Fibre integrals

Let \(\pi : E \rightarrow B\) be a smooth fibre bundle, let \(\alpha\) be a \(k\) -form on \(E\). Of course, \(\pi\) is locally a projection to base, and is thus a submersion. It follows that the fibres \(\pi^{-1}(b)\) will be \(\dim(E) - \dim(B) = e - b = n\) dimensional submanifolds for all \(b \in B\). Assume that they are compact oriented, and assume that \(k > n\).

Claim. Let \(\pi : E \rightarrow B\) be a smooth submersion, let \(X\) be a smooth vector field on \(B\). Then there exists a smooth vector field \(\widetilde{X}\) on \(E\) such that \(\pi_{*, p}(\widetilde{X}_p) = X_{\pi(p)}\) for all \(p \in E\).

Proof. Around \(p \in E\), pick coordinates \((U, \varphi) = (U, x^1, \dots, x^m)\) and \((V, \psi) = (V, y^1, \dots, y^m)\) so that \(\psi \circ \pi \circ \varphi^{-1}\) is a projection. Write \(X\) in \(V\) as \(\sum_{j = 1}^{m} X^j \frac{d}{dy^j}\) and define \(\widetilde{X}|_{U}\) as \(\sum_{j = 1}^{m} X^j \frac{d}{d x^j}\). We do this for some open cover \(U_{\alpha}\) and then combine via a partition of unity subordinate to the cover \(\varphi_{\alpha}\) to obtain \(\widetilde{X}\). \(\blacksquare\)

From here, let \(v_1, \dots, v_{k - n}\) be tangent vectors at some \(b \in B\). We can extend them to vector fields and lift via the above claim to obtain vector fields in \(E\), \(\widetilde{v_j}\). Let \(\beta\) be the \(n\) -form obtained from contracting \(\alpha\) with the lifted vector fields. Restrict \(\beta\) to submanifold \(\pi^{-1}(b)\), on which it is a top-form on an oriented compact manifold. We then define the map \(\pi_{*} \alpha : T_b B \times \cdots \times T_b B \rightarrow \mathbb{R}\) as

\begin{equation} (\pi_{*} \alpha)_b(v_1, \dots, v_{k - n}) = \int_{\pi^{-1}(b)} \iota_b^{*} \beta \end{equation}

where \(\iota_b : \pi^{-1}(b) \rightarrow E\) is inclusion. Of course, this map is alternating multilinear. We still must show that this map is well-defined independent of our choice of lifts, and that it is smooth in \(b\). To do so, we pass to coordinates. We have a trivialization of the fibre bundle which acts as a slice/submanifold chart for the fibres. In particular, we have open neighbourhood of \(U\) and a diffeomorphism \(\Phi : \pi^{-1}(U) \rightarrow U \times F\) such that \(\pi = \text{proj} \circ \Phi\). Assume without loss of generality that \((U, \varphi) = (U, x^1, \dots, x^b)\) is also a coordinate chart around \(b\). Note that the entire fibre \(\pi^{-1}(b)\) is contained in \(\pi^{-1}(U)\), let \((V, \psi) = (V, y^1, \dots, y^n)\) be coordinates around \(f \in F\) with \(\Phi(z) = (b, f)\) for some \(z \in \pi^{-1}(b)\). Then immediately \(W = \Phi^{-1}(U \times V)\) and

\begin{equation} (\varphi \times \psi) \circ \Phi = (x^1 \circ \Phi, \dots, x^b \circ \Phi, y^1 \circ \Phi, \dots, y^n \circ \Phi) = (\widetilde{x}^1, \dots, \widetilde{x}^b, \widetilde{y}^1, \dots, \widetilde{y}^n) \end{equation}

is a coordinate chart around \(z\). We write \(\widetilde{v}_j = X^q_j \frac{d}{d\widetilde{x}^q} + Y^r_j \frac{d}{d\widetilde{y}^r}\) in \(W\) as our lifted vector fields. If \(v_j = 0\) for some \(j\), then clearly \(X_j^q(p) = 0\) for any \(p \in \pi^{-1}(b)\), for all \(q\). Thus,

\begin{equation} \beta_p = \alpha_p \left( \cdots, \widetilde{v_{1, p}}, \dots, \widetilde{v_{k - n, p}} \right) = \alpha_p \left( \cdots, \widetilde{v_{1, p}}, \dots, Y^{r}_{j}(p) \frac{d}{d\widetilde{y}^{r}}, \dots, \widetilde{v_{k - n, p}} \right) \end{equation}

