The dilogarithm as a characteristic class for flat bundles. (English) Zbl 0624.57024

In a previous paper [ibid. 25, 159-195 (1982; Zbl 0496.52004)] the author with C.-H. Sah obtained an explicit homomorphism \(c: H_ 3(SL(2,{\mathbb{C}}), {\mathbb{Z}})\to {\mathbb{C}}/{\mathbb{Q}}\), involving the dilogarithmic function, where \(H_ 3(SL(2,{\mathbb{C}}), {\mathbb{Z}})\) is the 3rd homology group of the discrete group \(SL(2,{\mathbb{C}})\). Let \(\hat C_ 2\in H^ 3(SL(2,{\mathbb{C}}), {\mathbb{Q}}/{\mathbb{Z}})\) be the Cheeger-Chern-Simons class for flat \(SL(2,{\mathbb{C}})\)-bundles associated with the 2nd Chern polynomial \(C_ 2\). In the present paper, the author proves that \(2\hat C_ 2\) and c above are identical as elements in \(Hom(H_ 3(SL(2,{\mathbb{C}}),{\mathbb{Z}}), {\mathbb{C}}/{\mathbb{Q}}).\)
The homomorphism c is defined roughly as follows. Let \(z=((a_ 0-a_ 1)/(a_ 0-a_ 3))((a_ 1-a_ 3)/(a_ 1-a_ 2))=\{a_ 0,a_ 1,a_ 2,a_ 3\}\) be the cross-ratio of 4 points \(a_ 0,a_ 1,a_ 2,a_ 3\) on the Riemann sphere \({\mathbb{C}}\cup \{\infty \}\). Then the abelian group \({\mathcal P}_{{\mathbb{C}}}\) is the group generated by the symbols \(\{z\}=\{\{a_ 0,a_ 1,a_ 2,a_ 3\}\}\) subject to the relations \(\{z_ 1\}-\{z_ 2\}+\{z_ 2/z_ 1\}-\{(1-z_ 2)/(1-z_ 1)\}+\{(1- z_ 2)z_ 1/(1-z_ 1)z_ 2\}=0\) and \(\{z\}=0\) if there are two equals among \(a_ 0,a_ 1,a_ 2,a_ 3\). Then the maps \(\sigma: H_ 3(SL(2,{\mathbb{C}}), {\mathbb{Z}})\to {\mathcal P}_{{\mathbb{C}}}\) and \(\rho: {\mathcal P}_{{\mathbb{C}}}\to \Lambda^ 2_{{\mathbb{Z}}}({\mathbb{C}})\) are defined; \(\sigma\) is induced by \[ (g_ 1,g_ 2,g_ 3)\to \{\{\infty: g_ 1\infty: g_ 1g_ 2\infty: g_ 1g_ 2g_ 3\infty \}\},\quad (g_ 1,g_ 2,g_ 3)\in C_ 3(SL(2,{\mathbb{C}})) \] and \(\rho\) is defined by \[ \rho(\{z\})=(\log z)/2\pi i\wedge (\log (1-z))/2\pi i + 1\wedge (2\pi i)^{-2}\int^{z}_{0}\{\log ((1-t)/t)+\log (t/(1-t))\} dt \] (for suitable choice of branches of log). By choosing a splitting \(\alpha: \Lambda^ 2_{{\mathbb{Z}}}({\mathbb{C}})\to {\mathbb{C}}/{\mathbb{Q}}\) of the injection \(1\wedge id: {\mathbb{C}}/ {\mathbb{Q}}\to \Lambda^ 2_{{\mathbb{Z}}}({\mathbb{C}})\), c is represented by the cochain \(\tilde c=\alpha \circ \rho \circ \sigma\). Namely, \[ \tilde c(g_ 1,g_ 2,g_ 3)=\alpha (\rho (\{\infty: g_ 1\infty: g_ 1g_ 2\infty: g_ 1g_ 2g_ 3\infty \})). \] Since \({\mathbb{C}}/{\mathbb{Q}}= {\mathbb{R}}/{\mathbb{Q}}+i{\mathbb{R}}\), \(\hat C_ 2\) and c have the real part and the imaginary part, respectively. That \(2\hat C_ 2\) and c have the same imaginary part is proved rather easily and most part of the proof is devoted to proving that they have the same real parts, where some results on the homology groups of groups such as \(SL(2,{\mathbb{C}})\), \(SL(2,{\mathbb{R}})\) together with explicit calculations involving the dilogarithmic function are used.
Reviewer: T.Mizutani


57R20 Characteristic classes and numbers in differential topology
53C05 Connections (general theory)
20J99 Connections of group theory with homological algebra and category theory


Zbl 0496.52004
Full Text: DOI


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