Deformation groupoids and index theory

Deformation groupoids and index theory Karsten Bohlen Leibniz Universität Hannover GRK Klausurtagung, Goslar September 24, 2014

Contents 1 Groupoids 2 The tangent groupoid 3 The analytic and topological index 4 Connes-Thom 5 Proving Atiyah-Singer 6 Outlook 7 References

Groupoids Definition and examples Groupoids are generalized groups, a set endowed with a composition operation (G, ) but where not every two elements must necessarily be composable. They can also be viewed (interpreted) as generalizations of equivalence relations, as generalized spaces (with special regard to symmetry), substitutes for quotient spaces (from a group acting on a manifold), models for singular spaces etc..

Groupoids Definition and examples Groupoids are generalized groups, a set endowed with a composition operation (G, ) but where not every two elements must necessarily be composable. They can also be viewed (interpreted) as generalizations of equivalence relations, as generalized spaces (with special regard to symmetry), substitutes for quotient spaces (from a group acting on a manifold), models for singular spaces etc.. Formal definition: A groupoid is a small category in which every morphism is invertible. The standard notation: The category itself is denoted by G, the morphisms in this category are denoted by G (1) or by abuse of notation simply G. The objects are G (0) which are also called the units.

Groupoids We tend to write G G (0) is a groupoid. The two arrows are the range map r : G G (0) and source map s : G G (0). Two elements γ, η G are composable (i.e. γ η is defined and in G) iff s(γ) = r(η). The set of composable elements is usually denoted by G (2) = G s,r G = {(γ, η) G 2 : s(γ) = r(η)}.

Groupoids We tend to write G G (0) is a groupoid. The two arrows are the range map r : G G (0) and source map s : G G (0). Two elements γ, η G are composable (i.e. γ η is defined and in G) iff s(γ) = r(η). The set of composable elements is usually denoted by G (2) = G s,r G = {(γ, η) G 2 : s(γ) = r(η)}. It is useful to embellish this further: Denote the multiplication map by m : G (2) G, (γ, η) γ η, the inversion i : G G, γ γ 1 and the unit inclusion map u : G (0) G, x id x. This is summarized in the following exact sequence G (2) m G i G r,s G (0) u G.

Groupoids We tend to write G G (0) is a groupoid. The two arrows are the range map r : G G (0) and source map s : G G (0). Two elements γ, η G are composable (i.e. γ η is defined and in G) iff s(γ) = r(η). The set of composable elements is usually denoted by G (2) = G s,r G = {(γ, η) G 2 : s(γ) = r(η)}. It is useful to embellish this further: Denote the multiplication map by m : G (2) G, (γ, η) γ η, the inversion i : G G, γ γ 1 and the unit inclusion map u : G (0) G, x id x. This is summarized in the following exact sequence G (2) m G i G r,s G (0) u G. Another, useful definition: A groupoid is a tuple (G, G (0), G (1), m, s, r, i, u) such that certain axioms hold.

Groupoids Some examples:

Groupoids Some examples: Trivially, every space X can be viewed as a groupoid over itself (s = r = u = id, xx = x for each x X ).

Groupoids Some examples: Trivially, every space X can be viewed as a groupoid over itself (s = r = u = id, xx = x for each x X ). A group G {e} with only its unit e = e G as objects. Usual product and inverse.

Groupoids Some examples: Trivially, every space X can be viewed as a groupoid over itself (s = r = u = id, xx = x for each x X ). A group G {e} with only its unit e = e G as objects. Usual product and inverse. The pair groupoid is defined as follows. We are given a set M und we let G = M M with objects G (0) = M, source and range equal projections s = π 2, r = π 1. Composition, identities and inversion is given by (x, y) (y, z) = (x, z), u(x) = (x, x), (x, y) 1 = (y, x).

Groupoids Some examples: Trivially, every space X can be viewed as a groupoid over itself (s = r = u = id, xx = x for each x X ). A group G {e} with only its unit e = e G as objects. Usual product and inverse. The pair groupoid is defined as follows. We are given a set M und we let G = M M with objects G (0) = M, source and range equal projections s = π 2, r = π 1. Composition, identities and inversion is given by (x, y) (y, z) = (x, z), u(x) = (x, x), (x, y) 1 = (y, x). A group bundle is a bundle π : E M such that each E x = π 1 ({x}) is a group. Then G = E with s = r = π is also a groupoid with composition defined from group structure on each fiber. A (recurring) special case: The tangent bundle TM of a smooth manifold M where the fibers (T x M, +) have the additive group structure.

