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Rough index theory on spaces of polynomial growth and contractibility. (English) Zbl 1436.58019
The $$K$$-theory of the uniform Roe algebra $$C_u^*(M)$$ of a complete Riemannian manifold of bounded geometry $$M$$ is the receptacle for rough indices of elliptic operators over $$M$$. This paper’s title phrase “rough index theory” is to be understood rather broadly as the study of $$K_*(C_u^*(M))$$ in general.
The ubiquitous assumption for all of the main results is that $$M$$ has bounded geometry and polynomial volume growth and is either (A) polynomially $$k$$-connected for some $$k\in\mathbb{N}$$ or (B) polynomially contractible. Polynomial $$k$$-connected means that there is a polynomial $$P$$ such that for all $$i\leq k$$ every $$L$$-Lipschitz map $$S^i\to M$$ is contractible by a $$P(L)$$-Lipschitz contraction and polynomial contractibility means polynomial $$k$$-connectedness for all $$k$$.
The first main result of the paper is that in the case (A) for all $$q\leq k$$ every element in the coarse cohomology $$HX^q(M)$$ and the rough cohomology $$HR^q(M)$$ pairs continuously with $$K_*(C_u^*(M))$$. In the case (B), these pairings are used to detect the non-vanishing of the rough index of the Dirac operator and hence the non-existence of a metric of uniformly positive scalar curvature in the same strict quasi-isometry class. Furthermore, higher codimension obstructions to the existence of positive scalar curvature metrics are derived for closed connected manifolds with virtually nilpotent fundamental group.
As a main tool for the proofs but also as objects of independent interest, the author introduces “polynomial” versions of well-established objects: A smooth subalgebra $$C_{\mathrm{pol}}^*(M)$$ of $$C_u^*(M)$$ is constructed as the closure of the algebraic smooth uniform Roe algebra in a certain Fréchet topology, such that its continuous periodic cyclic homology is the target of a chern character for $$C_u^*(M)$$. Also, the chain complex of uniformly finite homology $$H_*^{uf}(Y)$$ is completed in a Fréchet topology, yielding homology groups $$H_*^{\mathrm{pol}}(Y)$$ ($$Y\subset M$$ a discretization).
The relation between all of these groups as well as the uniform $$K$$-homology $$K_*^u(M)$$ and the $$L^\infty$$-simplicial homology $$H^\infty_*(M)$$ are studied using assembly maps, chern characters and character maps and they are shown to be (topological) isomorphisms in certain cases. Most notably it is shown that if $$M$$ satisfies (B) and the rough Baum-Connes conjecture, then the chern character map $\mathrm{ch}_*: K_*(C_u^*(M))\mathbin{\overline{\otimes}}\mathbb{C}\to PHC_*^{\mathrm{cont}}(C_{\mathrm{pol}}^*(M))$ is an isomorphism. In contrast to the rough Baum-Connes isomorphism, this map does not go into but out of $$K_*(C_u^*(M))$$ which makes it useful for the study of particular elements.

##### MSC:
 58J22 Exotic index theories on manifolds 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K56 Index theory
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##### References:
 [1] J. M. Alonso, S. J. Pride, and X. Wang, Higher-dimensional isoperimetric (or Dehn) functions of groups, J. Group Theory, 2 (1999), no. 1, 81-112.Zbl 0927.20021 MR 1670329 · Zbl 0927.20021 [2] O. Attie, A surgery theory for manifolds of bounded geometry, 2004.arXiv:math/0312017 [3] T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.Zbl 0896.53003 MR 1636569 664A. Engel [4] D. P. Blecher, The geometry of the tensor product of C-algebras, Ph.D. thesis, University of Edinburgh, 1988. [5] J. Block and S. Weinberger, Aperiodic tilings, positive scalar curvature and amenability of spaces, J. Amer. Math. Soc., 5 (1992), no. 4, 907-918.Zbl 0780.53031 MR 1145337 · Zbl 0780.53031 [6] M. R. Bridson, Polynomial Dehn functions and the length of asynchronously automatic structures, Proc. London Math. Soc. (3), 85 (2002), no. 2, 441-466.Zbl 1046.20027 MR 1912057 · Zbl 1046.20027 [7] X. Chen and S. Wei, Spectral invariant subalgebras of reduced crossed product Calgebras, J. Funct. Anal., 197 (2003), no. 1, 228-246.Zbl 1023.46060 MR 1957682 · Zbl 1023.46060 [8] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), no. 3, 345-388.Zbl 0759.58047 MR 1066176 · Zbl 0759.58047 [9] J. Cuntz, Cyclic theory, bivariant K-theory and the bivariant Chern-Connes character, in Cyclic homology in non-commutative geometry, 1-71, Encyclopaedia Math. Sci., 121, Oper. Alg. Non-commut. Geom., II, Springer, Berlin, 2004.Zbl 1045.46043 MR 2052771 [10] J. Cuntz and A. Thom, Algebraic K-theory and locally convex algebras, Math. Ann., 334 (2006), no. 2, 339-371.Zbl 1095.19003 MR 2207702 · Zbl 1095.19003 [11] K. Dykema, T. Figiel, G. Weiss, and M. Wodzicki, Commutator structure of operator ideals, Adv. Math., 185 (2004), no. 1, 1-79.Zbl 1103.47054 MR 2058779 · Zbl 1103.47054 [12] A. Engel, Indices of pseudodifferential operators on open manifolds, Ph.D. thesis, University of Augsburg, 2014.arXiv:1410.8030 [13] A. Engel, Index theory of uniform pseudodifferential operators, 2015.arXiv:1502.00494 [14] A. Engel, Banach strong Novikov conjecture for polynomially contractible groups, 2017. arXiv:math/1702.02269v2 [15] A. Engel, Wrong way maps in uniformly finite homology and homology of groups, J. Homotopy Relat. Struct., 13 (2018), no. 2, 423-441.Zbl 1401.55006 MR 3802801 · Zbl 1401.55006 [16] H. Figueroa, J. M. Gracia-Bondía, and J. C. Várilly, Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2001.Zbl 0958.46039 MR 1789831 [17] G. Gong and G. Yu, Volume growth and positive scalar curvature, Geom. Funct. Anal., 10 (2000), no. 4, 821-828.Zbl 0979.53033 MR 1791141 · Zbl 0979.53033 [18] M. Gromov, Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits), Inst. Hautes Études Sci. Publ. Math., (1981), no. 53, 53-73.Zbl 0474.20018 MR 623534 [19] N. Große and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces, Math. Nachr., 286 (2013), no. 16, 1586-1613. Zbl 1294.46031 MR 3126616 · Zbl 1294.46031 [20] A. Grothendieck, Résumé de la théorie métrique des produis tensoriels topologiques, Bol. Soc. Math. São Paulo, 8 (1956), 1-79.Zbl 0074.32303 MR 94682 [21] B. Hanke, D. Pape, and T. Schick, Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble), 65 (2015), no. 6, 2681-2710.Zbl 1344.58012 MR 3449594 Rough index theory665 · Zbl 1344.58012 [22] A. Kumar and A. M. Sinclair, Equivalence of norms on operator space tensor products of C-algebras, Trans. Amer. Math. Soc., 350 (1998), no. 5, 2033-2048.Zbl 0906.46043 MR 1473449 · Zbl 0906.46043 [23] J.-L. Loday, Cyclic homology. Appendix E by María O. Ronco, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301, Springer-Verlag, Berlin, 1992.Zbl 0780.18009 MR 1217970 [24] B. Mavra, Bounded geometry index theory, Ph.D. thesis, University of Oxford, 1995. [25] T. Riley, Higher connectedness of asymptotic cones, Topology, 42 (2003), no. 6, 1289- 1352.Zbl 1038.20031 MR 1981358 · Zbl 1038.20031 [26] J. Roe, An index theorem on open manifolds. I, J. Differential Geom., 27 (1988), no. 1, 87-113.Zbl 0657.58041 MR 918459 [27] J. Roe, Partitioning non-compact manifolds and the dual Toeplitz problem, in Operator algebras and applications. Vol. 1, D. E. Evans and M. Takesaki (eds.), 187-228, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge, 1988. Zbl 0668.00014 MR 996446 · Zbl 0677.58042 [28] J. Roe, Hyperbolic metric spaces and the exotic cohomology Novikov conjecture, K- Theory, 4 (1990/91), no. 6, 501-512.Zbl 0756.55003 MR 1123175 · Zbl 0756.55003 [29] J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc., 104 (1993), no. 497, x+90pp.Zbl 0780.58043 MR 1147350 [30] J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, 90, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996.Zbl 0853.58003 MR 1399087 · Zbl 0853.58003 [31] J. Roe, Lectures on coarse geometry, University Lecture Series, 31, American Mathematical Society, Providence, RI, 2003.Zbl 1042.53027 MR 2007488 [32] L. M. Schmitt, Quotients of local Banach algebras are local Banach algebras, Publ. Res. Inst. Math. Sci., 27 (1991), no. 6, 837-843.Zbl 0759.46062 MR 1145667 · Zbl 0759.46062 [33] L. B. Schweitzer, A short proof that Mn.A/is local if A is local and Fréchet, Internat. J. Math., 3 (1992), no. 4, 581-589.Zbl 0804.46054 MR 1168361 · Zbl 0804.46054 [34] M. A. Shubin, Spectral theory of elliptic operators on noncompact manifolds. Méthodes semi-classiques. Vol. 1 (Nantes, 1991), Astérisque, 207 (1992), no. 5, 35-108. Zbl 0793.58039 MR 1205177 [35] J. Špakula, Uniform K-homology theory, J. Funct. Anal., 257 (2009), no. 1, 88-121. Zbl 1173.46049 MR 2523336 [36] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967.Zbl 0171.10402 MR 225131 · Zbl 0171.10402 [37] H. Triebel, Theory of Function Spaces. II. Reprint of the 1992 edition, Modern Birkhäuser Classics, Birkhäuser Verlag, 2010.Zbl 1235.46003 MR 3024598 [38] J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry, 2 (1968), 421-446.Zbl 0207.51803 MR 248688 666A. Engel · Zbl 0207.51803 [39] G. Yu, Cyclic cohomology and higher indices for noncompact complete manifolds, J. Funct. Anal., 133 (1995), no. 2, 442-473.Zbl 0849.58066 MR 1354039 · Zbl 0849.58066 [40] G. Yu, Zero-in-the-spectrum conjecture, positive scalar curvature and asymptotic dimension, Invent. Math., 127 (1997), no. 1, 99-126.Zbl 0889.58082 MR 1423027 · Zbl 0889.58082
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