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Register a new userLoday constructions on twisted products and on tori
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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 are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology.
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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 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 higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.
We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 times D^{2n-1}$, for $2n geq 6$, in a range of degrees up to $n-2$. Our analysis relies on understanding the homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian $K$-theory of the integral group ring of $pi_1(S^1)$. We also show that these homotopy groups vanish rationally.
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 $S_{bullet}$-constructions for exact categories and for stable and exact $(infty,1)$-categories, as well as the relative $S_{bullet}$-construction for exact functors.
We consider a twisted action of a discrete group G on a unital C*-algebra A and give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals of A and the maximal ideals in the associated reduced C*-crossed product.
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