Deformation Groupoids The tangent groupoid Deformations Consider a C manifold M with M =. The tangent groupoids (and similar constructions) admit to deform one space into another one while preserving a smooth structure. In the case of the Atiyah-Singer index we first glue the pair groupoid on M to the tangent bundle (viewed as a bundle of additive groups (T x M, +), hence a groupoid). TM C (TM) = C 0 (T M) deform M M C (M M) = K(L 2 (M)).

Deformation Groupoids The tangent groupoid Deformations Consider a C manifold M with M =. The tangent groupoids (and similar constructions) admit to deform one space into another one while preserving a smooth structure. In the case of the Atiyah-Singer index we first glue the pair groupoid on M to the tangent bundle (viewed as a bundle of additive groups (T x M, +), hence a groupoid). TM C (TM) = C 0 (T M) deform M M Definition: As a set the tangent groupoid is written G M = M M (0, 1] TM {0}. C (M M) = K(L 2 (M)).

The tangent groupoid Groupoid structure: At least as a set this is the union of the groupoids G 1 = M M (0, 1], G 2 = TM. Here G 1 has the pair groupoid structure on M M with the trivial set groupoid (0, 1]. And TM is the union of additive groups (T x M, +) x M.

The tangent groupoid Groupoid structure: At least as a set this is the union of the groupoids G 1 = M M (0, 1], G 2 = TM. Here G 1 has the pair groupoid structure on M M with the trivial set groupoid (0, 1]. And TM is the union of additive groups (T x M, +) x M. Topology: In local charts the topology is defined as follows. For (x n, y n ) M M and t n 0 we have (x n, y n, t n ) (x, v) TM iff x n x, y n x, x n y n t n v, n. One can prove that this gives G M the structure of a smooth manifold with boundary. [The boundary is simply TM = G 2 and the interior is M M (0, 1] = G 1.]

The tangent groupoid Groupoid structure: At least as a set this is the union of the groupoids G 1 = M M (0, 1], G 2 = TM. Here G 1 has the pair groupoid structure on M M with the trivial set groupoid (0, 1]. And TM is the union of additive groups (T x M, +) x M. Topology: In local charts the topology is defined as follows. For (x n, y n ) M M and t n 0 we have (x n, y n, t n ) (x, v) TM iff x n x, y n x, x n y n t n v, n. One can prove that this gives G M the structure of a smooth manifold with boundary. [The boundary is simply TM = G 2 and the interior is M M (0, 1] = G 1.] Switching to C -algebras generated by these (smooth / Lie) groupoids we obtain an important Lemma. Lem. There is a short exact sequence of C -algebras C 0 (0, 1] K C (G M ) e 0 C 0 (T M). This yields: K j (C (G M )) K j (C 0 (T M)) = K j (T M), j = 0, 1.

The tangent groupoid Proof. The homomorphism e 0 is induced by the restriction C c (G) C c (G 2 ) coming from the evaluation at t = 0 in G M. Then ker e 0 = C (G 1 ) which yields the exact sequence C (G 1 ) C (G) e 0 C (G 2 ). Now we notice two things: C (G 2 ) = C (TM) is isomorphic to (via fiberwise applied Fourier transform) C 0 (T M). Secondly, one easily verifies C (M M) = K(L 2 (M)) (the compact operators on L 2 ). And this yields: C (G 1 ) = C c (0, 1] C c (M M) = C 0 (0, 1] K. The K-theoretic assertion follows now immediately from applying the standard six-term exact sequence to the short exact sequence we just obtained. For this we only need to know that C 0 (0, 1] K is contractible and therefore has K-theory 0.

The analytic and topological index In order to prepare for the proof of Atiyah-Singer we define the analytic index and topological index in terms of the tangent groupoid. Denote by e 1 : C (G M ) C (M M {t}) = K the morphism defined by restricting C (G M ) to M M {t}. Then we define the analytic index ind a := (e 1 ) (e 0 ) 1 : K 0 (T M) Z. The following result is important in this.

The analytic and topological index In order to prepare for the proof of Atiyah-Singer we define the analytic index and topological index in terms of the tangent groupoid. Denote by e 1 : C (G M ) C (M M {t}) = K the morphism defined by restricting C (G M ) to M M {t}. Then we define the analytic index ind a := (e 1 ) (e 0 ) 1 : K 0 (T M) Z. The following result is important in this. Thm. The above defined analytic index coincides with the usual (Fredholm) index. Proof omitted. Requires either techniques from deformation quantization or pseudodifferential operators on groupoids.

The analytic and topological index Recall the definition of the topological index. 1 Fix a C embedding j : M R N for N sufficiently large. Denote by N M the normal bundle to this inclusion. Then we can obtain an embedding TM T R N and the pushforward T N TM. With this data we fix the well known Thom-isomorphism in K-theory τ : K 0 (TM) K 0 (T N ).

