Stefan Waner

Equivariant homotopy theory and Milnor’s theorem

The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex. This is the first of a series of three papers based on the authors thesis [Wal], the object of which is to discuss equivariant homotopy theory in general, and equivariant fibrations in particular, culminating in classification theorems for the various categories of equivariant fibrations and bundles. In the present paper, we discuss the foundations of equivariant homotopy theory and cellular theory and prove an equivariant version of Milnors theorem on spaces having the homotopy type of CW complexes, where we allow a compact Lie group G to act everywhere. The second paper in this series, Equivariant fibratons and transfer sets up the background for the study of (J-fiber spaces and equivariant stable homotopy theory and contains a description of the equivariant transfer for equivariant fibrations with compact fiber. In the third paper, Equivariant classifying spaces and fibrations, the geometric bar construction is used to construct explicit classifying spaces for equivariant bundles and fibrations, these results depending heavily on the equivariant cellular theory presented here. Also in preparation is a fourth paper which will sequel the present series and will deal with the classificaton of oriented G-spherical fibrations and bundles [Wa2]. The three papers are divided as follows: Equivariant homotopy theory and Milnors Theorem 1. Notations and definitions 2. Equivariant homotopy groups 3. Equivariant cellular theory 4. Milnors Theorem 5. Approximation of G-CW complexes by G-simplicial complexes 6. Finite dimensional G-simplicial complexes are G-equilocally convex 7. Reasonable GELC spaces are dominated by G-CW complexes Received by the editors September 7, 1978 and, in revised form, January 15, 1979. AMS (MOS) subject classifications (1970). Primary 54H15.

Equivariant RO(G)-graded bordism theories

Abstract Let G be a finite group. The RO( G )-graded bordism theories of Pulikowski [7] and Kosniowski [3] are studied. Representing equivariant Thom spectra are constructed, and the relevant transversality results proved. New methods for splitting away from the order of G are described, and behavior in the presence of a gap hypothesis is examined.

Principles of evolutionary learning design for a stochastic neural network.

Abstract The authors develop principles for evolutionary learning typical of biological systems and demonstrate how these principles can be realized with a formal stochastic network.

Equivariant ordinary homology and cohomology

Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredons ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of RO(G)-graded ordinary homology and cohomology, collecting scattered results and filling in gaps in the literature. In particular, we give details on change of groups and products that do not seem to have appeared elsewhere. We also discuss the relationship between ordinary homology and cohomology when the group is compact Lie, in which case the two theories are not represented by the same spectrum. The remainder of the work discusses the extension to grading on representations of fundamental groupoids, concentrating on those aspects that are not simple generalizations of the RO(G)-graded case. These theories can be viewed as defined on parametrized spaces, and then the representing objects are parametrized spectra; we use heavily foundational work of May and Sigurdsson on parametrized spectra. We end with a discussion of Poincare duality for arbitrary smooth equivariant manifolds.

The local structure of tangent G-vector fields

Abstract We introduce a notion of equivariant index in order to describe the behavior of tangent G -vector fields on smooth G -manifolds near isolated zeros. Our methods result in a calculation of the monoid of G -homotopy classes of self-maps of the unit sphere S ( V ) in a real orthogonal (finite dimensional) G -module V , this being the unstable analogue of a classical result of Segal. During the course of our calculation, we prove general position results on tangent G -vector fields and obtain canonical local structures for such fields.

Classifying pro-fibrations and shape fibrations

We use techniques of J.P. May to construct classifying spaces for fibrations in the category of inverse sequences of spaces (towers) and level-preserving maps. These spaces are used to classify fibrations in Top N , fibrations in pro-Top, and shape fibrations; the latter modulo certain compactness questions.

The equivariant Euler characteristic

Abstract We describe an equivariant version of the Euler characteristic in order to extend to the equivariant case classical results relating the Euler characteristic to vector field (Reinhart) bordism of smooth manifolds and controllable cut-and-paste equivalence. We show that the nonequivariant results continue to hold for an arbitrary finite ambient group G , both in the oriented and unoriented cases, and thereby extend work on this subject begun by several authors. We use a new definition of equivariant orientation in terms of a categorical notion of ‘groupoid representations’.

Equivariant SKK and vector field bordism

Abstract We use a notion of equivariant Euler characteristic in order to extend classical results on controllable cutting and pasting, and vector field bordism, to the case of manifolds acted on by an arbitrary finite group G , and modelled on a fixed virtual representation (in the sense of W. Pulikowski and C. Kosniowski). By restricting attention to such G -manifolds, one finds that classical results continue to hold in the oriented and unoriented case. This extends work of several authors.

Biologically motivated machine intelligence

Biological systems routinely solve problems involving pattern recognition and feature extraction. Such problems do not appear to admit similarly routine algorithmic solutions; the power of biological systems in this regard apparently arises from nonalgorithmic dynamics. It is our intention to explore, and to develop principles of, functional characteristics of natural (biological) intelligence with a view to their realization through an appropriate automation model.

History dependent stochastic automata: A formal view of evolutionary learning

We develop a theoretical framework for the study of evolutionary learning systems. the formalism we use is that of history dependent stochastic automata with suitable structure, as well as related structures. This formalism provides a natural setting in which to describe the learning of classification hierarchies, of control hierarchies and notions of selfreference, all of which are derived as consequences of the ability to learn by association.