Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general setup of $\mathbb{Z}/2$-Poincare duality spaces.

(From now on all cohomology coefficients are $\mathbb{Z}/2$.) Poincare duality induces, from the usual pullback, a pushforward map

$$ f_* \colon H^*(X) \rightarrow H^*(Y) .$$

While pullback commutes with Steenrod square, the pushforward unfortunately does not.

**Question**: can one "quantify" the failure of this commutativity? (i.e. some nice expression for the difference $f_* Sq - Sq f_*$?)

I have an answer to my question in terms of Wu classes, but I want something that doesn't involve them. Here's an example of the kind of thing that I'm looking for.

Suppose now that $X \hookrightarrow Y$ is a closed embedding of *smooth* manifolds. A reliable friend told me the following nice formula measuring the failure of this commutativity. If $N_{X/Y}$ is the normal bundle of $X$ in $Y$, then by the Gysin map, tubular neighborhood, excision etc. we get a composition

$$H^*(X) \xrightarrow{\text{Gysin}} H^{*+r}(N_{X/Y}, N_{X/Y}-X) \cong H^*(Y, Y-X) \xrightarrow{g^*} H^*(Y)$$

and the claim is that this composition is $f_*$. In particular, this can be used to write down a formula for the "error term" in commuting Sq with pushforward: if $u$ is the Thom class of $N_{Y/X}$ then

$$ f_*(Sq \alpha) = g^* (Sq \alpha \cup u) = Sq (g^* \alpha \cup Sq^{-1} u) $$

Now by definition $Sq^{-1} u = w^{-1} u$ where $w$ is the (total) Stiefel-Whitney class of $N_{Y/X}$, so

$$ f_* (Sq \alpha) = Sq ( g^* \alpha \cup w^{-1} u ) = Sq (f_*(\alpha \cup w^{-1})) $$

Unfortunately this doesn't make literally make sense if $X,Y$ are not smooth (since there's no $N_{X/Y}$). However, some things make sense; for instance, we can say that $g^* u = f_*(1)$. In my situation $f_*(1)$ is very simple; for instance I know that $Sq (f_*(1)) = f_*(1)$. (This is basically why I suspect that ``$w"=1$ in my case, so that $f_*$ and Sq should commute.)

**Question:** is there a "salvage" of the above formula to the case where $X,Y$ are not smooth?

I would prefer an answer that is formulated at the level of Poincare duality spaces, since that's more like my situation (and the question makes sense at that level).