K-7 - Homotopy, I

Abstract

This chapter gives a bird's-eye view of the basic facts in homotopy theory. H∗ and H∗ are singular homology and cohomology theories, respectively. When integral coefficient (co)homology groups are considered, the group ℤ of integers is not specified. The concept of homotopy may be found in Lagrange's method in the calculus of variations. Even in the 19th century, many mathematicians used the idea of deformation. In 1911, L.E.J. Brouwer gave the general definition of homotopy between two maps for the first time. For the spaces X and Y, if two maps ƒ0 and ƒ1 from X to Y are homotopic, then there exists a map F : X ×▪ →Y with F(x, 0) = ƒ0(x) and F(x, 1) = ƒ1(x) for all x ∈X, where ▪ is the unit interval. Such a map F is called a homotopy joining ƒ0 and ƒ1. The existence of homotopy clearly induces an equivalence relation in the set C(X, Y) of maps from X to Y. A compact space has the homotopy type of a CW-complex (not necessarily compact) if and only if (iff) it is a homotopy dominated by a finite CW-complex.