The Index of Dirac Operators on Incomplete Edge Spaces

We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a"geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.


Introduction
Ever since Cheeger's celebrated study of the spectral invariants of singular spaces [19][20][21] there has been a great deal of research to extend our understanding of geometric analysis from smooth spaces. Index theory in particular has been extended to spaces with isolated conic singularities quite successfully (beyond the papers of Cheeger see, e.g., [18,27,28,35]) and was used by Bismut and Cheeger to establish their families index theorem on manifolds with boundary [12][13][14].
The fact that Bismut and Cheeger used, [12,Theorem 1.5], is that for a Dirac operator (though not any Dirac-type operator) on a space with a conic singularity, the null space of L 2 sections naturally corresponds to the null space of the Dirac operator on the manifold with boundary obtained by excising the singularity and imposing the 'Atiyah-Patodi-Singer boundary condition' [7], provided an induced Dirac operator on the link has no kernel. Indeed Cheeger points out in [19] that one can recover the Atiyah-Patodi-Singer index theorem from the index formula for spaces with conic singularities.
In this paper we consider a Dirac operator on a space with non-isolated conic singularities, also known as an 'incomplete edge space', and the Dirac operator on the manifold with boundary obtained by excising a tubular neighborhood of the singularity and imposing the Atiyah-Patodi-Singer boundary condition. Although the relation between the domains of these two Dirac operators is much more complicated than in the case of isolated conic singularities, we show that under a "geometric Witt assumption" analogous to that used by Bismut-Cheeger, the index of these operators coincide. Thus we obtain a formula for the index of the Dirac operator on the singular space as the 'adiabatic limit' of the index of the Dirac operator with Atiyah-Patodi-Singer boundary conditions. Specifically, an incomplete edge space is a stratified space X with a single singular stratum Y. In keeping with Melrose's paradigm for analysis on singular spaces (see e.g., [41,45]) we resolve X by 'blowing-up' Y and obtain a smooth manifold with boundary M, whose boundary is the total space of a fibration of smooth manifolds A (product-type) incomplete edge metric is a metric that, in a collar neighborhood of the boundary, takes the form with x a defining function for ∂M, g Y a metric on Y and g Z a family of two-tensors that restrict to a metric on each fiber of φ. Thus we see that metrically the fibers of the boundary fibration are collapsed, as they are in X.
We also replace the cotangent bundle of M by a bundle adapted to the geometry, the 'incomplete edge cotangent bundle' T * ie M , see (1.3) below. This bundle is locally spanned by forms like dx, x dz, and dy, and the main difference with the usual cotangent bundle is that the form x dz is a non-vanishing section of T * ie M all the way to ∂M. We assume that (M, g) is spin and denote a spin bundle on M by S −→ M and the associated Dirac operator by ð. The operator ð does not induce an operator on the boundary in the usual sense, due to the degeneracy of the metric there, but we do have xð ∂M = c(dx)( 1 2 dim Z + ð Z ) where ð Z is a vertical family of Dirac operators on ∂M. It turns out, as in the conic case mentioned above, and the analogous study of the signature operator in [2,3], that much of the functional analytic behavior of ð is tied to that of ð Z . Indeed, in Section 2.4 below we prove the following theorem.
The spin bundle admits the standard Z/2Z grading into even and odd spinors S = S + ⊕ S − , and thus we have the chirality spaces D ± = D ∩ L 2 (M ; S ± ) and the restriction of the Dirac operator satisfies ð : D + −→ L 2 (M ; S − ). This map is Fredholm and our main result is an explicit formula for its index.
The metric (1) naturally defines a bundle metric on T * ie M, non-degenerate at ∂M, and the Levi-Civita connection of g naturally defines a connection ∇ on T * ie M. Our index formula involves the transgression of a characteristic class between two related connections. The restriction of T ie M to ∂M can be identified with N M ∂M ⊕ T ∂M/Y ⊕ φ * Y. Let where pt stands for 'product', are connections on N M ∂M ⊕ T ∂M/Y −→ ∂M.
Main Theorem. Let X be stratified space with a single singular stratum endowed with an incomplete edge metric g and let M be its resolution. If ð is a Dirac operator associated to a spin bundle S −→ M and ð satisfies the geometric Witt condition (2), then where A denotes the A-genus, T A(∇ v+ , ∇ pt ) denotes the transgression form of the A genus associated to the connections (3), and η the η-form of Bismut-Cheeger [11].
