PlanetMath: H-space

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H-space

(Definition)

A topological space is said to be an H-space (or Hopf-space) if there exists a continuous binary operation $\varphi\colon X\times X\to X$ and a point $p\in X$ such that the functions $X\to X$ defined by $x\mapsto\varphi(p,x)$ and $x\mapsto\varphi(x,p)$ are both homotopic to the identity map via homotopies that leave fixed. The element is sometimes referred to as an `identity', although it need not be an identity element in the usual sense. Note that the definition implies that $\varphi(p,p)=p$ .

Topological groups are examples of H-spaces.

If is an H-space with `identity' , then the fundamental group $\pi_1(X,p)$ is abelian. (However, it is possible for the fundamental group to be non-abelian for other choices of basepoint, if is not path-connected.)

"H-space" is owned by yark.

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Other names:

Hopf-space, H space, Hopf space

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Cross-references: path-connected, non-Abelian, abelian, fundamental group, topological groups, identity element, identity map, homotopic, functions, point, binary operation, continuous, topological space
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This is version 1 of H-space, born on 2006-10-07.
Object id is 8427, canonical name is HSpace.
Accessed 2063 times total.

Classification:

AMS MSC:

55P45 (Algebraic topology :: Homotopy theory :: $$-spaces and duals)

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