The *Pontryagin square* (at the prime 2) is a certain cohomology operation
$$
\mathfrak P_2: H^q(X;\Bbb Z_2) \to H^{2q}(X;\Bbb Z_4)
$$
which has the property that its reduction mod 2 coincides with $x\mapsto x^2$. Furthermore,
If $x\in H^q(X;\Bbb Z_2)$ is the reduction of an integral class $y$, then $\mathfrak P_2(x)$
is the mod 4 reduction of $y^2$.

For a definition of $\mathfrak P_2$, see e.g.:

Thomas, E.: A generalization of the Pontrjagin square cohomology operation. *Proc. Nat. Acad. Sci. U.S.A.,* **42** (1956), 266–269.

Suppose now that $X = M^{2q}$ is a closed smooth manifold of dimension $2q$. Then $x\in H^q(X;\Bbb Z_2)$ is Poincaré dual to a class $x' \in H_q(X;\Bbb Z_2)$. By Thom representability, $x'$ is represented by a map $f: Q^q\to M$ in which $Q$ is a (possibly unorientable) closed $q$-manifold (by taking the image of the fundamental class). By transversality, we can even assume that $f$ is an immersion.

**Question:** Is there an interpretation of $\mathfrak P_2(x)$ as a geometric operation on
$f$?

(*Note:* By the properties of the Pontryagin square, if $Q$ and $M$ are orientable, then
then $\mathfrak P_2(x)$ is represented by the self-intersection of $f$ reduced mod 4.)