Rough index theory on spaces of polynomial growth and contractibility.

*(English)*Zbl 1436.58019The \(K\)-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.

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.

Reviewer: Christopher Wulff (Göttingen)

##### MSC:

58J22 | Exotic index theories on manifolds |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

19K56 | Index theory |

##### 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 Calgebras, 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.