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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.