2 - Introduction to Lawson homology

Summary

Abstract

Lawson homology has quite recently been proposed as an invariant for algebraic varieties. Various equivalent definitions have been suggested, each with its own merit. Here we discuss these for projective varieties and we also derive some basic properties for Lawson homology. For the general case we refer to Paulo Lima-Filho's lectures (Chapter 3).

Keywords: Lawson homology, cycle spaces 2000 Mathematics subject classification: 14C25, 19E15, 55Qxx

Introduction

This paper is meant to serve as a concise introduction to Lawson homology of projective varieties. For another introduction the reader should consult.

It is organized as follows. In the first section we recall some basic topological tools needed for a first definition of Lawson homology. Then some basic examples are discussed. In the second section we discuss the topology of the so-called ‘cycle spaces’ in more detail in order to understand functoriality of Lawson homology. In the third and final section we relate various equivalent definitions. Here the language of simplicial spaces is needed and we only summarize some crucial results from the vast literature on this highly technical subject.

Basic notions

Homotopy groups

We start by recalling the definition and the basic properties of the homotopy groups. For any two pairs of topological spaces (X, A) and (Y, B) we use the notation [(X, A), (Y, B)] for the set of homotopy classes of maps X → Y sending A to B (any homotopy is supposed to send A to B as well).