Journal of Homotopy and Related Structures最新文献

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.

Motivated by the work of Gerstenhaber-Voronov and that of Malvenuto-Reuternauer, we define on pointed multiplicative operads in the category of vector spaces over an arbitrary ground field (mathbb {K}), a cosimplicial vector space structure. This permits us to construct on such operads some algebraic structures such as the homotopy G-algebra and the bicomplex algebra structures. Moreover we illustrate our constructions through some examples and explain or extend some well-known results.

Cazanave proved that the set of naive (mathbb {A}^1)-homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine (mathbb {A}^1)-homotopy classes of endomorphisms of the projective line. In this very short note we show that, over a field which is not quadratically closed, such a statement is never true for punctured affine space (mathbb {A}^nhspace{-0.1em}smallsetminus {0}) for (nge 2).

We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.

We develop the theory of nilpotent G-spaces and their localisations, for G a compact Lie group, via reduction to the non-equivariant case using Bousfield localisation. One point of interest in the equivariant setting is that we can choose to localise or complete at different sets of primes at different fixed point spaces—and the theory works out just as well provided that you invert more primes at (K le G) than at (H le G), whenever K is subconjugate to H in G. We also develop the theory in an unbased context, allowing us to extend the theory to G-spaces which are not G-connected.