The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory.Using that spectrum,we could surely produce a generalized homology theory.I call this Topological $K$-homology.

While,i noticed that there are "many" $K$-homologies in the literature:

Analytic $K$-homology

Geometric $K$-homology

$KK$-thoey

Is there a good survey paper on these "various" $K$-homology theories and their relationship?

there is a similar question in the equivaraint case.Atiyah and Segal gave the definition of equivaraint $K$-cohomology theory.

By constructing a $G$-spectrum,and then taking smash product with $G$ space,then take the stable $G$-homotopy groups,we could define generalized equivariant homology theory,in particular,equivariant $K$ homology(see Carlsson's paper"a survey of equivaraint stable homotopy theory" ).

Another equivaraint homology theory appeared in the literature is the Bredon type equivaraint homology,where we are given a functor from orbit category Or$G$ to abelian groups,using that as "coefficient system",we could produce an "equivaraint homology theory".

I guess there are some other equivaraint $K$-homologies.say, index theorists usually use the notation $KO^G_i(M)$ for closed $G$-manifold $M$, which is the home of some signature operator classes.

what's the relationship between these equivaraint $K$-homology theories? What's the correct reference for $KO^G_i(M)$ mentioned above?

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology theory.Using that spectrum,we could surely produce a generalized homology theory.I call this Topological $K$-homology.

While,i noticed that there are "many" $K$-homologies in the literature:

Analytic $K$-homology

Geometric $K$-homology

$KK$-theory

Is there a good survey paper on these "various" $K$-homology theories and their relationship?

there is a similar question in the equivariant case.Atiyah and Segal gave the definition of equivariant $K$-cohomology theory.

By constructing a $G$-spectrum,and then taking smash product with $G$ space,then take the stable $G$-homotopy groups,we could define generalized equivariant homology theory,in particular,equivariant $K$ homology(see Carlsson's paper"a survey of equivariant stable homotopy theory" ).

Another equivariant homology theory appeared in the literature is the Bredon type equivariant homology,where we are given a functor from orbit category Or$G$ to abelian groups,using that as "coefficient system",we could produce an "equivariant homology theory".

I guess there are some other equivariant $K$-homologies.say, index theorists usually use the notation $KO^G_i(M)$ for closed $G$-manifold $M$, which is the home of some signature operator classes.

what's the relationship between these equivariant $K$-homology theories? What's the correct reference for $KO^G_i(M)$ mentioned above?