Spaces with multiplication of points

Geometric structure Algebraic structure
H-space Topological space Continuous multiplication, identity
Topological group Hausdorff space Continuous group operations
Lie group Differentiable manifold Differentiable group operations

An H-space (AKA Hopf space) is not a group; it may lack inverses or even associativity. An H-space is sometimes defined with \({a\mapsto\mathbf{1}a}\) and \({a\mapsto a\mathbf{1}}\) only homotopic to the identity, sometimes through basepoint preserving maps. These alternate definitions are equivalent for H-spaces that are cell complexes. The unit vectors in a normed real division algebra have a continuous multiplication and identity, and form the spaces \({S^{1}}\), \({S^{3}}\), and \({S^{7}}\); thus these are H-spaces, and in fact are the only spheres that can be made into H-spaces. \({S^{1}}\) and \({S^{3}}\) are also Lie groups, but \({S^{7}}\) is not even a topological group since it is non-associative.

Since we view manifolds and groups as our most basic geometric and algebraic objects, the structures short of a Lie group have limited interest for us: any group that “looks like a manifold” is automatically a Lie group. More precisely, any group can be made into a topological group, and any topological group that is locally Euclidean can be identified with a single Lie group. The second is because for any topological manifold with continuous group operation, there exists exactly one differentiable structure that turns it into a Lie group.

In any topological group \({G}\) the identity component \({G^{e}}\), the connected component containing the identity, is a normal subgroup. Thus we can view the connected components as equal-sized copies of \({G^{e}}\); in a Lie group, these copies are in fact diffeomorphic. A connected Lie group is sometimes called an analytic group. Cartan’s theorem states that any closed subgroup of a Lie group is also a Lie group.

An Illustrated Handbook