If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace $K(\mathcal{A}) \to TC(\mathcal{A})$ is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision.

The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the $\mathbf{T}$-Tate spectrum of topological Hochschild homology, where $\mathbf{T}$ is the circle group.

Suppose $(C,x)$ is a pointed locally connected small Grothendieck site, and let $(X,z)$ denote any connected locally fibrant simplicial sheaf $X$ equipped with a "geometric" point $z$. Following Artin-Mazur, an étale homotopy type of $X$ may then be defined via the geometrically pointed hypercovers of $X$ to yield a pro-object of the homotopy category, but this is not the only possible definition. In Étale homotopy of simplicial schemes, Friedlander defined another étale homotopy type of a simplicial scheme $X$ by taking diagonals of geometrically pointed bisimplicial hypercovers. In this paper, these two types are shown to be pro-isomorphic by means of a direct comparison of the associated cocycle categories. Friedlander's construction of étale homotopy types as actual pro-simplicial sets relies on a rigidity property of the étale topology that may not always be available for arbitrary sites; the cocycle methods employed here do not have this limitation. By consequence, the associated homotopy types constructed from hypercovers and bisimplicial hypercovers are shown to be pro-isomorphic on any locally connected small Grothendieck site, and the comparison at the level of cocycles shows, in particular, that both abelian and non-abelian sheaf cohomology may be computed via bisimplicial hypercovers on arbitrary small Grothendieck sites.

Given a continuous function $f\colon \mathbb{X} \to \mathbb{R}$ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case $\mathbb{X} = \mathbb{R}^3$ has ramifications in the fields of medical imaging and scientific visualization.

A semisimplicial set has face maps but not degeneracies. A basic fact, due to Rourke and Sanderson, is that a semisimplicial set satisfying the Kan condition can be given a simplicial structure. The present paper gives a combinatorial proof of this fact and a generalization to multisemisimplicial sets.

Let $T_{\operatorname{poly}}(\mathbb{R}^d)$ denote the space of skew-symmetric polyvector fields on $\mathbb{R}^d$, turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any "hand" or any "foot". As an application, we explicitly compute the cohomology of the "ascending graphs" quotient complex.

In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the non-triviality of the desingularization morphism. Using direct computations, we prove that some of these products are indeed commutative.

We determine the mod $p$ cohomology of the loop space and the double loop space of the symmetric space of exceptional type $EI$ exploiting the Serre spectral sequence and the Eilenberg-Moore spectral sequence.

Let $X$ be a connected CW complex and let $K(G,n)$ be an Eilenberg-Mac Lane CW complex where $G$ is abelian. As $K(G,n)$ may be taken to be an abelian monoid, the weak homotopy type of the space of continuous functions $X \to K(G,n)$ depends only upon the homology groups of $X$. The purpose of this note is to prove that this is true for the actual homotopy type. Precisely, the space $\mathrm{map}_* \big(X, K(G,n)\big)$ of pointed continuous maps $X \to K(G,n)$ is shown to be homotopy equivalent to the Cartesian product \[ \prod_{i \leq n} \mathrm{map}_* \big(M_i, K(G,n)\big). \] Here, $M_i$ is a Moore complex of type $M\big(H_i(X), i\big)$. The spaces of functions are equipped with the compact open topology.

We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. The motivation is to provide a natural framework in a subsequent paper for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous homotopy fixed point spectral sequences for Lubin-Tate spectra under the action of the extended Morava stabilizer group.

We provide a new and conceptually simplified construction of continuous homotopy fixed point spectra for Lubin-Tate spectra under the action of the extended Morava stabilizer group. Moreover, our new construction of a homotopy fixed point spectral sequence converging to the homotopy groups of the homotopy fixed points of Lubin-Tate spectra is isomorphic to an Adams spectral sequence converging to the homotopy groups of the spectra constructed by Devinatz and Hopkins. The new idea is built on the theory of profinite spectra with a continuous action by a profinite group.

A space $X$ is called $W$-trivial if for every vector bundle $\xi$ over $X$, the total Stiefel-Whitney class $W(\xi) = 1$. In this article we shall investigate whether the suspensions $\Sigma^k D(m,n)$ of Dold manifolds are $W$-trivial or not.

We give a description of real equivariant bordism for the group $G = Z/2 \times \cdots \times Z/2$, which is similar to the description of complex equivariant bordism for the group $S^1 \times \cdots \times S^1$ given by Hanke.

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic, where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results.

We show that the connectivity at infinity for configuration spaces on complete graphs is determined by the connectivity of chessboard complexes.

We formulate the axioms of an orbifold theory with power operations. We define orbifold Tate $K$-theory, by adjusting Devoto’s definition of the equivariant theory, and proceed to construct its power operations. We calculate the resulting sym- metric powers, exterior powers and Hecke operators and put our work into context with orbifold loop spaces, level structures on the Tate curve and generalized Moonshine.

There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology groups, and in some cases expectations of the Betti numbers; however, little seems known so far about limiting distributions of random Betti numbers.

In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complexes: Erd˝os–R´enyi random clique complexes, random Vietoris-Rips complexes, and random ˇCech complexes. These results may be of practical interest in topological data analysis.