# Push-Pull formula for currents

Let $$f$$ be transverse to $$g$$ so that $$\require{AMScd}$$ $$\begin{CD} V@>{f'}>> Z\\ @V{g'}VV @VV{g}V\\ Y @>{f}>> X \end{CD}$$ is a Cartesian square of smooth manifolds. Then, if $$g$$ is proper and oriented, and $$g'$$ has the induced orientation, we get the formula

$$$$\label{eq} f^*\circ g_* = g'_*\circ f'^* : H^*(Z;\mathbb R)\to H^{*+dim(X)-dim(Z)}(Y;\mathbb R).$$$$

There is nothing special in using real coefficients. Now, I'm going to follow de Rham and compute my real cohomology groups using currents. The space of currents on $$M$$, which I'll denote by $$\mathcal D^*(M)$$, is the space of continuous linear functionals on the space of smooth, compactly supported forms on $$M$$. We have the pushforward $$g_*:\mathcal D^{*}(Z)\to \mathcal D^{*+dim(X)-dim(Z)}(X).$$ We cannot pull back any current along $$f$$. However, I believe since $$f$$ is transverse to $$g$$, $$f^*$$ is defined on $$g_*$$ of forms. I've so far failed to find a concrete reference for that statement, but I think it follows from [Theorem 8.2.4, 1]. Let me explain why I think that (as a response to Shifrins comment): The theorem says that $$f^*T$$ can be defined if $$N(f)\cap WF(T)=\emptyset$$. Here $$N(f)$$ is the ''set of normals" of $$f$$, defined by $$N(f)=\{(x,\xi_x)\in X\times T^*X\backslash 0: \exists y\in Y\ s.t.\ \forall v\in T_yY \text{ we have } \xi(D_yfv)=0 \}.$$ The wave front set I do not have a good grasp of. But I think it is true that $$WF(g_*\omega)\subset N(g)$$ when $$\omega$$ is a smooth form. If that is true, one must only observe that when $$f$$ and $$g$$ are transverse, $$N(f)\cap N(g)=\emptyset$$. That's easy to see: If $$(x,\xi)\in N(f)\cap N(g)$$, we can choose $$z\in g^{-1}(x)$$ and $$y\in f^{-1}(x)$$ such that $$\xi(D_yf T_yY+D_zg T_zZ)=0$$. But then $$\xi=0,$$ since the sum we are applying it to is $$T_xX$$ since $$f\pitchfork g$$.

My main question is: does the equation $$f^*\circ g_*=g'_*\circ f'^*$$ hold as maps $$\Omega^*(Z)\to \mathcal D^{*+dim(X)-dim(Z)}(Y)$$?

[1] Hörmander, Lars, The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis., Moskva: ”Mir”. 464 p. (1986). ZBL0619.35001.

• I don't understand how $\dim(Y)$ enters into $g_*$. The fact that $f\pitchfork g$ merely tells us that when $f(y)=g(z)=x$, $f_*T_yY + g_*T_zZ = T_xX$. Why is this relevant? Since $g$ pulls back $k$-forms on $X$ to $k$-forms on $Z$, shouldn't $g_*$ map $\mathcal D^{\dim Z-k}(Z)$ to $\mathcal D^{\dim X-k}(X)$, hence $\mathcal D^j(Z)$ to $\mathcal D^{\dim X-\dim Z+\ell}(X)$? Commented Jan 27, 2022 at 19:20
• Oops. $\ell$ should be $j$ in that final formula. Commented Jan 27, 2022 at 19:41
• If all we need is that the diagram commutes, I’m happy to assume just that! Regarding your comment about dim(Y), that was a typo. Thank you for pointing it out. I’m sorry I didn’t notice it prior to posting. Commented Jan 27, 2022 at 23:12
• Let me ponder the rest of the question. :) Commented Jan 27, 2022 at 23:34
• I have edited the question to include why I think transversality is relevant :) Commented Jan 28, 2022 at 13:37