The simplest setting of incomplete edge spaces occurs when Z is a sphere, as then X is a smooth manifold and the singularity at Y is entirely in the metric. Atiyah and LeBrun have recently studied the case where Z = S 1 and X is four-dimensional, so that Y is an embedded surface, and the metric g asymptotically has the form The cone angle 2πβ is assumed to be constant along Y. In [6] they find formulas for the signature and the Euler characteristic of X in terms of the curvature of this incomplete edge metric. In this setting, assuming that g is Einstein and self-dual or anti-self-dual, Lock and Viaclovsky [36] compute the index of the 'anti-self-dual deformation complex'. Using work of Dai [24] and Dai-Zhang [26], we recover the Atiyah-LeBrun formula for the signature (see Theorem 5.2 below) and show that our formula for the index of the Dirac operator (4) simplifies substantially in this case.
Corollary 2. If ð is a Dirac operator on a smooth four-dimensional manifold X, associated to an incomplete edge metric with constant cone angle 2πβ ≤ 2π along an embedded surface Y, then its index is given by where [Y ] 2 is the self-intersection number of Y in X.
The formulas in (4) and (5), and indeed our proof, are obtained by taking the limit of the index formula for the Dirac operators on the manifolds with boundary M ε = {x ≥ ε}, so in particular the contribution from the singular stratum Y is the adiabatic limit [11,24,52] of the η-invariant from the celebrated classical theorem of Atiyah, Patodi, and Singer [7], which we review in Section 5. It is important to note that the analogous statement for an arbitrary Dirac-type operator is false and the general index formula requires an extra contribution from the singularity. We will return to this in a subsequent publication.
One very interesting aspect of the spin Dirac operator is its close relation to the existence of positive scalar curvature metrics. Most directly, the Lichnerowicz formula shows that the index of the Dirac operator is an obstruction to the existence of such a metric. This is still true among metrics with incomplete edge singularities. Analogously to the results of Chou for conic singularities [22] we prove the following theorem in §6.
Theorem 3. Let (M, g) be a spin space with an incomplete edge metric. a) If the scalar curvature of g is non-negative in a neighborhood of ∂M then the "geometric Witt assumption" (2) holds. b) If the scalar curvature of g is non-negative on all of M, and positive somewhere, then Ind(ð) = 0.
Notice that the first part of this theorem, as indeed Theorem 1, show that our geometric Witt assumption is a natural assumption on ð. Now let us indicate in more detail how these theorems are proved. For convenience we work throughout with a product-type incomplete edge metric as described above, but removing this assumption would only result in slightly more intricate computations below. The proof of Theorem 1 follows the arguments employed in [2,3] to prove the analogous result for the signature operator. Thus we start with the two canonical closed extensions of ð from C ∞ comp (M ), namely D max := u ∈ L 2 (M ; S) : ðu ∈ L 2 (M ; S) , where the convergence in the second definition is in L 2 (M ; S), and we show that under Assumption (2), these domains coincide Since ð is a symmetric operator, this shows that it is essentially self-adjoint. One difference between the case of isolated conic singularities (dim Y = 0) and the general incomplete edge case is that in the former, even if Assumption (2) does not hold, D max /D min is a finite dimensional space. In contrast, when dim Y > 0, this space is generally infinitedimensional.
We prove (7) by constructing a parametrix Q for ð in Section 2. From the mapping properties of Q, we deduce both that ð is essentially self-adjoint, and that it is a Fredholm operator from the domain of its unique self-adjoint extension to L 2 . The relationship between the mapping properties of Q and the stated conclusions can be seen largely through (2.31) below, which states that the maximal domain has 'extra' vanishing, i.e. sections in D max lie in weighted spaces x δ L 2 with weight δ higher than generically expected. This shows that the inclusion of the domain into L 2 is a compact operator, which in particular gives that the kernel of ð on the maximal domain is finite dimensional.
Once this is established we give a precise description of the Schwartz kernel of the generalized inverse Q of ð using the technology of [37,39]. In Section 3, we use Q and standard methods from layer potentials to construct a family of pseudodifferential projectors is Fredholm and has the same index as (ð, D). This domain is constructed so that the boundary values coincide with boundary values of 'ð−harmonic' L 2 -sections over the excised neighborhood of the singularity, M − M ε .
