Let $\Sigma$ denote a finite product of unit simplices and $E:\mathbb{R}^k \rightrightarrows \Sigma$ an upper-hemicontinuous and compact valued correspondence with graph $\Gamma$. By $\pi: \Gamma \to \mathbb{R}^k$ denote the projection from the graph of $E$ to the underlying Euclidean space.
For a locally compact space $L$, by $\bar{L}$ denote its one-point compactification, and by $\bar{\pi}$ denote the extension of $\pi$ via continuity from $\bar{\Gamma}$ to $\bar{\mathbb{R}}^k$ (where $\bar{\pi}(\infty)=\infty$).
Now, let $E$ have the property that $\bar{\pi}$ is homotopic to a homeomorphism, in the sense that there exists a homeomorphism $\phi$ from $\mathbb{R}^k$ to $\Gamma$ such that $\pi \circ \phi$ is homotopic to the identity on $\mathbb{R}^k$, under a homotopy that extends to $\bar{\mathbb{R}}^k$.
Question: How do I conclude that for any $x\in \mathbb{R}^k$ that $\pi^{-1}(x)$ is generically finite and odd?
Geometrically, this makes sense: the graph $\bar{\Gamma}$ is a deformed rubber sphere floating above the sphere of $\bar{\mathbb{R}}^k$. I just don't quite know how to go the last step and formally conclude that the preimages under $\pi$ are generically finite and odd.
If $\bar{\Gamma}$ was smooth, I would imagine this would be a matter of appealing to Sard's theorem, but this is not the case, and I'm not sure how/if one can pass to a smooth approximation here. This also feels like one ought to be able to resolve it using winding numbers and the fact that $\pi \circ \phi$ is homotopic to the identity and hence of degree 1, but again, I'm not comfortable enough to formally finish this on my own, though I suspect this is rather trivial for someone familiar.
The intuition for this problem is a structure theorem from game theory. For a fixed, finite set of players, each with fixed finite sets of actions, the space of all possible normal form games is just a sufficiently high-dimensional Euclidean space. The correspondence $E$ is the set of Nash equilibrium points for each game. This relates to Theorem 1 in this paper (p. 1021), albeit with the notation somewhat changed.