If we then take any \(n\) tangent vectors at \(p \in \pi^{-1}(b)\), and push forward by inclusion into \(E\), each will be a linear combination of the tangent vectors \(\frac{d}{d\widetilde{y}^j}\) at \(p\), which when plugged into the above expression, will always yield \(0\), as we will always have a repeated index (particularly, there are \(n\) different coordinates \(\widetilde{y}^j\) and \(n + 1\) slots in the form where we are plugging in tangent vectors of the form \(\frac{d}{d\widetilde{y}^j}\)). Thus, \(\beta_p = 0\) for all \(p \in \pi^{-1}(b)\), in this case. This immediately proves that our map is well-defined, independent of the chosen extension and lift of \(v_1, \dots, v_{k - n}\), via multilinearity of the differential form.

Finally, let us prove smoothness. The chart \((W, (\varphi \times \psi) \circ \Phi)\) around \(z\) gives rise to the chart around \(z\) for \(\pi^{-1}(b)\), \((W \cap \pi^{-1}(b), \theta) = (W \cap \pi^{-1}(b), \widetilde{y}^1 \circ \iota_b, \dots, \widetilde{y}^n \circ \iota_b)\). We can write \(\beta\), a smooth form on \(E\), as \(\beta = \sum_{I, J} C_{I, J} d\widetilde{x}^I \wedge d\widetilde{y}^J\). It then follows that

\begin{equation} (\iota_b^{*} \beta)_{\Phi^{-1}(b, f)} = \sum_{I, J} (C_{I, J} \circ \Phi^{-1})(b, f) \iota_b^{*} d\widetilde{x}^I \wedge \iota_b^{*} d\widetilde{y}^J = (C \circ \Phi^{-1})(b, f) d(\widetilde{y}^1 \circ \iota_b) \wedge \cdots \wedge d(\widetilde{y}^n \circ \iota_b) \end{equation}

where all terms with non-trivial \(d\widetilde{x}^I\) are eliminated when pulled-back by \(\iota_b\), as the functions \(\widetilde{x}^j \circ \iota_b\) are constant. We have the diffeomorphism \(\Phi : W \rightarrow \Phi(W) \subset U \times F\). Inside \(U \times F\), we have submanifold \(\{b\} \times V\) and the smooth embedding \(\Phi \circ \iota_b\) from submanifold \(W \cap \pi^{-1}(b)\) to \(\{b\} \times V\). These manifolds are of the same dimension, so this is actually a diffeomorphism, and as a result, we have a diffeomorphism of \(V\) with \(W \cap \pi^{-1}(b)\), where we take \(f\) to \(\Phi^{-1}(b, f)\). Call this diffeomorphism \(\Psi\). We have

\begin{equation} \int_{\pi^{-1}(b) \cap W} \iota_b^{*} \beta = \int_{V} (\Psi)^{*} \iota_b^{*} \beta = \int_F (C \circ \Phi^{-1})(b, \cdot) dy^1 \wedge \cdots \wedge dy^n \end{equation}

The right hand side will clearly be smooth in \(b\), as in coordinates, we can always differentiate under the integral sign. The integral over all of \(\pi^{-1}(b)\) is a sum over terms of the above form, glued together via a partition of unity. Thus, such a sum is also smooth in \(b\). We have therefore shown that \(\pi_{*} \alpha\) is a \(k - n\) form on the base space.

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Remark. It is pretty easy to see that if forms \(\alpha\) and \(\omega\) on \(E\) agree inside \(\pi^{-1}(U)\), then their fibre integrals will agree on \(U\). This is simply due to the fact that for \(b \in U\), \(\pi^{-1}(b) \subset \pi^{-1}(U)\), which is the region over which we integrate.

