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The context is Guillemin and Pollack, Chapter 4.6, Cohomology with Forms.

Let $U$ be an open subset of $\mathbf{R}^k$ and let $\omega$ be a $p$-form on $\mathbf{R} \times U$, represented as $$ \omega = \sum_I f_I(t, x) dt \wedge dx_I + \sum_J g_J(t, x) dx_J. $$ Define an operator $P$ sending $p$-forms on $\mathbf{R} \times U$ to $(p-1)$-forms on $\mathbf{R} \times U$ by $$P \omega = \sum_I \left( \int_0^t f_I(s, x) ds \right) dx_I. $$

Lemma. Let $X$ be a $k$-dimensional manifold, $\pi \colon \mathbf{R} \times X \to X$ the natural projection, and $i_a \colon X \to \mathbf{R} \times X$ the embedding $x \mapsto (a, x)$ for any fixed $a \in \mathbf{R}$. Then locally, $$ dP\omega + Pd\omega = \omega - \pi^* i_a^* \omega. $$

Question. As stated, is this formula true only for $a = 0$?

Letting $\omega$ be locally represented as above, the authors note that $$ \pi^* i_a^* \omega = \sum_J g_J(a, x) dx_J $$ and my calculations verify this. However, I am calculating $$dP \omega = \sum_{i, I} \left( \int_0^t \frac{\partial}{\partial x_i} f_I(s, x)ds \right) dx_i \wedge dx_I + \sum_I f_I(t, x) dt \wedge dx_I $$ and $$ Pd\omega = -\sum_{i, I} \left( \int_0^t \frac{\partial}{\partial x_i} f_I(s, x)ds \right) dx_i \wedge dx_I + \sum_J \left( g_J(t, x) - g_J(0, x) \right) dx_j$$ hence $$dP \omega + P d\omega = \omega - \sum_J g_J(0, x) dx_J = \omega - \pi^* i_0^* \omega. $$

Assuming this calculation is correct, if I am not mistaking the lemma can be remedied by letting the operator $P$ vary with $a$ as well, i.e. $$P_a \omega = \sum_I \left( \int_a^t f_I(t, x) \right) dx_I$$ so that

$$dP_a \omega + P_a d\omega = \omega - \pi^* i_a^* \omega. $$

Do my calculations check out? Is this what Guillemin and Pollack probably intended?

Many thanks.

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  • $\begingroup$ It is at least suspect in the original formula that the right side depends on $a$, whereas the left side does not. $\endgroup$ – Lukas Geyer Jan 5 '16 at 0:21
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I believe I am correct. It doesn't matter all that much that $P$ is actually parametrized by $a$. The point is that for each $a$ this provides a homotopy of chain maps between $\text{id}^\#$ and $\pi^\# i_a^\#$, which is the essential lemma in this context.

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