2 Background Material 2 2.1 Homotopy Theory of Simplicial Vector Spaces . . . . . . . . . 2 2.1.1 The Dold-Kan correspondence . . . . . . . . . . . . . . 2 2.1.2 Homotopy theory . . . . . . . . . . . . . . . . . . . . . 4 2.1.3 Tensor Products and the Eilenberg-Zilber Theorem . . 5 2.2 Representation Theory of Finite Groups . . . . . . . . . . . . 6 2.2.1 Coinvariants and Invariants of kG-modul...

Deleanu, Frei, and Hilton have developed the notion of generalized Adams completion in a categorical context. In this paper, we have obtained the Postnikov-like approximation of a module, with the help of a suitable set of morphisms.

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the alge...

Classifying endotrivial kG-modules, i.e., elements of the Picard group stable module category for an arbitrary finite G, has been a long-running quest. By deep work Dade, Alperin, Carlson, Thevenaz, and others, it reduced to understanding subgroup consisting modular representations that split as trivial k direct sum projective when restricted Sylow p-subgroup. In this paper we identify first co...

In (2003), we proved the injective homotopy exact sequence of modules by a method that does not refer to any elements of the sets in the argument, so that the duality applies automatically in the projective homotopy theory (of modules) without further derivation. We inherit this fashion in this paper during our process of expanding the homotopy exact sequence. We name the resulting doubly infin...

We set up foundations of representation theory over S, the stable sphere, which is the “initial ring” of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, HarishChandra pairs and Zuckermann functors. As an application, we construct a Khovanov slk-stable homotopy type with a large prime hypothesis, which is...

The relative homotopy theory of modules, including the (module) homotopy exact sequence, was developed by Peter Hilton (1965). Our thrust is to produce an alternative proof of the existence of the injective homotopy exact sequence with no reference to elements of sets, so that one can define the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, ...

Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersectio...