Let $G$ be a compact lie group. We introduce the set ${\mathbf{S}}_G\left(Y\right)$ for every smooth $G$ manifold $Y$. It consists of equivalence classes of pair $(X,f)$ where $f:X\to Y$ is a $G$ map which defines a homotopy equivalence from $X$ to $Y$. Two pairs $(X_i,f_i)$, for $i=0,1$, are equivalent if there is a $G$ homotopy equivalence $\phi :X_0\to X_1$ such that $f_0$ is $G$ homotopic to $f_1\circ \phi$.Properties of the set ${\mathbf{S}}_G\left(Y\right)$ and related to the representation of $G$ on the tangent spaces of $X$ and $Y$ at the fixed points. For the case $G=S^1$ and $Y$ is the $S^1$ manifold defined by a “linear” $S^1$ action on complex...