  Let $B$ be a C$^*$-algebra and $X$ a C$^*$ Hilbert $B$-module. If $p\in B$ is
a projection, denote by $S_p =\{x\in X : < x,x> =p\}$, the $p$-sphere of $X$.
For $\phi$ a state of $B$ with support $p$ in $B$ and $x\in S_p$, consider the
state $\phi_x$ of $L_B(X)$ given by $\phi_x(t)= \phi(< x,t(x)>)$. In this paper
we study certain sets associated to these states, and examine their topologic
properties. As an application of these techniques, we prove that the space of
states of the hyperfinite II$_1$ factor $R_0$, with support equivalent to a
given projection $p\in R_0$, regarded with the norm topology (of the conjugate
space of $R_0$), has trivial homotopy groups of all orders.
  The same holds for the space $$ S_p(R_0)=\{v\in R_0:v^*v=p\}\subset R_0 $$ of
partial isometries with initial space $p$, regarded with the ultraweak
topology.
