Timothy Porter

Abstract homotopy and simple homotopy theory

Abstract Homotopy Theory Case Studies Exact Sequences Elementary Homotopy Coherence Abstract Simple Homotopy Theory Additive Simple Homotopy Theories.

N-types of simplicial groups and crossed N-cubes

As IS well known, simplicial groups model all connected homotopy types. In particular certain simplicial groups, namely those with vanishing Moore complex in dimensions greater than n, provide algebraic models for n-types of simplicial groups and thus for connected (n + 1)-types of spaces. In [ 121, Loday gave the foundation of a theory of another algebraic model for (n + 1)-types that generalized that given for 2-types by MacLane and Whitehead [13]. His models, called cat”-groups, have very pleasant properties and in work with Brown [3], [4] have been shown to satisfy a form of generalized Van Kampen theorem. These cat”-groups form a category equivalent to that of the crossed n-cubes of the title of this paper (c.f. Ellis and Steiner [9]). Loday’s original result was stated using a theory of n-cubes of fibrations. This made it difficult to generalize to other algebraic contexts which clearly may be useful for further development in both homotopy theory and homotopical algebra. The aim of this paper is thus to provide full, detailed, purely algebraic proofs of Loday’s main results and in the process prove stronger results that should prove of independent interest. The key result, missed by Loday, although he knew all the ingredients to prove it, is that any crossed module can be obtained as n 0 of a normal inclusion, N + G, of simplicial groups. This generalizes easily to crossed n-cubes: each crossed n-cube is isomorphic to 7co of a simplicial inclusion crossed n-cube determined by a simplicial group G. and n-normal subgroups N(1) . , . . . , N(n). Thus any crossed square can be obtained by taking rro of a square of the form:

Iterated Peiffer Pairings in the Moore Complex of a Simplicial Group

We introduce a pairing structure within the Moore complex NG of a simplicial group G and use it to investigate generators for NGn∩Dn where Dn is the subgroup generated by degenerate elements. This is applied to the study of algebraic models for homotopy types.

Homology of commutative algebras and an invariant of simis and vasconcelos

Abstract Simis and Vasconcelos [8, and 9] have introduced for any ideal,I, of finite type in a commutative ring,R, an associated module,δ(I), which is an invariant ofI. (δ(I) measures the extent to which the Koszul complex of a set of generators ofI gives one information on the syzygies of the conormal moduleI/I2.) In this paper we show thatδ(I) is isomorphic toH2(R,R/I,R/I), the second Andre´-Quillen homology of theR-algebra,R/I (see [1]). One can obtain special cases of this result by combining alternative invariant formulations ofδ(I) (from [9]) with results of Andre´, however the general case seems to be most easily proved by using crossed module techniques adapted from ideas of Lichtenbaum and Schlessinger [4] and from the corresponding theory in the category of groups. The only relatively deep results from the homology of commutative algebras that we shall use are necessary to identify the Lichtenbaum-Schlessinger homology group,T2, with the corresponding Andre´-Quillen homology,H2. (A reference for this result is [1, p. 206, Proposition 12].)

Topological Quantum Field Theories from Homotopy n-Types

Using simplicial methods developed in an earlier note, the paper constructs topological quantum field theories using an algebraic model of a homotopy n -type as initial data, generalising constructions of Yetter for n =1 and n =2.

Varieties of simplicial groupoids I: Crossed complexes

Abstract It is usual to use algebraic models for homotopy types. Simplicial groupoids provide such a model. Other partial models include the crossed complexes of Brown and Higgins. In this paper, the simplicial groupoids that correspond to crossed complexes are shown to form a variety within the category of all simplicial groupoids and the corresponding verbal subgroupoid is identified.

CHAPTER 3 – Proper Homotopy Theory

This chapter iscusses the proper homotopy theory. The Proper homotopy theory is both an old and a new area of algebraic topology. Its origins go back to the classification of noncompact surfaces by Kerekjarto in 1923, but it is probably fair to say that it got off the ground as a distinct area of algebraic topology as a result of the geometric work of Larry Siebenmann in 1965. The methods and perspectives of the shape theory interacted with those of proper homotopy theory for the mutual benefit of both. This led, in 1976, to the publication by Edwards and Hastings of their lecture notes, which laid down a theoretical framework for studying the proper homotopy theory that is still actively used today. Furthermore, the other approach to the proper homotopy theory currently being developed is based on the algebraic homotopy theory of Baues, and Baues himself has collected material from earlier sources, together with a wealth of new material, in a draft manuscript. The main point of this approach is the use of the language and results of the theory of cofibration categories.

Joins for (augmented) simplicial sets

Abstract We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category Δ.

Spaces of maps into classifying spaces for equivariant crossed complexes

Abstract We give an equivariant version of the homotopy theory of crossed complexes. The applications generalize work on equivariant Eilenberg—Mac Lane spaces, including the non abelian case of dimension 1, and on local systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk and Svensson. Further, we give results not just on the homotopy classification of maps but also on the homotopy types of certain equivariant function spaces.

Freeness Conditions for 2-Crossed Modules of Commutative Algebras

In this paper we give a construction of free 2-crossed modules. By the use of a ‘step-by-step’ method based on the work of André, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.