The bicategory of parametrized spectra has a remarkably rich structure. We can take traces in this bicategory, giving classical invariants that count fixed points. We can also take C_n -equivariant external smash powers and equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these smash powers and traces for parametrized spectra depends on technical statements about the bicategory that can be difficult to verify directly, especially if one wants the construction to have a direct geometric interpretation. In this paper, we demonstrate the effectiveness of two tools—rigidity and deformable functors—by using them to establish formal structures on this bicategory directly from point-set level data.
We describe the main properties of the RO(C2 & times;Sigma 2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RO(C_2 imes \Sigma _2)$$\end{document}-graded cohomology ring of a point and apply the results to compute the subring of motivic classes given by the Bredon motivic cohomology of real numbers and to compute RO(C2 & times;Sigma 2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RO(C_2 imes \Sigma _2)$$\end{document}-graded cohomology ring of E Sigma 2C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\Sigma _2}C_2$$\end{document}. This generalizes Voevodsky's identification of motivic cohomology of real numbers with the positive cone of RO(C2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RO(C_2)$$\end{document} graded cohomology of a point.
We introduce a parametrized version of the homotopic distance between maps, extending the notion introduced by Macías-Virgós and Mosquera-Lois to the fibrewise setting. Working in the category of fibrewise spaces over a fixed base space B , we define the parametrized homotopic distance D_B(f,g) between fibrewise maps and show that it admits an equivalent description in terms of the fibrewise sectional category of a canonical projection. This interpretation allows us to relate D_B(f,g) to the fibrewise Lusternik–Schnirelmann category and to the parametrized topological complexity. We establish the fundamental properties of the invariant, including fibre-homotopy invariance, behaviour under composition, triangle inequality, product inequalities, and cohomological lower bounds. For fibrewise fibrations, we obtain estimates for the parametrized homotopic distance in terms of the corresponding invariants on the fibres and the fibrewise LS category of the base, extending classical results of Varadarajan and Farber-Grant to the parametrized context. We also study maps into fibrewise H-spaces admitting a fibrewise division, deriving sharp estimates for the parametrized topological complexity in this setting and presenting explicit examples arising from sphere bundles. Finally, we introduce a pointed version of the parametrized homotopic distance, compare it with the unpointed invariant, and identify conditions under which both notions coincide.
The Morava E -theories, $$E_{n}$$ E n , are complex-oriented 2-periodic ring spectra, with homotopy groups $$W_{{{\mathbb {F}}}_{p^{n}}}[[u_{1}, u_{2},\ldots , u_{n-1}]][u,u^{-1}]$$ W F p n [ [ u 1 , u 2 , … , u n - 1 ] ] [ u , u - 1 ] . Here W denotes the ring of Witt vectors. $$E_{n}$$ E n is a Landweber exact spectrum and hence uniquely determined by its homotopy groups as $$BP_{*}$$ B P ∗ -algebra. Algebraic K -theory of $$E_{n}$$ E n is a key ingredient towards analyzing the layers in the p -complete Waldhausen’s algebraic K -theory chromatic tower. One hopes to use the machinery of trace methods to get results towards algebraic K -theory once the computation for $$THH(E_{n})$$ T H H ( E n ) is known. In this paper we describe $$THH(E_{2})$$ T H H ( E 2 ) as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of $$E_{2}$$ E 2 . For these results, we first calculate K ( i )-homology of $$THH(E_{2})$$ T H H ( E 2 ) using a Bökstedt spectral sequence and then lift the generating classes of K (1)-homology to fundamental classes in homotopy group of $$THH(E_{2})$$ T H H ( E 2 ) . These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to $$THH(E_{2})$$ T H H ( E 2 ) .
