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We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $Sigma X$. We prove several structural properties about different notions of stability, corresponding to different levels of invariance required of $Xotimes A$, and establish stability in important cases, such as complex and real periodic topological K-theory, $KU$ and $KO$.
We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions
We define a relative version of the Loday construction for a sequence of commutative S-algebras $A rightarrow B rightarrow C$ and a pointed simplicial subset $Y subset X$. We use this to construct several spectral sequences for the calculation of hig
In previous work, we develop a generalized Waldhausen $S_{bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give
Let $G=SU(2)$ and let $Omega G$ denote the space of based loops in SU(2). We explicitly compute the $R(G)$-module structure of the topological equivariant $K$-theory $K_G^*(Omega G)$ and in particular show that it is a direct product of copies of $K^