Analytical signatures and proper actions
Keywords
Bordism invariance, Coarse index, Hilbert-Poincaré complexesAbstract
We compare Mishchenko's definition of non-commutative signature for an oriented manifold with an orientation preserving proper action of a discrete, countable group $G$ with the (more analytical) counterpart defined by Higson and Roe in the series of articles ``Mapping surgery to analysis''. A generalization of the bordism invariance of the coarse index is also addressed.References
O. Attie, Quasi-isometry classification of some manifolds of bunded geometry, K-theory 33 (2004), 277–299.
N. Higson and J. Roe, Mapping surgery to analysis I, K-theory 33 (2004), 277–299.
N. Higson and J. Roe, Mapping surgery to analysis II, K-theory 33 (2004), 301–324.
N. Higson and J. Roe, Mapping surgery to analysis III, K-theory 33 (2004), 301–324.
N. Higson and J. Roe Analytic K-homology, Oxford Science Publications, 2000.
S. Illman, Existence and uniqueness of equivariant triangulations of smooth proper Gmanifolds with some applications to equivariant Whitehead torsion, J. Reine Angew. Math. 524 (2000), 129–183.
G.G. Kasparov, The generalized index of elliptic operators, Functional Analysis and Its Applications 7 (1973), no. 3, 82–83.
G.G. Kasparov, Topological invariants of elliptic operators I: K-homology, Math. USSR Izvestiya 9 (1975), no. 4, 751–792.
G.G. Kasparov, The operator K-functor and extensions of C ∗ -algebras, Math. USSR Izvestija 16 (1981), no. 3, 513–572.
T. Korppi, Equivariant triangulations of differentiable and real-analytic manifolds with a properly discontinuous action, Annales Academi Acientiarum Fennic Matematica Dissertationes, number 141, Suomalainen Tiedeakatemia, Helsinki, 2005.
A.S. Mishchenko, The theory of almost algebraic Poincaré complexes and local combinatorial Hirzebruch formula, Acta Appl. Math. 68 (2001), no. 1–3, 5–37.
A.S. Mishchenko, Homotopy invariants of non-simply connected manifolds I. Rational invariants, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 501–514.
A.S. Mishchenko, Signature of manifolds with proper action of a discrete group and the Hirzebruch type formula, Conference: “Topology, Geometry, and Dynamics: Rokhlin Memorial”, Saint Petersburg, 2010.
R. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323.
J. Roe, Comparing analytical assembly maps, Quart. J. Math. 53 (2002), 241–248.
C. Wulff, Bordism invariance of the Coarse index, Proc. Amer. Math. Soc. 140 (2020), no. 8, 2693–2697.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
