The main goal of the present project is to give a sketch of the calculation of the algebraic K-theory of a finite field. Not all the details are developed, but several references are given where detailed proofs may be found. However, the main source of the project is D.J. Benson’s book on ”Representations and Cohomolgy: Cohomolgy of groups and modules” [4]. Section 1 is an introduction to princ...

Let X be a path-connected p-local CW -complex and let X be the n-fold self smash product of X. Decompositions of ΣX have been largely studied (See for instance [11, 19, 21, 22, 27].) with various applications in homotopy theory such as the exponents problem of homotopy groups [3, 4, 5], Morava K-theory [17, 22] and Steenrod modules [27]. For a general space X, a natural way to study decompositi...

This document consists of lecture notes for four lectures given by Brooke Shipley at the 2017 Young Topologists Meeting at the Swedish Royal Institute of Technology (KTH) in Stockholm, Sweden. The original conference abstract for the lecture series is below. This sequence of lectures will explore the connections between the differential graded world and the spectral world. There will first be a...

There is a well-known equivalence between the homotopy types of connected CW-spaces X with πnX=0 for n 6= 1, 2 and the quasi-isomorphism classes of crossed modules ∂ : M → P [16]. When the homotopy groups π1X and π2X are finite one can represent the homotopy type of X by a crossed module in which M and P are finite groups. We define the order of such a crossed module to be |∂| = |M | × |P |, an...

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding “synthetic homotopy theory”. Here we consider the Seifert–van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in...

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could...