {Q}). Let (Sh_bullet rightarrow St) denote the sharbly resolution of the Steinberg module. By Borel–Serre duality, the one-dimensional (mathbb {Q})-vector space (H^0(textrm{SL}_n(mathbb {Z}), mathbb {Q})) is isomorphic to (H_t(textrm{SL}_n(mathbb {Z}),St)). We find an explicit generator of (H_t(textrm{SL}_n(mathbb {Z}),St)) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of (textrm{SL}_n(mathbb {Z})).

J. Milnor introduced a specific class of codimension-1 submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides exact values for this invariant for numerous Milnor manifolds. Furthermore, we improve the upper bounds on the higher equivariant topological complexity. As an application, we obtain sharper bounds on the higher equivariant topological complexity of Milnor manifolds with free (mathbb {Z}_2) and (S^1)-actions.

We are interested in computing the Bredon cohomology with coefficients in the constant Mackey functor (underline{{mathbb {F}}_2}) for equivariant (text {Rep}(C_2)) spaces, in particular for Grassmannian manifolds of the form (operatorname {Gr}_k(V)) where V is some real representation of (C_2.) It is possible to create multiple distinct (text {Rep}(C_2)) constructions of (and hence multiple filtration spectral sequences for) a given Grassmannian. For sufficiently small examples one may exhaustively compute all possible outcomes of each spectral sequence and determine if there exists a unique common answer. However, the complexity of such a computation quickly balloons in time and memory requirements. We introduce a statistic on (mathbb {M}_2)-modules valued in the polynomial ring (mathbb Z[x,y]) which makes cohomology computation of Rep((C_2))-complexes more tractable, and we present some new results for Grassmannians.