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Let \(M\) be a smooth manifold, consider the trivial fibre bundle \(\pi : [0, 1] \times M \rightarrow M\). We have the maps \(j_s : M \rightarrow [0, 1] \times M\) given by \(j_s(p) = (s, p)\) for \(s = 0, 1\). If \(M\) is \(n\) -dimensional, then \([0, 1] \times M\) is \((n + 1)\) -dimensional. Given some \((s, p) \in [0, 1] \times M\), we have coordinates \((U, \varphi) = (U, x^1, \dots, x^n)\) around \(p\) in \(M\), and global coordinate \(t\) for \([0, 1]\), which we combine to form a chart \(([0, 1] \times U, t, x^1 \circ \pi, \dots, x^n \circ \pi)\) around \((s, p)\). It follows that if \(\alpha\) is a \((k + 1)\) -form on \([0, 1] \times M\), then inside \([0, 1] \times U\), it can be written as

\begin{equation} \alpha = \sum_{I} f_I(s, p) dt \wedge \pi^{*}( dx^{i_1} \wedge \cdots \wedge dx^{i_k}) + \sum_{J} g_J(s, p) \pi^{*}(dx^{j_1} \wedge \cdots \wedge dx^{j_{k + 1}}) \end{equation}

From here, it follows fairly immediately from the definition, and the above remark explaining that the fibre integral is "local", that for some \(p \in U\),

\begin{equation} (\pi_{*} \alpha)_p = \sum_{I} \left( \int_{[0, 1]} f_I(s, p) \ ds \right) dx^{i_1} \wedge \cdots \wedge dx^{i_k} \end{equation}

Clearly, \(d\alpha\) is given by

\begin{align} d\alpha = -\sum_{I, i} \frac{d f_I}{dx^i}(s, p) dt \wedge \pi^{*}( dx^i \wedge dx^I) + \sum_{J} \frac{d g_J}{dt}(s, p) dt \wedge \pi^{*} dx^J +\sum_{J, j} \frac{d g_J}{dx^j}(s, p) \pi^{*}(dx^j \wedge dx^J) \end{align}

and, in addition, we have via differentiating under the integral sign,

\begin{equation} d \pi_{*} \alpha = \sum_{I, i} \left( \int_{[0, 1]} \frac{d f_I}{dx^i}(s, p) \ ds \right) dx^i \wedge dx^I \end{equation}

so it is easy to see that

\begin{equation} d \pi_{*} \alpha + \pi_{*} d\alpha = \sum_{J} \left( \int_{[0, 1]} \frac{d g_J}{ds}(s, p) \ ds \right) dx^J = \sum_{J} (g_J(1, p) - g_J(0, p)) dx^J = j_1^{*} \alpha - j_0^{*} \alpha \end{equation}

This holds in each coordinate chart, and thus holds everywhere. It follows immediately that \(\pi_{*}\) is a chain homotopy of the maps \(j_0^{*}\) and \(j_1^{*}\). As a result, \(j_0^{*} = j_1^{*}\) as maps on cohomology, \(H^k([0, 1] \times M) \rightarrow H^k(M)\). Suppose \(f : M \rightarrow N\) is a smooth homotopy equivalence, so there is \(g : N \rightarrow M\) where \(f \circ g \simeq \text{id}_N\) and \(g \circ f \simeq \text{id}_M\). If we let \(F : [0, 1] \times N \rightarrow N\) be a homotopy between \(f \circ g\) and \(\text{id}_N\), we have \(F \circ j_0 = f \circ g\) and \(F \circ j_1 = \text{id}_N\), and from above,

\begin{equation} g^{*} \circ f^{*} = (f \circ g)^{*} = (F \circ j_0)^{*} = (F \circ j_1)^{*} = \text{id}_N^{*} \end{equation}

Similarly, \(f^{*} \circ g^{*} = \text{id}_M^{*}\). Thus, \(f^{*} : H^k(N) \rightarrow H^k(M)\) is an isomorphism of cohomology groups, and we have the classic result of homotopy invariance of cohomology. This immediately implies the Poincare lemma, for example.