I'd like to solve or find a reference for Exercise 2 (a), Chapter 9 in the book Curvature and Characteristic Classes by Johan L. Dupont. https://mathscinet.ams.org/mathscinet/article?mr=500997
- Statement of the exercise:
Exercise 2. Let G be a Lie group with finitely many components and let $\alpha: \Gamma \to G$ be a homomorphism from a discrete group. Let $K \subseteq G$ be a maximal compact subgroup. For $\omega \in Inv_{G} A^{*}(G / K)$ ,the element $J_{*} \omega \in H^{*}(B G_{d}, \mathbb{R})$ , defines a characteristic class for flat G-bundles.
a) Let $\pi: \tilde{M} \to M$ be a differentiable principal $\Gamma$-covering and let $\pi_{\alpha}: E_{\alpha} \to M$ be the corresponding flat G-bundle (see Exercise 1a) and let $\bar{\pi}_{\alpha}: \tilde{M}\times_{\Gamma} G / K \to M$ be the associated fibre bundle with fibre $G/K$. Show that $\bar{\pi}_{\alpha}$ induces an isomorphism in cohomology and that the pull-back $\bar{\pi}_{\alpha}^{*}(J_{*}(\omega)(E_{\alpha})) \in H^{*}(\tilde{M} \times_{\Gamma} G / K, \mathbb{R})$ of the characteristic class $J_{*}(\omega) \in H^{*}(M, \mathbb{R})$ is represented in $A^{*}(\tilde{M} \times_{\Gamma} G / K)$ by the unique form whose lift to $\tilde{M} ×G / K$ is just $\omega$ pulled back under the projection $\tilde{M} ×G / K \to G / K$.
- Some necessary definitions:
Definition 9.5. A filling of $G/K$ is a family of $C^{\infty}$ singular simplices $\sigma\left(g_{1}, ..., g_{p}\right): \Delta^{p} \to G / K, g_{1}, ..., g_{p} \in G, p=0,1,2, ...$ (so for $p=0$ $\sigma(ø)=0$ is some "base point",usually $0=\{K\})$ such that for $p=1,2, ...,$ $$ \sigma\left(g_{1}, ..., g_{p}\right) \circ \varepsilon^{i}= \begin{cases}L_{g_{1}} \circ \sigma\left(g_{2}, ..., g_{p}\right), & i=0, \\ \sigma\left(g_{1}, ..., g_{i} g_{i+1}, ..., g_{p}\right), & 0<i<p, \\ \sigma\left(g_{1}, ..., g_{p-1}\right), & i=p. \end{cases}$$
The merit of a filling σ of $G/K$ is that it enables us to construct explicit Eilenberg-MacLane cochains: Consider the subcomplex $Inv_{G}(A^{*}(G / K))$ of the de Rham complex $A^*(G / K)$ consisting of G -invariant forms (where the G-action is induced by the left G-action on $G/K$). Define the map $J: Inv_{G}(A^*(G / K)) \to C^{*}\left(NG_{d}\right)$ by $$ J(\omega)\left(g_{1}, ..., g_{p}\right)= \int_{\Delta^p} \sigma\left(g_{1}, ..., g_{p}\right)^* \omega, g_{1}, ..., g_{p} \in G, \omega \in A^{p}(G / K), p=0,1,2, ...$$
Proposition 9.10. a) J is a chain map. b) The induced map on homology $J_{*}: H\left(Inv_{G} A^{*}(G / K)\right) \to H\left(C^{*}\left(NG_{d}\right)\right)=H^{*}\left(BG_{d}, \mathbb{R}\right)$ is independent of the choice of filling.
- My attempt:
The first statement that $\bar{\pi}_{\alpha}$ induces an isomorphism in cohomology is obvious. The point is to find out why the classifying map for $E_\alpha$ pulls back the universal class $J_{*}(\omega)$ to the specified class on $M$. Here $M$ is not even required to be a CW-complex, but assumptions such as being a manifold is ok.
I believe that the proof of part b) of Proposition 9.10 is crucial, as it provides another equivalent construction of $J_*$ for the universal case, which is seemingly close to the construction in the exercise. The proof involves various definitions and propositions proved before Chapter 9, e.g. theory of simplicial manifolds and simplicial de Rham cohomology, and is also lengthy.
Besides, in the paper Simplicial de Rham cohomology and characteristic classes of flat bundles by Dupont, https://mathscinet.ams.org/mathscinet/article?mr=413122
the proof of Theorem 1.1 shows that for a homomorphism $f:\Gamma\to G$ from a discrete group $\Gamma$, the construction in the exercise above applied to the universal principal $\Gamma$-bundle (considered as a simplicial manifold) gives $j_\Gamma(\phi)$ in the Eilenberg--Maclane cohomology of $\Gamma$ (here $\phi$ is the form $\omega$), which is proved to be equal to
$$j_{\Gamma}(\phi)\left(\gamma_{1}, ..., \gamma_{q}\right)=\int_{\Delta\left(\gamma_{1}, ..., \gamma_{q}\right)} \phi, $$
a formula expressing the pullback of $J_*(\phi)$ to the cohomology of $\Gamma$.
However, I cannot show the pullback of the universal class $J_*(\omega)$ to the cohomology of $M$ is given by the construction in the exercise. The difference from $B\Gamma$ is that the classifying map for $B\Gamma$ case is just given by $Bf$ so everything goes as expected, but for $M$ it is much harder.
Besides, $M$ is not a simplicial manifold, but there is a technique replacing it with a simplicial manifold (w.r.t. a principal $G$-bundle) in the book page 80, around diagram 5.12. This might be useful.
- Ending remarks:
I am not an expert in this book, but I need this exercise in my research. Originally I came across such a statement in Chern—Simons theory and cohomological invariants of representation varieties by Nicolas Tholozan https://mathscinet.ams.org/mathscinet/article?mr=4748247 , Theorem 3.8. The commutative diagram in Theorem 3.8 is somehow a "folklore fact" and finally I found the related exercise above which is exactly what I need.
I think either this is far more than an exercise, or I am missing something that I am not familiar with/ I am going the wrong way. If there's any reference solving this exercise, please let me know. Thanks!
