On Extension Dimension and [L]-Homotopy

by Karassev, Alexandre V.

There is a new approach in dimension theory, proposed by A. N. Dranish­nikov and based on the concept of extension types of complexes. Following Dranishnikov, for a CW-complex L we introduce the definition of exten­sion type [L] of this complex. Further, for a space X we define the notion of extension dimension e - dim of X, which generalizes both Lebesgue and cohomological dimensions.

An adequate homotopy and shape theories, which are specifically designed to work for at most [L]-dimensional spaces, have also been developed. Fol­lowing A. Chigogidze, we present the concept of [L]-homotopy. This concept generalizes the concept of standard homotopy as well as of n-homotopy, intro­duced by R. H. Fox and studied by J.H.C. Whitehead. We also investigate the class of spaces which play a significant role in [L]-homotopy theory, namely, absolute (neighborhood) extensors modulo a complex (shortly A(N)E([L])-­spaces). Observe that A(N)E([Sn])-spaces are precisely A(N)E(n)-spaces. The first result of the present thesis describes A(N)E([L])-spaces in terms of local properties and provides an extension-dimensional version of Dugundji theorem.

Another result of the present work is related to the theory of continuous selections. The finite-dimensional selection theorem of E. Michael is very useful in geometric topology and is one of the central theorems in the theory of continuous selections of multivalued mappings. In the thesis we present the proof of an extension-dimensional version of the finite dimensional selection theorem. This version contains Michael's original finite dimensional theorem as a special case.

The concept of [L]-homotopy naturally leads us to the definition of alge­braic [L]-homotopy invariants, and, in particular, [L]-homotopy groups. We give a detailed description of [L]-homotopy groups introduced by Chigogidze.

The notion of closed model category, introduced by D. Quillen, gives an axiomatic approach to homotopy theory. It should be noted that while there exist several important examples of closed model category structures on the category of topological spaces TOP, the associated homotopies in all cases are very closely related to the ordinary homotopy. Based on the above men­tioned [L]-homotopy groups we, in this thesis, provide the first examples of model category structures on TOP whose homotopies are substantially dif­ferent from the ordinary one. Namely, we show that [L]-homotopy is indeed a homotopy in the sense of Quillen for each finite CW-complex L.

Observe that [L]-homotopy groups may differ from the usual homotopy groups even for polyhedra. The problem which arises in a natural way is to describe [L]-homotopy groups in terms of "usual" algebraic invariants of X and L (in particular, in terms of homotopy and homology groups). In the present work we compute the n-th [L]-homotopy group of Sn for a complex L whose extension type lies between extension types of Sn and Sn+l.