H-space

In the topology of a is H-space of a topological space X (often referred to as coherent provided) and a continuous mapping of a unit in the sense that the Endomorphisms ${\displaystyle \mu \colon X\times X\to X}$ ${\displaystyle e\in X}$

${\displaystyle \mu (\cdot ,e)\colon X\to X}$ and ${\displaystyle \mu (e,\cdot )\colon X\to X}$

homotop for the identical mapping to are relative to . ${\displaystyle id_{X}}$ ${\displaystyle X}$ ${\displaystyle e}$

There are also definitions in which stronger or weaker demands are made on this homotopy: Sometimes the homotopy is only demanded relatively , sometimes even relatively . These three variants are equivalent when is CW complex . ${\displaystyle e}$ ${\displaystyle X}$ ${\displaystyle X}$

The name H-Raum 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. The cohomology ring of a path-connected H-space with finitely generated free cohomology groups is a Hopf algebra . In addition, one can explain the Pontryagin product on the homology groups of an H-space .

The fundamental group of an H-space is Abelian: Let an H-space with unity , and be and loops with a base point . Then we can explain a picture through . Now to homotopic and to . This corresponds to a homotopy of the concatenation of loops . ${\displaystyle X}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle e}$ ${\displaystyle F\colon \left[0,1\right]\times \left[0,1\right]\to X}$ ${\displaystyle F(a,b)=f(a)g(b)}$ ${\displaystyle F(.,0)=F(.,1)=fe}$ ${\displaystyle f}$ ${\displaystyle F(0,.)=F(1,.)=eg}$ ${\displaystyle g}$ ${\displaystyle F}$ ${\displaystyle f\cdot g}$ ${\displaystyle g\cdot f}$

Examples

JF Adams has shown that among the spheres there are only and H-spaces; the multiplication is respectively of the multiplication in , , ( quaternions ) and ( octonions ) induced. ${\displaystyle S^{0},S^{1},S^{3}}$ ${\displaystyle S^{7}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {O} }$

Let be a unitary ring , the group of invertible matrices above and the classifying space of . Then the plus construction provides an H-space . Its fundamental group is the Abelization of . ${\displaystyle R}$ ${\displaystyle GL(R)=\bigcup _{n\geq 0}GL(n,R)}$ ${\displaystyle R}$ ${\displaystyle BGL(R)}$ ${\displaystyle GL(R)}$ ${\displaystyle BGL^{+}(R)}$ ${\displaystyle GL(R)}$

literature

• Edwin H. Spanier: Algebraic Topology. 1. corrected Springer edition, reprint. Springer, Berlin et al. 1995, ISBN 3-540-90646-0 .