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Pushback Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle

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Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be aan oriented $m$-dim ball bundle, i.e. each fiber is aan oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\partial N\to B$, where $\iota: \partial N\to N$ is the boundary of $N$.

Let $\omega\in \Omega^n(N)$ be a closed form. By the pullback of $\iota$ and pushforward of $\pi_\partial$, we have a closed form $$ \omega_b=\frac{1}{\mathrm{Vol}(S^{m-1})}\pi_{\partial,*}(\iota^*\omega\wedge \mathrm{vol}_{S^{m-1}}), $$ where

  • $\mathrm{vol}_{S^{m-1}}$ denotes the standard volume form of $S^{m-1}$,
  • $\mathrm{Vol}(S^{m-1})$ denotes the standard volume of $S^{m-1}$,
  • $\pi_{\partial,*}$ denotes the pushforward map, i.e. the integral over the fiber.

Question Can we say that $\int_B\omega_b=\int_B \omega$? In particular, when $N$ is ball bundle of an oriented vector bundle over $B$, is this true?

PS: The version is modified by Daniele Tampieri, I just removed an extral word.

Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be a $m$-dim ball bundle, i.e. each fiber is a $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\partial N\to B$, where $\iota: \partial N\to N$ is the boundary of $N$.

Let $\omega\in \Omega^n(N)$ be a closed form. By the pullback of $\iota$ and pushforward of $\pi_\partial$, we have a closed form $$ \omega_b=\frac{1}{\mathrm{Vol}(S^{m-1})}\pi_{\partial,*}(\iota^*\omega\wedge \mathrm{vol}_{S^{m-1}}), $$ where

  • $\mathrm{vol}_{S^{m-1}}$ denotes the standard volume form of $S^{m-1}$,
  • $\mathrm{Vol}(S^{m-1})$ denotes the standard volume of $S^{m-1}$,
  • $\pi_{\partial,*}$ denotes the pushforward map, i.e. the integral over the fiber.

Question Can we say that $\int_B\omega_b=\int_B \omega$?

PS: The version is modified by Daniele Tampieri, I just removed an extral word.

Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\partial N\to B$, where $\iota: \partial N\to N$ is the boundary of $N$.

Let $\omega\in \Omega^n(N)$ be a closed form. By the pullback of $\iota$ and pushforward of $\pi_\partial$, we have a closed form $$ \omega_b=\frac{1}{\mathrm{Vol}(S^{m-1})}\pi_{\partial,*}(\iota^*\omega\wedge \mathrm{vol}_{S^{m-1}}), $$ where

  • $\mathrm{vol}_{S^{m-1}}$ denotes the standard volume form of $S^{m-1}$,
  • $\mathrm{Vol}(S^{m-1})$ denotes the standard volume of $S^{m-1}$,
  • $\pi_{\partial,*}$ denotes the pushforward map, i.e. the integral over the fiber.

Question Can we say that $\int_B\omega_b=\int_B \omega$? In particular, when $N$ is ball bundle of an oriented vector bundle over $B$, is this true?

PS: The version is modified by Daniele Tampieri, I just removed an extral word.

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