Let $M,M'$ be oriented connected compact smooth manifolds of the same dimension, let $S$ be a smooth manifold,
and let $\nu : S\times M\rightarrow M'$ be some smooth map.
Let $\nu_s : M\rightarrow M'$ denote $\nu_s(\cdot) = \nu(s,\cdot)$ for $s\in S$.
If I'm not mistaken, we can formalize the claim that the per-$s$ degree map $s \mapsto \mathrm{degree}(M\xrightarrow{\nu_s} M')$,
as a map $S \rightarrow \mathbb{Z}$, should depend continuously on $s \in S$ (hence be locally constant) by using e.g. de Rham theory.
(Details: Fix some top-degree differential form $\omega$ on $M'$ which is non-exact; then $\mathrm{deg}(\nu_s) = (\int_M \nu_s^*\omega)/(\int_{M'}\omega)$, which we can probably argue depends continuously on $s$.)
More generally, if $M,M'$ are Poincaré duality spaces (with chosen fundamental classes), and $S$ is a topological space and "smooth" everywhere above is replaced by "continuous", does the same claim hold? How can the proof be formalized?
Best Answer
What you are considering is the adjoint map $f:S \rightarrow \operatorname{Map}(M, M')$ and asking if for any $x,y \in S$, $\operatorname{deg}(f(x))=\operatorname{deg}(f(y))$. Since the degrees of homotopic maps are equal, it suffices to find a path from $f(x)$ to $f(y)$, but by assumption $M$ is connected, so it is path connected. So we may take the image of any path $x$ to $y$ under $f$.
The same argument works for Poincare duality spaces.