Generalised eta invariants, end-periodic manifolds, and their applications to positive scalar curvature

Abstract

This thesis studies the applications of index theory to positive scalar curvature (PSC), in particular questions of existence and number of path components of the moduli space of PSC metrics. After Atiyah-Singer proved their legendary index theorem [AS63, AS68a, AS68b], many fruitful applications to positive scalar curvature were discovered, for instance Lichnerowicz's obstruction to positive scalar curvature metrics on spin manifolds with nonvanishing A-hat genus [Lic63]. The theorem of Lichnerowicz relies notably on the existence of a spin Dirac operator on any spin manifold|a self-adjoint, elliptic, first order differential operator having marvellous connections to the geometry of the underlying Riemannian manifold. In 1975, Atiyah, Patodi and Singer [APS75a, APS75b, APS76] proved their index theorem for a Dirac operator D on a manifold Z with boundary @Z = Y . This takes a similar form to the Atiyah-Singer index theorem but notably has a correction term n(A) = (nA(0)+h)=2, called the eta invariant, appearing for the boundary. The eta invariant is defined solely in terms of the spectrum of the Dirac operator A on the boundary, so is a spectral invariant. As it stands, this invariant is not at all robust; if one slightly perturbs the metric on Y then most likely one will produce a change in the eta invariant. A notable exception to this is conformal deformations, which leave the Dirac operator unchanged. There is, however, a more robust invariant which can be procured from the eta invariant. If one twists the Dirac operator A on Y by two unitary representations δ1; δ2 : π1(Y ) → U(N) of the fundamental group of Y , one obtains two twisted Dirac operators A1 and A2 on Y which are locally isomorphic. Subtracting the eta invariants of these twisted Dirac operators yields a modified invariant p(δ1; δ2;A), called the rho invariant. The rho invariant still isn't quite robust, but upon taking the mod Z reduction of the rho invariant, many striking invariance properties emerge. For example, writing = 1(Y ), the rho invariant descends to a well-defined map on geometric K-homology [HR10]: p(δ1; δ2) : K1(Bπ) → R=/Z. The rho invariant can be used to further study the properties of PSC metrics on manifolds. Whereas the Atiyah-Singer theorem is mostly useful for even-dimensional manifolds, the APS index theorem allows one to obtain results for odd-dimensional manifolds by considering them as boundaries of even-dimensional manifolds. In particular, one can obtain obstructions to PSC [HR10], and study the number of path components of the moduli space of PSC metrics on a manifold [BG95]. More recently, Mrowka, Ruberman and Saveliev [MRS16] discovered and proved a new index theorem for manifolds with periodic ends. Roughly speaking, these are manifolds Z∞ which have a compact piece Z, attached to which are one or more ends which repeat themselves periodically off to infinity. Such manifolds were first studied by Taubes [Tau87], who used them to prove (following work of Donaldson and Freedman) that R4 admits an uncountable family of mutually non-diffeomorphic smooth structures. The index theorem of MRS involves, like the APS index theorem, a correction term Nep(D) appearing for the periodic ends. The main contribution of this thesis is the development of a new analogue of geometric K-homology that is tailored to the setting of manifolds with periodic ends. The group is called Kep 1 (Bπ), and as in the APS case there is an analogous rho invariant descending to a well-defined map: ...