Marcelo A. Aguilar

Algebraic Topology from a Homotopical Viewpoint

Introduction.- Basic Concepts and Notation.- Function Spaces.- Connectedness and Algebraic Invariants.- Homotopy Groups.- Homotopy Extension and Lifting Properties.- CW-Complexes Homology.- Homotopy Properties of CW-Complexes.- Cohomology Groups and Related Topics.- Vector Bundles.- K-Theory.- Adams Operations and Applications.- Relations Between Cohomology and Vector Bundles.- Cohomology Theories and Brown Representability.- Appendix A: Proof of the Dold-Thom Theorem.- Appendix B: Proof of the Bott Periodicity Theorem.- References.- Index.- Glossary.

Topological Abelian Groups and Equivariant Homology

We prove an equivariant version of the Dold–Thom theorem by giving an explicit isomorphism between Bredon–Illman homology and equivariant homotopical homology π*(F G (X, L)), where G is a finite group and L is a G-module. We use the homotopical definition to obtain several properties of this theory and we do some calculations.

Transfers for ramified covering maps in homology and cohomology

Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a new cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. Our transfers can also be defined in Karoubis approach to homology and cohomology. Furthermore, we show that one can define mixed transfers from other homology or cohomology theories to the ordinary ones.

Quasifibrations and Bott periodicity

We give a proof of the Bott periodicity theorem, along the lines proposed by McDuff, based on the construction of a quasifibration over U with contractible total space and Z BU as fiber.

The transfer for ramified covering G-maps

Abstract Let G be a finite group. The main objective of this paper is to study ramified covering G-maps and to construct a transfer for them in Bredon–Illman equivariant homology with coefficients in a homological Mackey functor M. We show that this transfer has the usual properties of a transfer.

CW-Complexes and Homology

We start this chapter by defining and studying a very important class of spaces, known as the CW-complexes; in the next chapters these will be the spaces with which we shall mainly work.

Homotopy Extension and Lifting Properties

We already saw in the previous chapter that the inclusion X ↪ CX of a space X into its (reduced) cone has a homotopy extension property (see 3.1.6); we also saw that the projection PY ↠ Y of the (pointed) path space of a space onto the space Y has, dually, a homotopy lifting property (see 3.3.17). In this chapter we shall study systematically these two properties. More precisely, we analyze families of maps that have one of the two essentially dual properties, generally known as the homotopy extension and homotopy lifting properties. These topics are of great importance in algebraic topology and will be used in subsequent chapters.

Cohomology Groups and Related Topics

In this chapter we shall use the Eilenberg-Mac Lane spaces introduced in the previous chapter in order to define cohomology groups. Then, using the homotopy properties proved for Moore spaces, we shall introduce a multiplicative structure on cohomology groups.

Connectedness and Algebraic Invariants

In this chapter we shall introduce the concepts of path connectedness and of homotopy of continuous maps between two spaces. We shall study the sets of homotopy classes of maps and relate this with path connectedness. Finally, we shall define the homotopy groups of a topological space, which are important algebraic invariants for such spaces.

Adams Operations and Applications

In this chapter we shall define the important Adams operations in complex K-theory and see how they are applied to prove a central theorem of mathematics, namely, to determine the dimensions n for which ℝ n admits the structure of a division algebra.