To compute this index, we consider the operators where π AP S,ε , is the projection onto the positive spectrum of ð ∂Mε . From [7] we know that these are Fredholm operators and where L ε is a local integral over ∂M ε compensating for the fact that the metric is not of product-type at ∂M ε . These domains depend fundamentally on ε. Not only does D AP S,ε vary as ε → 0, it does not limit to a fixed subspace of L 2 (∂M ) with any natural metric.
(More precisely, the boundary value projectors π AP S,ε which define the boundary condition do not converge in norm.) Through a semiclassical analysis, which we carry out using the adiabatic calculus of Mazzeo-Melrose [38], we show that the projections E ε and π AP S,ε are homotopic for small enough ε, with a homotopy through operators with the same principal symbol. The adiabatic calculus technology boils this down to an explicit analysis of modified Bessel functions, which we carry out in the appendix. Then we can appeal to arguments from Booss-Bavnbek-Wojciechowski [16] to see that the two boundary value problems have the same index. Having shown that the index of (ð, D) is equal to the adiabatic limit of the index formula (8), the Main Theorem follows as shown in Section 5.
1.1. Incomplete edge metrics and their connections. Let M be the interior of a compact manifold with boundary. Assume that ∂M = N participates in a fiber bundle Let X be the singular space obtained from M by collapsing the fibers of the fibration φ. If we want to understand the differential forms on X while working on M, it is natural to restrict our attention to Following Melrose's approach to analysis on singular spaces [44] let be the vector bundle whose space of sections is (1.2). We call T * ie M the 'incomplete edge cotangent bundle', and its dual bundle T ie M, the 'incomplete edge tangent bundle'. (Note that T ie M is simply a rescaled bundle of the (complete) 'edge tangent bundle' of Mazzeo [37].) The incomplete edge tangent bundle is, over M, canonically isomorphic to T M, but its extension to M is not canonically isomorphic to T M (though they are of course isomorphic bundles).
Let x be a boundary defining function on M, meaning a smooth non-negative function x ∈ C ∞ (M ; [0, ∞)) such that {x = 0} = N and |dx| has no zeroes on N. We will typically work in local coordinates x, y, z (1.4) where y are coordinates along Y and z are coordinates along Z. In local coordinates the sections of T * ie M are spanned by dx, x dz, dy where dz denotes a φ-vertical one form and dy a φ-horizontal one form. The crucial fact is that x dz vanishes at N as a section of T * M , but it does not vanish at N as a section of T * ie M because the 'x' is here part of the basis element and not a coefficient. Similarly, in local coordinates the sections of T ie M are spanned by x ∂ z , ∂ y , and, in contrast to T M , the vector field 1 x ∂ z is non-degenerate at N as a section of T ie M.
Next consider a metric on M that reflects the collapse of the fibers of φ. Let C be a collar neighborhood of N in M compatible with x, C ∼ = [0, 1] x × N. Fix a splitting A 'product-type incomplete edge metric' is a Riemannian metric on M that, for some choice of collar neighborhood and splitting, has the form where g Z + φ * g Y is a submersion metric for φ independent of x. Note that this metric naturally induces a bundle metric on T ie M with the advantage that it extends non-degenerately to M . We will consider this as a metric on T ie M from now on. (A general incomplete edge metric is simply a bundle metric on T ie M −→ M .) To describe the asymptotics of the Levi-Civita connection of g ie , let us start by recalling the behavior of the Levi-Civita connection of a submersion metric. Endow N = ∂M with a submersion metric of the form g N = φ * g Y + g Z . Given a vector field U on Y, let us denote its horizontal lift to N by U . Also let us denote the projections onto each summand by The connection ∇ N differs from the connections ∇ Y on the base and the connections ∇ N/Y on the fibers through two tensors. The second fundamental form of the fibers is defined by and the curvature of the fibration is defined by The behavior of the Levi-Civita connection (cf. [32,Proposition 13]) is then summed up in the table: We want a similar description of the Levi-Civita connection of an incomplete edge metric. The splitting of the tangent bundle of C induces a splitting in terms of which a convenient choice of vector fields is ∂ x , 1 x V, U where V denotes a vertical vector field at {x = 0} extended trivially to C and U denotes a vector field on Y, lifted to ∂M and then extended trivially to C . Note that, with respect to g ie , these three types of vector fields are orthogonal, and that their commutators satisfy . We define an operator ∇ on sections of the ie-bundle through the Koszul formula for the Levi-Civita connection where W 1 and W 2 are ie-vector fields.
