arXiv subjects
Publications and source records attributed to Bernard Badzioch.
In a recent work Malkiewich and Merling proposed a definition of the equivariant $K$-theory of spaces for spaces equipped with an action of a finite group. We show that the fixed points of this spectrum admit a tom Dieck-type splitting. We also show that this splitting is compatible with the splitting of the equivariant suspension spectrum. The first of these results has been obtained independently by John Rognes.
Given a unipotent bundle of smooth manifolds we construct its secondary transfer map and show that this map determines the higher smooth torsion of the bundle. This approach to higher torsion provides a new perspective on some of its properties. In particular it yields in a natural way a formula for torsion of a composition of two bundles.
Given a fixed object $A$ in a suitable pointed simplicial model category $\C$, we study the problem of recovering the target $Y$ from the pointed mapping space \w{\mapa(A,Y)} (up to $A$-equivalence). We describe a recognition principle, modelled on the classical ones for loop spaces, but using the more general notion of an \emph{\Ama[.]} It has an associated transfinite procedure for recovering \w{\CWA Y} from \w[,]{\mapa(A,Y)} inspired by Dror-Farjoun's construction of \ww{\CWA{}}-approximations.
We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer-Weiss-Williams torsion is proportional to the higher torsion of Igusa and Klein.
We show that if A is a simply connected, finite, pointed CW-complex then the mapping spaces Map(A, -) are preserved by the localization functors only if A has the rational homotopy type of a wedge of spheres of a fixed dimension.
We give a construction of a torsion invariant of bundles of smooth manifolds which is based on the work of Dwyer, Weiss and Williams on smooth structures on fibrations.
Dwyer, Weiss, and Williams have recently defined the notions of parametrized topological Euler characteristic and parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and Reidemeister torsion of finite CW-complexes.
An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space $X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the results establishing equivalence of homotopy theories of loop spaces and simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids etc.
We suggest a new delooping machine, which is based on recognizing an n-fold loop space by a collection of operations acting on it, like the traditional delooping machines of Stasheff, May, Boardman-Vogt, Segal, and Bousfield. Unlike in the traditional delooping machines, which carefully select a nice space of such operations, we consider all natural operations on n-fold loop spaces, resulting in the algebraic theory Map (V_. S^n, V_. S^n). The advantage of this new approach is that the delooping machine is universal in a certain sense, the proof of the recognition principle is more conceptual, works the same way for all values of n, and does not need the test space to be connected.
The paper examines machines of the type of the $Γ$-spaces of Segal which describe homotopy structures on topological spaces. The main result of the paper shows that for any such machine one can find an algebraic theory characterizing the same structure on spaces as the original machine.
We describe a perfect group whose localization is not perfect.
We give a homotopy theoretical characterization of generalized Eilenberg-Mac Lane spaces, modeled after Segal's characterization of infinite loop spaces via Gamma spaces.