Dai Tamaki

Higher topological complexity and its symmetrization

We develop the properties of the n th sequential topological complexity TCn , a homotopy invariant introduced by the third author as an extension of Farber’s topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of TCn.X/ to the Lusternik‐Schnirelmann category of cartesian powers of X , to the cup length of the diagonal embedding X ,! X n , and to the ratio between homotopy dimension and connectivity of X . We fully compute the numerical value of TCn for products of spheres, closed 1‐connected symplectic manifolds and quaternionic projective spaces. Our study includes two symmetrized versions of TCn.X/. The first one, unlike Farber and Grant’s symmetric topological complexity, turns out to be a homotopy invariant of X ; the second one is closely tied to the homotopical properties of the configuration space of cardinality-n subsets of X . Special attention is given to the case of spheres. 55M30; 55R80

A dual Rothenberg-Steenrod spectral sequence

Motivation One of the most convenient computational tools for the mod p homology of loop spaces is the cobar-type Eilenberg-Moore spectral sequence (EMSS, for short). The computation of H,(P’C”X; F,) using the path loop fibration nnznx -+ pn”-‘x:“x + R”_‘Z”_X (1) illustrates the power of EMSS. In this case, the spectral sequence E2 = CotorH*(R’~‘r”x;F,)(Fp, F,) =sH,(QTX; F,) collapses at E2, reducing the calculation to an easy homological algebra. On the other hand, the Serre spectral sequence for the fibration (1) contains infinitely many nontrivial differentials and does not compute H,(Q”C”X; F,) directly. It would be very useful to have an EMSS for any generalized homology theory, for example, Morava K-theory.

Totally normal cellular stratified spaces and applications to the configuration space of graphs

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced by the third author by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrists theorem on the homotopy dimension of the configuration space of graphs. We also make sample calculations of the fundamental group of ordered and unordered configuration spaces of two points for small graphs.

On the E 1 -term of the gravity spectral sequence

The author constructed a spectral sequence strongly converging to h_*(Omega^n Sigma^n X) for any homology theory in [Topology 33 (1994) 631-662]. In this note, we prove that the E^1-term of the spectral sequence is isomorphic to the cobar construction, and hence the spectral sequence is isomorphic to the classical cobar-type Eilenberg-Moore spectral sequence based on the geometric cobar construction from the E^1-term. Similar arguments can be also applied to its variants constructed in [Contemp Math 293 (2002) 299-329].

Twisting Segal's K-homology theory

The first half of this article is based on talks delivered by the author during the conference “Noncommutative Geometry and Physics 2008 – K-theory and D-brane –”. The basic idea of twisting generalized cohomology theories already appeared in the paper [AS04] by Atiyah and Segal, in which a modern treatment of twisted K-theory was introduced. Their construction is based on a homotopy theoretic point of view, i.e. as cohomology theories twisted by automorphisms of representing spectra. Nowadays algebraic topologists regard twisted (co)homology theories as (co)homology theories defined by bundles of spectra. See, for example, a paper by C.L. Douglas [Dou06]. A more systematic study was done by Waldmüller in [Wal]. The first half of this article is intended to be an exposition of basic ideas behind these abstract approaches to twisted (co)homology theories for those who are not familiar with homotopy theory. Descriptions of K-theory depend on the context. The periodic cohomological K-theory of a compact Hausdorff space X can be described in terms of

The Salvetti complex and the little cubes

We study how the combinatorial structure of the Salvetti complexes of the braid arrangements are related to homotopy theoretic properties of iterated loop spaces. We prove the skeletal filtrations on the Salvetti complexes of the braid arrangements give rise to the cobar-type Eilenberg-Moore spectral sequence converging to the homology of

Homology operations and vn-Bockstein operations

\Omega^2\Sigma^2 X

Cellular Stratified Spaces I: Face Categories and Classifying Spaces

. We also construct a new spectral sequence that computes the homology of

Cellular Stratified Spaces II: Basic Constructions

\Omega^{\ell}\Sigma^{\ell} X

Discrete Morse theory and classifying spaces

for