The analytic and topological index Recall the definition of the topological index. 1 Fix a C embedding j : M R N for N sufficiently large. Denote by N M the normal bundle to this inclusion. Then we can obtain an embedding TM T R N and the pushforward T N TM. With this data we fix the well known Thom-isomorphism in K-theory τ : K 0 (TM) K 0 (T N ). 2 We can identify T N with a tubular neighborhood of TM in T R N and this yields an inclusion of T N in T R N as an open subset. The natural extension induces a map in K-theory which we also fix ψ : K 0 (T N ) K 0 (T R N ).

The analytic and topological index Recall the definition of the topological index. 1 Fix a C embedding j : M R N for N sufficiently large. Denote by N M the normal bundle to this inclusion. Then we can obtain an embedding TM T R N and the pushforward T N TM. With this data we fix the well known Thom-isomorphism in K-theory τ : K 0 (TM) K 0 (T N ). 2 We can identify T N with a tubular neighborhood of TM in T R N and this yields an inclusion of T N in T R N as an open subset. The natural extension induces a map in K-theory which we also fix ψ : K 0 (T N ) K 0 (T R N ). 3 Finally the space T R N = R 2N is contracted to a point which yields the isomorphism β : K 0 (R 2N ) Z.

The analytic and topological index Recall the definition of the topological index. 1 Fix a C embedding j : M R N for N sufficiently large. Denote by N M the normal bundle to this inclusion. Then we can obtain an embedding TM T R N and the pushforward T N TM. With this data we fix the well known Thom-isomorphism in K-theory τ : K 0 (TM) K 0 (T N ). 2 We can identify T N with a tubular neighborhood of TM in T R N and this yields an inclusion of T N in T R N as an open subset. The natural extension induces a map in K-theory which we also fix ψ : K 0 (T N ) K 0 (T R N ). 3 Finally the space T R N = R 2N is contracted to a point which yields the isomorphism β : K 0 (R 2N ) Z. 4 This enables us to define the topological index ind t := β ψ τ : K 0 (T M) Z.

Connes-Thom Some standard tools What do we need? A smooth groupoid G G (0) and homomorphism h : (G, ) (R N, +). A space Z = G (0) R N on which G can be made to act (via h). The associated semi-direct product groupoid G h := Z h G Z.

Connes-Thom Some standard tools What do we need? A smooth groupoid G G (0) and homomorphism h : (G, ) (R N, +). A space Z = G (0) R N on which G can be made to act (via h). The associated semi-direct product groupoid G h := Z h G Z. Secondly: orbit space, a topological space that has good properties if G h has good properties. Generally, it is the quotient space BG := Z/. [For the curious: (x, y) (y, w) : r(γ) = x, s(γ) = y, v = w + h(γ).]

Connes-Thom Some standard tools What do we need? A smooth groupoid G G (0) and homomorphism h : (G, ) (R N, +). A space Z = G (0) R N on which G can be made to act (via h). The associated semi-direct product groupoid G h := Z h G Z. Secondly: orbit space, a topological space that has good properties if G h has good properties. Generally, it is the quotient space BG := Z/. [For the curious: (x, y) (y, w) : r(γ) = x, s(γ) = y, v = w + h(γ).] Special instances of a general result in K-theory. Thm. (Elliot-Natsume-Nest) There exists a morphism in K-theory T C h : K j (C (G)) K j (C (G h )), j = 0, 1. Such that: i) T C h is natural. ii) For G = G (0) (the trivial set groupoid) we obtain T C h = B the so-called Bott morphism.

Proving Atiyah-Singer Thm. (Atiyah-Singer, 1963) We have equality of indices ind a = ind t.

Proving Atiyah-Singer Thm. (Atiyah-Singer, 1963) We have equality of indices ind a = ind t. Basic idea: Reduce the groupoids (via Morita equivalence) to spaces. Find geometric interpretation of the index. The pair groupoid M M reduces to a point. Problem: The tangent space TM is not Morita equivalent (as groupoid) to a space.

Proving Atiyah-Singer Thm. (Atiyah-Singer, 1963) We have equality of indices ind a = ind t. Basic idea: Reduce the groupoids (via Morita equivalence) to spaces. Find geometric interpretation of the index. The pair groupoid M M reduces to a point. Problem: The tangent space TM is not Morita equivalent (as groupoid) to a space. Proof. Set G = G M from now on. Write down a definition of h : G R 2N, determine G h and calculate BG. j(x) j(y) h(x, y, t) =, t > 0, t h(x, v) = j (v), v T x M, t = 0. For Z := G (0) R 2N = I M R 2N the groupoid G h := Z G Z. The source / range maps and multiplication in this groupoid are s(γ, v) = (s G (γ), v + h(γ)), r(γ, v) = (r G (γ), v) (γ, v) (η, v + h(γ)) = (γ η, v).