and otherwise We point out a few consequences of these computations. First note that the defines a connection on the incomplete edge tangent bundle. Also note that this connection asymptotically preserves the splitting of T ie C into two bundles in that if W 1 , W 2 ∈ V ie are sections of the two different summands then In fact, let us denote the projections onto each summand of (1.7) by Denote by j ε : {x = ε} → C (1.8) the inclusion, and identify {x = ε} with N = {x = 0}, note that the pull-back connections j * ε ∇ v+ and j * ε ∇ h are independent of ε and In terms of the local connection one-form ω and the splitting (1.7), we have where P V + ω is the projection onto the dual bundle of ∂ x ⊕ 1 x T N/Y , P H ω is the projection onto the dual bundle of φ * T Y, and the forms ω N/Y , ω S , ω Y , ω v+ are defined by these equations. Finally, consider the curvature R ie of ∇. If W 1 , W 2 ∈ V ie are sections of two different summands of (1.7) and W 3 , We will be interested in the curvature along the level sets of x. Schematically, if Ω denotes the End(T ie M )-valued two-form corresponding to the curvature of ∇, then with respect to the splitting (1.7) we have where Ω v+ is the tangential curvature associated to ω N/Y + ω S and Ω Y is the curvature associated to ω Y , and analogously to (1.9), (1.11) Following [12] and [32], it will be convenient to use the block-diagonal connection ∇ on T ie M from the splitting (1.6). Thus satisfies ∇∂ x = 0, ∇ ∂x = 0, and The connection ∇ is a metric connection and preserves the splitting (1.7).

Clifford bundles and Clifford actions.
The incomplete edge Clifford bundle, denoted Cl ie (M, g), is the bundle obtained by taking the Clifford algebra of each fiber of T ie M . Concretely, This is a smooth vector bundle on all of M . We assume that M is spin and fix a spin bundle S −→ M. Denote Clifford multiplication by We denote the connection induced on S by the Levi-Civita connection ∇ by the same symbol. Let ð denote the corresponding Dirac operator.
denote a local orthonormal frame consistent with the splitting (1.6). In terms of this frame and the connection ∇ from (1.12), the Dirac operator ð decomposes as Proof. Consider the difference of connections (on the tangent bundle) From [15] we have and otherwise which establishes (1.15).
1.3. The APS boundary projection. We now define the APS boundary condition discussed in the introduction. We will make use of a simplified coordinate system near the boundary of M , namely, let (x, x ) be coordinates near a point on ∂M for which x ∈ R n−1 are coordinates on ∂M and x is the same fixed boundary defining function used in (1.4). For the cutoff manifold M ε = {x ≥ ε}, consider the differential operator on sections of S over ∂M ε defined by choosing any orthonormal frame e p , p = 1, . . . n − 1 of the distribution of the tangent bundle orthogonal to ∂ x and setting where ∇ is the connection from (1.12). The operator ð ε is defined independently of the choice of frame, so we may take frames as in (1.14) to obtain We refer to ð ε below as the tangential operator, since for every ε it acts tangentially along the boundary ∂M ε . The operator ð ε is self-adjoint on L 2 (∂M ε , S).
We denote the dual coordinates on T * x,x M by (ξ, ξ ). Using the identification of T * M with T M induced by the metric g, the principal symbol of ð is given by Note that using coordinates as in (1.19), where π ε,±, ξ (x ) are orthogonal projections onto ± eigenspaces of σ( ð ε )(x , ξ ). We will define a boundary condition for ð on the cutoff manifolds M ε , where V −,ε is the direct sum of eigenspaces of ð ε with negative eigenvalues. We recall basic facts about π AP S,ε . 7,49]). For fixed ε, the operator π AP S,ε is a pseudodifferential operator of order 0, i.e. π AP S,ε ∈ Ψ 0 (∂M ε ; S). Its principal symbol satisfies Consider the domains for ð on L 2 (M ε ; S) defined as follows In one of the main results of this paper, we will show that a Dirac operator satisfying the geometric Witt assumption (2) has a unique closed extension. Denoting this domain by D and its restriction to positive spinors by D + we will show that, for ε > 0 sufficiently small, ). Indeed, this will follow from Theorems 3.1 and 4.1 below.

