Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBounded cohomology and homotopy colimits
342
0
0.0
(
0
)
Authors
George Raptis
Ask ChatGPT about the research
No Arabic abstract
We discuss an approach to the emph{covering} and emph{vanishing} theorems for the comparison map from bounded cohomology to singular cohomology, based on the observation that the comparison map is the coassembly map for bounded cohomology.
rate research
Read More
In this short note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by a finite category, but the behavior is more uniform when we index by finite quasicategories instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of $mathbb{R}$ in $mathbb{R}^n$ with fixed behavior outside a compact set and such that the images of the copies of $R$ are disjoint -- even for $n=3$. We further study the case $n=3$ in degree zero and show that our integrals represent a universal finite type invariant of the space of classical homotopy string links. As a consequence, we obtain configuration space integral expressions for Milnor invariants of string links.
We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams.
We develop foundations for the category theory of $infty$-categories parametrized by a base $infty$-category. Our main contribution is a theory of indexed homotopy limits and colimits, which specializes to a theory of $G$-colimits for $G$ a finite group when the base is chosen to be the orbit category of $G$. We apply this theory to show that the $G$-$infty$-category of $G$-spaces is freely generated under $G$-colimits by the contractible $G$-space, thereby affirming a conjecture of Mike Hill.
There are two main approaches to the problem of realizing a $Pi$-algebra (a graded group $Lambda$ equipped with an action of the primary homotopy operations) as the homotopy groups of a space $X$. Both involve trying to realize an algebraic free simplicial resolution $G_bullet$ of $Lambda$ by a simplicial space $W_bullet$ and proceed by induction on the simplicial dimension. The first provides a sequence of Andr{e}-Quillen cohomology classes in $H_{AQ}^{n+2}(Lambda;Omega^{n}Lambda)$ for $n geq 1$ as obstructions to the existence of successive Postnikov sections for $W_bullet$ by work of Dwyer, Kan and Stover. The second gives a sequence of geometrically defined higher homotopy operations as the obstructions by earlier work of Blanc; these were identified with the obstruction theory of Dwyer, Kan and Smith in earlier work of the current authors. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of $Lambda$. In this paper we 1) provide an explicit construction of the cocycles representing the cohomology obstructions; 2) provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to long Toda brackets), and 3) show that these two constructions correspond under an evident map.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
