H-space

H -space is a generalization of the concept of a topological group of a certain type.

Definition

Connected topological space $X$ ${\ displaystyle X}$ $X$ together with continuous display

\mu \colon X\times X\to X

{\ displaystyle \ mu \ colon X \ times X \ to X}

\mu \colon X\times X\to X

with a single element , i.e. an element $e\in X$ ${\ displaystyle e \ in X}$ $e\in X$ such that

\mu (e,x)=\mu (x,e)=x

{\ displaystyle \ mu (e, x) = \ mu (x, e) = x}

\mu (e,x)=\mu (x,e)=x

for anyone $x\in X$ ${\ displaystyle x \ in X}$ $x\in X$ called an H-space .

Sometimes limited to the weaker condition that the mappings $x$ $\mapsto$ $μ$ $($ $e$ $,$ $x$ $)$ {\ displaystyle x \ mapsto \ mu (e, x)} and $x$ $\mapsto$ $μ$ $($ $x$ $,$ $e$ $)$ {\ displaystyle x \ mapsto \ mu (x, e)} homotopic to the identity (sometimes with a fixed $e$ $\in$ $X$ {\ displaystyle e \ in X} )
- These three definitions are equivalent for CW complexes .

Each topological group is an H- space.
For an arbitrary topological space $X$ {\ displaystyle X} space $H$ $X$ {\ displaystyle {\ mathcal {H}} _ {X}} all continuous mappings $X$ $\to$ $X$ {\ displaystyle X \ to X} homotopic to the identity is an H- space.
- Wherein $\mu \colon {\mathcal {H}}_{X}\times {\mathcal {H}}_{X}\to {\mathcal {H}}_{X}$ ${\ displaystyle \ mu \ colon {\ mathcal {H}} _ {X} \ times {\ mathcal {H}} _ {X} \ to {\ mathcal {H}} _ {X}}$ $\mu \colon {\mathcal {H}}_{X}\times {\mathcal {H}}_{X}\to {\mathcal {H}}_{X}$ can be defined as composition $\mu (f,g)=f\circ g$ ${\ displaystyle \ mu (f, g) = f \ circ g}$ $\mu (f,g)=f\circ g$ .

Among the spheres , only $S$ $0$ {\ displaystyle \ mathbb {S} ^ {0}} , $S$ $one$ {\ displaystyle \ mathbb {S} ^ {1}} , $S$ $3$ {\ displaystyle \ mathbb {S} ^ {3}} and $S$ $7$ {\ displaystyle \ mathbb {S} ^ {7}} are H- spaces. Wherein
- Each of these spaces forms a subset of elements with a unit norm among real numbers , complex numbers , quaternions and octonions, respectively.
- $\mathbb {S} ^{0}$ ${\ displaystyle \ mathbb {S} ^ {0}}$ $\mathbb {S} ^{0}$ , $\mathbb {S} ^{1}$ ${\ displaystyle \ mathbb {S} ^ {1}}$ $\mathbb {S} ^{1}$ and $\mathbb {S} ^{3}$ ${\ displaystyle \ mathbb {S} ^ {3}}$ $\mathbb {S} ^{3}$ are Lie groups , and $\mathbb {S} ^{7}$ ${\ displaystyle \ mathbb {S} ^ {7}}$ $\mathbb {S} ^{7}$ - not.

Hatcher, Allen (2002), Algebraic Topology , Cambridge: Cambridge University Press, ISBN 0-521-79540-0 , < http://www.math.cornell.edu/~hatcher/AT/ATpage.html > . Section 3.C