Mapping properties of ð
In this section we will use the results and techniques in [2,3] to prove Theorem 1. We proceed by constructing a parametrix for ð and analyzing the mapping properties of this parametrix.
Let D denote the domain of the unique self-adjoint extention of ð. At the end of this section, we analyze the structure of the generalized inverse Q for ð, that is, the map where π ker is L 2 −orthogonal projection onto the kernel of ð. Here the adjoint Q * is taken with respect to the pairing defined for sections φ, ψ by where G is the Hermitian inner product on S.

2.1.
The "geometric Witt condition". The proof of Theorem 1 relies on an assumption on an induced family of Dirac operators on the fiber Z which we describe now. By Lemma 1.1, on a collar neighborhood of the boundary, The operator ð Z y defines a self-adjoint operator on the fiber over y ∈ Y in the boundary fibration N φ −−→ Y acting on sections of the restriction of the spin bundle S y . We will assume the following "geometric Witt condition" discussed in the introduction.
Review of edge and incomplete edge operators. A vector field on M is an 'edge vector field' if its restriction to N = ∂M is tangent to the fibers of φ [37]. A differential operator is an edge differential operator if in every coordinate chart it can be written as a polynomial in edge vector fields. Thus if E and F are vector bundles over M, we say that P is an m th order edge differential operator between sections of E and F, denoted P ∈ Diff m e (M ; E, F ), if in local coordinates we have where α denotes a multi-index (α 1 , . . . , α b ) with |α| = α 1 + . . . + α b and similarly for γ = (γ 1 , . . . , γ f ), and each a j,α,γ (x, y, z) is a local section of hom(E, F ). A differential operator P is an 'incomplete edge differential operator' of order m if P = x m P is an edge differential operator of order m. Thus, symbolically, and in local coordinates The (incomplete edge) principal symbol of P is defined on the incomplete edge cotangent bundle, In local coordinates it is given by We say that P is elliptic if this symbol is invertible whenever (ξ, η, ζ) = 0.
Remark 2.2. Clearly ellipticity of the incomplete edge symbol is equivalent to ellipticity of the usual principal symbol of a differential operator. The advantage of the former is that it will be uniformly elliptic over M , whereas the latter typically will not.
Lemma 2.3. The Dirac operator ð on an incomplete edge space is an elliptic incomplete edge differential operator of order 1, i.e. is an elliptic element of Diff 1 ie (M ; S). In particular, xð is an elliptic element of Diff 1 e (M ; S).

2.3.
Parametrix of xð on weighted edge spaces. Lemma 2.3 shows that xð is an elliptic edge operator. By the theory of edge operators [37], this implies that xð is a bounded operator between appropriate weighted Sobolev spaces, whose definition we now recall.
In particular, u ∈ H 1 e (M ; S) if and only if, for any edge vector field V ∈ C ∞ (M ; T e M ), ∇ V u ∈ L 2 (M ; S). The weighted edge Sobolev spaces are defined by is bounded for all δ ∈ R, k ∈ N, in fact for k ∈ R by interpolation. We will prove the following In fact we will need more than Proposition 2.4; the proof of the Main Theorem requires a detailed understanding of the structure of parametrices for xð. To understand these, we must recall of edge double space M 2 e , depicted heuristically in Figure 3 below, and edge pseudodifferential operators, defined in [37] with background material in [42]. The edge double space M 2 e is a manifold with corners, obtained by radial blowup of where the notation is that in [42]. Here where φ : N −→ Y is fiber bundle projection onto Y. Whereas M × M is a manifold with corners with two boundary hypersurfaces, M 2 e has a third boundary hypersurface introduced by the blowup.
Let ff be the boundary hypersurface of M 2 e introduced by the blowup. Furthermore we have a blowdown map.
