H-space

In topology, an H - space of a topological space X is (often considered to be continuous provided ) and a continuous map with a unit in the sense that the endomorphisms

Homotopic to the identity map idX on X are relative to e.

There are also definitions in which stronger or weaker demands are placed on these homotopy: Sometimes the homotopy is e, sometimes called only relatively relatively X. These three variants are equivalent if X CW complex.

The name H- space was proposed by Jean -Pierre Serre in honor of Heinz Hopf.

Properties

The multiplicative structure of an H- space enriches the structure of its homology and cohomology. Thus, the cohomology ring of a path-connected H- space with finitely generated free cohomology is a Hopf algebra. Also, can you explain the Pontryagin product on the homology groups of an H - space.

The fundamental group of a space H is abelian: Let X be an H - space with unit e, and f and g loops with base point e Then we can find a mapping F: by [0,1] × [0,1 ] → X F (a, b) = f (a) g (b) explain. Now F is (-, 0 ) = f ( - 1) = fe homotopic to f and F (0, -) = F ( 1 -) = EC corresponds to an R to G. Then, the concatenation of homotopy f · g of loops to g · f

Examples

JF Adams has shown that, among the spheres only S0, S1, S3 and S7 H- spaces; the multiplication is induced by the multiplication on each, ( quaternions ) and ( octonions ).