Mathematics

We show that the smooth theta divisors of general principally polarised abelian varieties can be chosen as irreducible algebraic representatives of the coefficients of the Chern-Dold character in complex cobordisms and describe the action of the Landweber-Novikov operations on them. We introduce a quantisation of the complex cobordism theory with the dual Landweber-Novikov algebra as the deformation parameter space and show that the Chern-Dold character can be interpreted as the composition of quantisation and dequantisation maps. Some smooth real-analytic representatives of the cobordism classes of theta divisors are described in terms of the classical Weierstrass elliptic functions. The link with the Milnor-Hirzebruch problem about possible characteristic numbers of irreducible algebraic varieties is discussed.

In the early 1940's, P.A.Smith showed that if a finite p-group G acts on a finite complex X that is mod p acyclic, then its space of fixed points, X^G, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study chromatic versions of this statement, with the question: given H<G and n, what is the smallest r such that if X^H is acyclic in the (n+r)th Morava K-theory, then X^G must be acyclic in the nth Morava K-theory? Barthel this http URL. then answered this when G is abelian, by finding general lower and upper bounds for these `blue shift' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E.E.Floyd, which replaces acyclicity by bounds on dimensions of homology, and thus applies to all finite G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. In one direction, we are able to use classic constructions and representation theory to search for blue shift number lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that don't follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. As samples of new applications, we offer a new result about involutions on the 5-dimensional Wu manifold, and a calculation of the mod 2 K-theory of a 100 dimensional real Grassmanian that uses a C_4 chromatic Floyd theorem.

We define H -fibration sequences as fibrations where the holonomy action of the fundamental group of the base on the fiber lies in a given subgroup H of E(F) , where E(F) is the homotopy automorphism group of the fiber. Furthermore, we classify these H -fibration sequences via a universal H -fibration sequence.

Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4 -dimensional cases: in these cases standard spheres are characterized. Canonical projections of unit spheres are special generic. In suitable cases, it is easy to construct special generic maps on manifolds represented as connected sums of products of spheres for example. It is an interesting fact that these maps restrict the topologies and the differentiable structures admitting them strictly in various cases. For example, exotic spheres, which are not diffeomorphic to standard spheres, admit no special generic map into some Euclidean spaces in considerable cases. In general, it is difficult to find (families of) manifolds admitting no such maps of suitable classes. The present paper concerns a new result on this work where key objects are products of cohomology classes of the manifolds. We can see that closed symplectic maifolds, real projective spaces, and so on, admit no special generic map into a connected open manifold in considerable cases for example.

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal ∞ -categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondance of coalgebras in ∞ -categories. Moreover it shows that Shipley's zig-zag of Quillen equivalences provides an explicit symmetric monoidal equivalence of ∞ -categories for the stable Dold-Kan correspondance. We study homotopy coherent coalgebras associated to a monoidal model category and we show that these coalgebras cannot be rigidified. That is, their ∞ -categories are not equivalent to the Dwyer-Kan localizations of strict coalgebras in the usual monoidal model categories of spectra and of connective discrete R -modules.

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in 3 -space. Cocycles of them can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call G -Alexander biquandles and G -Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle of them from a group cocycle.

We prove that the genuine C 2 n -spectrum N C 2 n C 2 M U R is cofree, for all n . Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal conjecture for C 2 , independent of Lin's theorem.

In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds X and X ′ , the following conditions are equivalent: (1) the upper triangular matrix associated to X can be transformed into that of X ′ by those three operations; (2) X and X ′ are diffeomorphic; (3) the integral cohomology rings of X and X ′ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.

The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano d -folds with d=3,4 or with Picard number ≥2d−2 . In particular, we show that those manifolds except for two toric Fano 4 -folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano 4 -folds (their ID numbers are 50 and 57 on a list of Øbro) have isomorphic cohomology rings and their total Pontryagin classes are preserved under an isomorphism between their cohomology rings, but we do not know whether they are diffeomorphic or homeomorphic.

We compute the cohomology ring of a generalised type of configuration space of points in R r . This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and Cohen. However, our computations give a generalisation to any graph and an alternative proof of the classical result. Moreover, we show that there are deletion-contraction short exact sequences for this cohomology rings.