The edge front face, ff, is the radial compactification of the total space of a fiber bundle This bundle is the fiber product of two copies of ∂M and the tangent bundle T Y . Choosing local coordinates (x, y, z) as in (1.4), and our fixed bdf x, and letting (x, y, z, x , y , z ) denote coordinates on M × M , the functions define coordinates near ff in the set 0 ≤ σ < ∞, and in these coordinates x is a boundary defining function for ff, meaning that {x = 0} coincides with ff on 0 ≤ σ < ∞, and x has non-vanishing differential on ff. When x = 0, σ gives coordinates on the R + fiber, Y on the R b fiber, and z, z on the Z 2 fiber. Below we will also use polar coordinates near ff. These have the advantage that they are defined on open neighborhoods of sets in ff which lie over open sets V in the base Y . With (x, y, z) as above, let so (ρ, φ, y , z, z ) form polar coordinates (in the sense that |φ| 2 = 1) near the lift of V to ff and in the domain of validity of y, z.
We now define the calculus of edge pseudodifferential operators with bounds, which is similar to the large calculus of pseudodifferential edge operators defined in [37]. Thus, Ψ m e,bnd (M ; S) will denote the set of operators A mapping C ∞ comp (M ; S) to distributional sections D (M ; S), whose Schwartz kernels have the following structure. Let End(S) denote the bundle over M × M whose fiber at (p, q) is End(S q ; S p ). The Schwartz kernel of A, K A is a distributional section of the bundle End(S) over M × M satisfying that for a section φ ∈ C ∞ comp (M ; S), where dV ol g is the volume form of an incomplete edge metric g asymptotically of the form (1.5). Moreover, , ∆ e ; β * End(S)), K 1 is supported near ∆ e , and K 2 ∈ C ∞ (M int ; End(S)) satisfies the bounds where ρ lf , ρ rf , and ρ ff are boundary defining functions for lf, rf, and ff respectively, and the bound is in the norm on End(S) over M × M, see [37] for details. Since the bounds a and b in (2.10) will be of some importance, we let denote the subspace of Ψ m e,bnd (M ; S) of pseudodifferential edge operators whose Schwartz kernels satisfy (2.10) with bounds a and b.
The bounds in (2.10) determine the mapping properties of A on weighted Sobolev spaces. From [37, Theorem 3.25], we have Remark 2.7. In Mazzeo's paper [37] the convention used to describe the weights (orders of vanishing) of the Schwartz kernels of elements in Ψ m e is slightly different from ours. There one chooses a half-density µ on M which looks like √ dxdydz near ∂M . The choice of µ gives an isomorphism between the sections of S and the sections of S ⊗ Ω 1/2 (M ) where Ω 1/2 (M ) is the half-density bundle of M (simply by multiplying by µ), and the Schwartz kernel of an edge pseudodifferential operator, A, in this context is the section κ A of End(S) ⊗ Ω 1/2 (M 2 e ) with the property that (2.12) One nice feature of (2.12) is that κ A is smooth (away from the diagonal) down to ff. With our convention in (2.9), it is singular of order −f due to the factors of x in the volume form of g.
Given an elliptic edge operator P ∈ Diff m e (M ; S), to construct a parametrix for P one must study two models for P , the indicial family I y ( P , ζ) and the normal operator N ( P ) y .
First we discuss the indicial operator. For each y in the base Y , the indicial family I y ( P , ζ) is an elliptic operator-valued function on C obtained by taking the Mellin transform (see [37,Sect. 2]) of the leading order part of P in x. By (2.2), the leading order part of Remark 2.8. The shift by f /2 + 1/2 in (2.14) comes from the following considerations. We want to understand the mapping properties of xð on L 2 (M ; S) with the natural measure dV ol g given by the incomplete edge metric g. On the other hand, the values of iζ for which (2.13) fails to be invertible give information about the mapping properties of xð on the Sobolev spaces defined with respect to b−measure In particular, the Fredholm property in Proposition 2.4 is equivalent to xð being a Fredholm map from the space where the Sobolev spaces are now defined with respect to the b−measure. Alternatively, as in [3] we could define P = x −f /2−1/2 (xð)x f /2+1/2 take the Mellin transform and use the values of iζ as the indicial roots, but we would get the same answer as in (2.14). Now we discuss the normal operator N ( P ). Letting P act on the left on M × M (i.e., in the coordinates (x, y, z) in (2.8)), P restricts to an operator on ff acting on the fibers of ff and parametrized by the base Y . That is, for every y ∈ Y we have an operator N ( P ) y acting on the fiber ff y over y.