Proving Atiyah-Singer The unit is u(x, v) = (x, v), since h is a homomorphism hence h(x) = 0, x G (0). The inverse is given by from h(γ) + h(γ 1 ) = 0. As a set BG looks like this: (γ, v) 1 = (γ 1, v + h(γ)) BG = T N {0} R 2N (0, 1]. [The locally compact topology is similarly obtained as for the tangent groupoid.] Apply the theorem. Consider the morphism T C h : K j (C (G M )) K j (C (G h )). in this special case. If G is endowed with a so-called Haar system this induces a Haar system on the groupoid G h. Then R N acts on C (G) by automorphisms. Namely, for each character χ R 2N we have an automorphism R 2N v α v ( ): C c (G) C c (G). E.g. set for each f C c (G). α v (f )(γ) = e iv h(γ) f (γ)

Proving Atiyah-Singer This yields an isomorphism of C -algebras C (G h ) = C (G) α R 2N. Now the Connes-Thom morphism yields for G = G M T C h : K j (C (G)) K j (C (Z h G)) = K j (C (G) R 2N ). The Connes-Thom theorem now states that T C h in this case is an isomorphism. The space Z induces a so-called Morita equivalence BG M Z h G which furnishes an isomorphism in K-theory K j (C (Z h G)) = K j (BG), j = 0, 1. Inserting this in the above iso T C h and evaluating on both sites at t = 0 yields T C t=0 h : K j (C (G t=0 M ) = K j (TM) K j (BG t=0 M ) = K j (T N ). It can be proved that this map recovers the usual Thom isomorphism in the topological index. So in particular it holds that τ = T C t=0 h.

Deformation Groupoids Proving Atiyah-Singer The equality of indices is established from the naturality of the Connes-Thom morphisms. We have the commutativity of the following diagram (and subdiagrams). K 0 (C (M M {t})) T C M M K 0 (R 2N {t}) Z e 1 K 0 (C (G M )) e 0 K 0 (T M) This ends the proof of AS. e 1,h T C h K 0 (C (G h )) }{{} T C t=0 h =τ =K 0 (BG) ψ e 0,h K 0 (T N ) β K 0 (R 2N )

Outlook Let G M smooth grpd, M possibly with boundary or corners.

Outlook Let G M smooth grpd, M possibly with boundary or corners. First exact sequence yields the analytic index C (G) Ψ 0 (G) C 0 (S G) ind a : K 1 (C 0 (S G)) K 0 (C (G)).

Outlook Let G M smooth grpd, M possibly with boundary or corners. First exact sequence yields the analytic index C (G) Ψ 0 (G) C 0 (S G) ind a : K 1 (C 0 (S G)) K 0 (C (G)). Second approach general tangent groupoid technique. Let G ad = A(G) G (0, 1] G (0) [0, 1] the adiabatic groupoid (endowed with a so-called normal cone topology). Evaluations at 0 and 1 yields maps in K-theory e 0 : K j (C (G ad )) K j (A(G)), e 1 : K j (C (G ad )) K j (C (G)). We obtain the analytic index (e 0 is again an isomorphism by contractability) ind a = e 1 e 1 0 : K (A(G)) K (C (G)).

Outlook Groupoids provide a very good model for spaces with geometric singularities as demonstrated by A. Connes. In general the latter index does not yield the Fredholm index. But still in many cases it does. Cases that used to be very much intractable (e.g. manifolds with boundary). Let s list a small part of related work where groupoid techniques are used: Monthubert / Nistor: The case of manifolds with corners (approaches 1 and 2 combined), no boundary conditions. Lescure / Monthubert / Rouse: Manifolds with boundary (and corners work in progress), APS boundary conditions (approach 2). Work in progress: Manifolds with corners and embedded hypersurface plus boundary conditions of Boutet de Monvel type (cf. last status report).

References [1] A. Connes Noncommutative Geometry, Academic Press, 1995 [2] C. Debord, J.-M. Lescure Index theory and Groupoids, Preprint, arxiv:0801.3617 [3] J. M. Lescure, B. Monthubert, P. C. Rouse A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary, arxiv:1207.3514v3 [4] B. Monthubert, V. Nistor A topological index theorem for manifolds with corners, Compositio Math., Vol 148, 02, 640-668, 2012

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