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**The index of signature operators on Lipschitz manifolds.**
*(English)*
Zbl 0531.58044

This paper is a part of a general program leading to index theory of elliptic operators on compact Lipschitz manifolds. The ultimate goal of this program (by now successfully realized in the joint work of the author and D. Sullivan [see the following review] is an analytic proof of the topological invariance of the rational Pontryagin classes. The feasibility of such an approach follows from Sullivan’s theorem on existence of essentially unique Lipschitz structure on every compact topological manifold of dimension 4.

The Atiyah-Singer index theorem asserts that two integers (the analytic index and the topological index) associated to an elliptic differential operator on a compact smooth manifold are equal. A central part in this theory is played by the (twisted) signature operators. The paper under review shows how to do analysis on Lipschitz manifolds so that the signature operators can be defined. It is then proved that these operators are Fredholm between appropriately defined Sobolev spaces of forms with coefficients in Lipschitz vector bundles, so that the analytic index is defined. Finally this analytic index is shown to be a Lipschitz invariant. This last fact is a consequence of the excision theorem, which states that under certain conditions indices of two signature operators on two different Lipschitz manifolds having a common open part are equal. To carry this out the author has to define basic objects, e.g. Riemannian metrics, differential forms, exterior derivative etc., and redo much of ”standard elliptic machinery” on Lipschitz manifolds. Of course, not all the pieces can be defined, and, for those that can, not all results remain true. However, in his earlier paper [Invent. Math. 61, 227-249 (1980; Zbl 0456.58027)] the author did much of the required analysis in a different non-smooth setting. As a matter of fact, the treatment of the signature operator (without twisting) in the paper under review is very close to this earlier work. Genuinely new ideas were required to define the twisted signature operators, to prove that they are Fredholm (Rellich-type result), and to compute their index. The difficulties encountered are illustrated e.g. by the fact that the Sobolev spaces (of forms with first derivatives in \(L^ 2)\) vary drastically when the Riemannian metric is changed.

The Atiyah-Singer index theorem asserts that two integers (the analytic index and the topological index) associated to an elliptic differential operator on a compact smooth manifold are equal. A central part in this theory is played by the (twisted) signature operators. The paper under review shows how to do analysis on Lipschitz manifolds so that the signature operators can be defined. It is then proved that these operators are Fredholm between appropriately defined Sobolev spaces of forms with coefficients in Lipschitz vector bundles, so that the analytic index is defined. Finally this analytic index is shown to be a Lipschitz invariant. This last fact is a consequence of the excision theorem, which states that under certain conditions indices of two signature operators on two different Lipschitz manifolds having a common open part are equal. To carry this out the author has to define basic objects, e.g. Riemannian metrics, differential forms, exterior derivative etc., and redo much of ”standard elliptic machinery” on Lipschitz manifolds. Of course, not all the pieces can be defined, and, for those that can, not all results remain true. However, in his earlier paper [Invent. Math. 61, 227-249 (1980; Zbl 0456.58027)] the author did much of the required analysis in a different non-smooth setting. As a matter of fact, the treatment of the signature operator (without twisting) in the paper under review is very close to this earlier work. Genuinely new ideas were required to define the twisted signature operators, to prove that they are Fredholm (Rellich-type result), and to compute their index. The difficulties encountered are illustrated e.g. by the fact that the Sobolev spaces (of forms with first derivatives in \(L^ 2)\) vary drastically when the Riemannian metric is changed.

Reviewer: J.Dodziuk

### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |

47A53 | (Semi-) Fredholm operators; index theories |

57R20 | Characteristic classes and numbers in differential topology |

58J22 | Exotic index theories on manifolds |

58A14 | Hodge theory in global analysis |

57Q99 | PL-topology |

### Keywords:

topological 4-manifold; ZFM 531.58045; rational Pontryagin classes; Lipschitz structure; Atiyah-Singer index theorem### Citations:

Zbl 0456.58027
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\textit{N. Teleman}, Publ. Math., Inst. Hautes Étud. Sci. 58, 251--290 (1983; Zbl 0531.58044)

### References:

[1] | M. F. Atiyah, R. Bott, V. K. Patodi, On the heat equation and the Index Theorem,Inventiones Math. 19. (1973), 279–330. · Zbl 0257.58008 |

[2] | M. F. Atiyah, I. M. Singer, The Index of Elliptic Operators, Part I,Annals of Math. 87 (1968), 484–530. · Zbl 0164.24001 |

[3] | M. F. Atiyah, I. M. Singer, The Index of Elliptic Operators, Part III,Annals of Math. 87 (1968), 546–604. · Zbl 0164.24301 |

[4] | J. Cheeger, On the Hodge theory of Riemannian Pseudo-manifolds,Proc. Symp. Pure Math. 36 (1980), 91–146. · Zbl 0461.58002 |

[5] | J. Dodziuk, De Rham-Hodge theory for L2-cohomology of infinite coverings,Topology 15 (1977), 157–165. · Zbl 0348.58001 |

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[8] | G. G. Kasparov, Topological Invariants of Elliptic Operators, I: K-homology,Math. U.S.S.R. Izvestija (English Transl.)9 (1975), 751–792. · Zbl 0337.58006 |

[9] | I. M. Singer, Future Extensions of Index Theory and Elliptic Operators, inProspects in Mathematics, Annals Math. Studies 70, Princeton (1971), 171–185. · Zbl 0247.58011 |

[10] | D. Sullivan, Hyperbolic geometry and Homeomorphisms, inGeometric Topology, Proc. Georgia Topology Conf. Athens, Georgia 1977, 543–555, ed. J. C. Cantrell, Academic Press, 1979. |

[11] | N. Teleman,Global Analysis on P. L. Manifolds, M.I.T. Ph. D. Thesis, 1977. · Zbl 0381.46035 |

[12] | N. Teleman, Global Analysis on P. L. Manifolds,Trans. Amer. Math. Soc. 256 (1979), 49–88. · Zbl 0448.57007 |

[13] | N. Teleman, Combinatorial Hodge Theory and Signature Theorem,Proc. Symp. Pure Math. 36 (1980), 287–292. · Zbl 0456.58026 |

[14] | N. Teleman, Combinatorial Hodge Theory and Signature Operator,Inventiones Math. 61 (1980), 227–249. · Zbl 0456.58027 |

[15] | H. Whitney,Geometric Integration Theory, Princeton, 1966. · Zbl 0083.28204 |

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