Mathematics

We construct a zig-zag from the once delooped space of pseudoisotopies of a closed 2n -disc to the once looped algebraic K -theory space of the integers and show that the maps involved are p -locally (2n−4) -connected for n>3 and large primes p . The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik--Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa's stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of BDiff ∂ ( D 2n+1 ) in degrees up to 2n−5 .

We construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories. We show that the functor H ≃ :2Cat→DblCat , a more homotopical version of the usual horizontal embedding H , is right Quillen and homotopically fully faithful when considering Lack's model structure on 2Cat . In particular, H ≃ exhibits a levelwise fibrant replacement of H . Moreover, Lack's model structure on 2Cat is right-induced along H ≃ from the model structure for weakly horizontally invariant double categories. We also show that this model structure is monoidal with respect to Böhm's Gray tensor product. Finally, we prove a Whitehead Theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalence.

Given a compact Lie group G and a commutative orthogonal ring spectrum R such that R[G ] ∗ = π ∗ (R∧ G + ) is finitely generated and projective over π ∗ (R) , we construct a multiplicative G -Tate spectral sequence for each R -module X in orthogonal G -spectra, with E 2 -page given by the Hopf algebra Tate cohomology of R[G ] ∗ with coefficients in π ∗ (X) . Under mild hypotheses, such as X being bounded below and the derived page R E ∞ vanishing, this spectral sequence converges strongly to the homotopy π ∗ ( X tG ) of the G -Tate construction X tG =[ EG ˜ ∧F(E G + ,X) ] G .

Dwyer, Miller and Wilkerson proved that at the prime 2, the classifying spaces of SU(2) and SO(3) can be obtained as a homotopy pushout of the classifying spaces of certain subgroups. In this paper we show explicitly how these decompositions arise from the fusion systems of SU(2) and SO(3) over maximal discrete 2-toral subgroups.

In this note, we present a formula for the action of the primitive Steenrod-Milnor operations on generators of algebra of invariants of the general linear group G L n =GL(n, F p ) in the polynomial algebra with p an arbitrary prime number.

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅ . Assume that σ is a complex conjugation, i.e, the differential of σ anti-commutes with J . The space P(m,X):= S m ×X/∼ where (v,x)∼(−v,σ(x)) is known as a generalized Dold manifold. Suppose that a group G≅ Z s 2 acts smoothly on X such that g∘σ=σ∘g for all g∈G . Using the action of the diagonal subgroup D=O(1 ) m+1 ⊂O(m+1) on the sphere S m for which there are only finitely many pairs of antipodal points that are stablized by D , we obtain an action of G=D×G on S m ×X , which descends to a (smooth) action of G on P(m,X) . When the stationary point set X G for the G action on X is finite, the same also holds for the G action on P(m,X) . The main result of this note is that the equivariant cobordism class [P(m,X),G] vanishes if and only if [X,G] vanishes. We illustrate this result in the case when X is the complex flag manifold, σ is the natural complex conjugation and G≅( Z 2 ) n is contained in the diagonal subgroup of U(n) .

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT's starting from a Hopf bc k -valued Brown functor where Hopf bc k is the category of bicommutative Hopf algebras over a field k : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT's. In another application, we reproduce the Dijkgraaf-Witten TQFT and the Turaev-Viro TQFT from an ordinary Hopf bc k -valued homology theory.

Consider a connected topological space X with a point x∈X and let K be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on X and its Tannakian dual π K (X,x) with respect to the fibre functor in x . The maximal pro-étale quotient of π K (X,x) is the étale fundamental group of X studied by Kucharczyk and Scholze. For well behaved topological spaces, π K (X,x) is the pro-algebraic completion of the ordinary fundamental group π 1 (X,x) . We obtain some structural results on π K (X,x) by studying (pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate π K (X,x) for some generalized solenoids.

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R , and outputs an explicit system of piecewise linear motion planners for R . The algorithm is designed in such a way that the cardinality of the outputed system is probabilistically close (with parameters chosen by the user) to minimal possible. This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber's topological complexity TC. Besides its relevance toward technological applications, our work revels that, unlike other discrete approaches to TC, the SC model can recast Farber's invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois' notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

We prove a comparison isomorphism between singular cohomology and sheaf cohomology.