Guruprasad, K.; Kumar, Shrawan A new geometric invariant associated to the space of flat connections. (English) Zbl 0698.53016 Compos. Math. 73, No. 2, 199-222 (1990). Let \(\pi\) : \(E\to M\) be a principal G-bundle, \({\mathcal C}\) the space of all connections on E, \({\mathcal F}\) the subspace of the flat connections, \(\Delta_{\bullet}({\mathcal C})\), \(\Delta_{\bullet}({\mathcal F})\) denote the smooth singular chain complexes of \({\mathcal C}\) and \({\mathcal F}\) respectively, \(\Lambda_{dR}(M)\) the de Rham complex of M. Fix a G- invariant homogeneous polynomial P of degree k on \({\mathfrak g}=Lie G\). The aim of the paper under review is to define a certain functorial map \({\bar \chi}_ p: \Delta_{\bullet}({\mathcal C})\to \Lambda^{2k- \bullet}(M)\) and study its properties. For a smooth n-simplex \(\sigma\) : \(\Delta^ n\to {\mathcal C}\) let \(\nu_{\gamma}\) be a “tautological” connection on \(\pi \times id: E\times \Delta^ n\to M\times \Delta^ n,\) where for any \(t\in \Delta^ n\) \(\nu_{\sigma}\) on \(E\times \{t\}\) is \(\sigma\) (t). Then: \[ {\bar \chi}_ P(\sigma)=(- 1)^{n(n+1)/2}\int_{\Delta^ n}P(\Omega_{\sigma}) \] where \(\Omega_{\sigma}\) is the curvature of \(\sigma\). The authors also define a map \({\bar \psi}_ P: \Delta_{\bullet -1}({\mathcal F})\to \Lambda^{2k-\bullet}(E)\) by \[ {\bar \psi}_ P(\sigma)=(- 1)^{n(n+1)/2}\int_{\Delta^{n-1}}TP(\nu_{\sigma}) \] where \(TP(\nu_{\sigma})\) is the Chern-Simons secondary form. In general \({\bar \psi}{}_ P\) is not a chain map, but its restriction to the truncated chain complex \(T_ k({\mathcal F})\), where \((T_ k({\mathcal F}))_ n=\Delta_ n({\mathcal F})\) if \(n<k\), and \(=0\) if \(n\geq k\), is a chain map giving rise to a map in homology \[ \psi_{P,\bullet}: H_{\bullet - 1}(T_ k({\mathcal F}))\quad \to \quad H_{dR}^{2k-\bullet}(E). \] By the contractibility of \({\mathcal C}\) the authors show that \(\psi_ P\) is essentially the same as \(\chi_ P.\) For a simply connected M \(\chi_ P\) may be identified with the slant product in the following way: for the trivial bundle \(\pi\) : \(M\times G\to M\) let \({\mathcal L}_ 0=Map(M,G)\) denote the space of all pointed smooth maps \(M\to G\). Taking the orbit of the trivial connection on \(\pi\) we get an embedding \(\alpha\) : \({\mathcal L}_ 0\to {\mathcal F}\). If TP is the universally transgressive cohomology class in \(H_{dR}^{2k-1}(G)\), then \[ \xi_{P,n}(\sigma)=(-1)^{n(n+1)/2}\int_{\sigma}ev^*(TP) \] where ev: \({\mathcal L}_ 0\times M\to G\) is the evaluation map. It is proved that \[ s^*\circ \psi_{P,n}\circ \alpha_*:\quad H_{n- 1}(T_ k({\mathcal L}_ 0))\quad \to \quad H_{dR}^{2k-n}(M) \] factors through \(H_{n-1}({\mathcal L}_ 0)\) and is equal to \(\xi_{P,n}\) for all \(n\leq k\), where \(s^*\) is induced from the section \(s(x)=(x,e)\) of \(\pi\), and that \(\xi_{P_ j,n}\) is surjective onto \(H_{dR}^{2k_ j-n}(M)\) for any “primitive generator” \(P_ j\in I^{k_ j}(G).\) As a trivial consequence of the above results the authors also obtain a result due to Quillen which asserts that in general “roughly half” the generators of \(H_{dR}({\mathcal L}^ e_ 0)\) can be represented by left invariant forms C where \({\mathcal L}^ e_ 0\) is the identity component of \({\mathcal L}_ 0)\). Reviewer: V.B.Marenich Cited in 3 Documents MSC: 53C05 Connections (general theory) 57N65 Algebraic topology of manifolds Keywords:cohomology generators; principal G-bundle; flat connections; de Rham complex; functorial map; Chern-Simons secondary form; slant product; trivial connection PDFBibTeX XMLCite \textit{K. Guruprasad} and \textit{S. Kumar}, Compos. Math. 73, No. 2, 199--222 (1990; Zbl 0698.53016) Full Text: Numdam EuDML References: [1] S.S. Chern and J. Simons : Characteristic forms and geometric invariants . Annals of Math. 99 (1974) 48-69. · Zbl 0283.53036 [2] J.L. Koszul : ” Fibre bundles and differential geometry ”. TIFR lecture notes, Springer-Verlag, (1986). · Zbl 0607.53001 [3] S. Kumar : Non-representability of cohomology classes by bi-invariant forms (Gauge and Kac-Moody groups) . Comm. Math. Phys. 106 (1986) 177-181. · Zbl 0615.57025 [4] J. Milnor : Remarks on infinite-dimensional Lie groups . In: Relativity, groups and topology II , DeWitt, B.S., Stora, R. (eds.). 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