We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord(n, X),...

In ?-tilting theory, it is often difficult to determine when a set of bricks forms 2-simple minded collection. The aim this paper contained in collection for finite algebra. We begin by extending the definition mutation from collections more general sets (which we call semibrick pairs). This gives us an algorithm check if pair then use show that gentle algebra (whose quiver contains no loops or...

A successful generalization of phantom map theory to the equivariant case for all compact Lie groups is obtained in this paper. One of the key observations is the discovery of the fact that homotopy fiber of equivariant completion splits as product of equivariant Eilenberg-Maclane spaces which seems impossible at first sight by the example of Triantafillou[19].

* This note arose from consultations during the tenure of a John Simon Guggenheim Memorial Fellowship by MacLane. ' Whitehead, J. H. C., "Combinatorial Homotopy I and II," Bull. A.M.S., 55, 214-245 and 453-496 (1949). We refer to these papers as CH I and CH II, respectively. ' By a complex we shall mean a connected CW complex, as defined in §5 of CH I. We do not restrict ourselres to finite com...

The Hopkins-Mahowald theorem realizes the Eilenberg-Maclane spectra $H\mathbb F_p$ as Thom for all primes $p\in\mathbb N_{>0}$. In this article, we record a known proof of generalization theorem, realizing $Hk$ perfect rings $k$, and provide further by $HR$ perfectoid $R$. We also discuss even generalizations to prisms $(A,I)$ indicates how adapt our proofs Breuil-Kisin case.

We prove that stable f -localizations (where f is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg–MacLane spectrum HZ. As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a...

We study Voevodsky’s slice tower for S-spectra, and raise a number of questions regarding its properties. We show that the 0th slice does not in general admit transfers, although it does for a P-loop-spectrum. We define a new tower for each of the higher slices, and show that the layers in these towers have the structure of Eilenberg-Maclane spectra on effective motives. 2010 Mathematics Subjec...