Homotopy invariance of higher signatures and 3-manifold groups
Bulletin de la Société Mathématique de France, Volume 136 (2008) no. 1, p. 1-25

For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3-manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients holds. The non-oriented case is also discussed.

Nous démontrons que pour des variétés fermées et orientées les signatures qui proviennent des groupes fondamentaux d’une large classe de variétés orientables de dimension 3 sont des invariants homotopiques. Cette classe, que nous décrivons soigneusement, contient en particulier les variétés géométriques par morceaux au sens de Thurston. Si la conjecture de géométrisation de Thurston s’avère vraie cette classe coïncide alors avec celle des groupes fondamentaux de variétés de dimension 3 orientables. Plus précisément nous démontrons que tous les groupes dans cette classe satisfont la conjecture de Baum-Connes avec coefficients. Nous discutons également le cas non-orientable.

DOI : https://doi.org/10.24033/bsmf.2547
Classification:  19K35,  57R20,  57M50,  46L80
Keywords: Baum-Connes conjecture, JSJ decomposition, Thurston geometrization conjecture
@article{BSMF_2008__136_1_1_0,
     author = {Matthey, Michel and Oyono-Oyono, Herv\'e and Pitsch, Wolfgang},
     title = {Homotopy invariance of higher signatures and $3$-manifold groups},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {136},
     number = {1},
     year = {2008},
     pages = {1-25},
     doi = {10.24033/bsmf.2547},
     zbl = {1179.19004},
     mrnumber = {2415334},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2008__136_1_1_0}
}
Matthey, Michel; Oyono-Oyono, Hervé; Pitsch, Wolfgang. Homotopy invariance of higher signatures and $3$-manifold groups. Bulletin de la Société Mathématique de France, Volume 136 (2008) no. 1, pp. 1-25. doi : 10.24033/bsmf.2547. http://www.numdam.org/item/BSMF_2008__136_1_1_0/

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