To obtain an expression for N ( P ) y in coordinates, write where a i,α,β ∈ C ∞ (M ; End S), and use the projective coordinates in (2.8) The mapping properties of P are deduced from mapping properties of the N ( P ) y . In particular, to prove Proposition 2.4 we will need Lemma 2.10 below, which shows that the Fourier transform of N y (xð) is invertible on certain spaces. Edge pseudodifferential operators also admit normal operators. Given A ∈ Ψ m e,bds (M ; S), the restriction N (A) := ρ f ff K A | ff is well defined, and in fact meaning that N (A) = κ 1 + κ 2 where κ 1 is a distribution on ff conormal to ∆ e ∩ ff of order m and κ 2 is a smooth function on ff int with bounds in (2.10) (with the point p restricted to ff). Using (2.2) and the projective coordinates in (2.8), and letting c ν denote the operator induced by c(∂ x ) on the bundle S y , the restriction of the spin bundle to the fiber over y, the normal operator of xð satisfies where ð Y can be written locally in terms of the limiting base metric h y = g Y | y in (1.5) as The operator N (xð) acts on sections of S y . The remainder of this subsection consists in establishing the following theorem. for any k. Here Π ker,δ (resp. Π coker,δ ,) is x δ L 2 (M ; S) orthogonal projection onto the kernel (resp. cokernel) of xð. The Schwartz kernels satisfy the following bounds.
In particular this establishes that xð : We will see that Theorem 2.9 can be deduced from the work of Mazzeo in [37] and its modifications in [2,3,40]. In order to see that the results of those papers apply, we must prove that the normal operator N (xð) is invertible in a suitable sense. Taking the Fourier transform of N y (xð) in (2.15) in the Y variable gives and for each y, one considers the mapping of weighted edge Sobolev spaces defined by picking a positive cutoff function φ : [0, ∞) σ −→ R that is 1 near zero and 0 near infinity and letting where, in terms of k y = g N/Y y , i.e. it is the standard Sobolev space on [0, ∞) σ × Z whose sections take values in the bundle S restricted to the boundary over the base point y. Consider Proof of Lemma 2.10. Given y ∈ Y and η ∈ T y Y with η = 0, writing η = η/ |η|, we have (ic( η)) 2 = id. Furthermore, so these operators are simultaneously diagonalizable on L 2 (Z; S 0,y , k 0 ). Thus for each y and η we have an orthonormal basis Note that the existence of such an orthonormal basis is automatic from the existence of any simultaneous diagonalization φ i . Indeed, since c ν is the operator on the bundle S y induced by c(∂ x ), we have ic( η)c ν φ i = −c ν φ i , so we can reindex to obtain φ i,± satisfying the two equations on the right in (2.20). But then since c ν ð Z y = −ð Z y c ν , the first equation in (2.20) follows automatically. Using the φ i,± , we define subspaces of H r,δ,l by where H r,δ,l (σ f dσ) is defined as in (2.19) in the case that Z is a single point. In particular, for all η and i, Note that multiplication by c ν defines a unitary isomorphism of W r,δ,l i . We consider the map L(y, η) on each space individually. We claim that Thus, writing elements in W r,δ,l i as vector valued functions (a, b) T = aφ i,+ + bφ i,− , we see that L(y, η) indeed maps W r,δ,l i to W r−1,δ,l i , acting as the matrix From this, one checks that that the kernel of L(y, η) can be written using separation of variables as superpositions of sections given, again in terms of the φ i,± by where I ν (z) and K ν (z) are the modified Bessel functions [1]. By the asymptotic formulas [1, 9.7], only the sections involving the K µ,η are tempered distributions, and since K ν (z) ∼ z −ν as z → 0 for ν > 0, by Assumption 2.1, On the other hand, the ordinary differential operator in (2.23) admits an explicit right inverse if δ < 1. Specifically, consider the matrix (2.24) Then the operator Q y,µ on W r−1,δ,l i defined by acting on elements a(σ)φ i, which are equations (9.6.15) and (9.6.26) from [1].) That Q y,µ : W r,δ,l i −→ W r−1,δ,l i is bounded for δ < 1 can be seen using [37], but one can also check it directly using the density of polyhomogeneous functions. Invertibility on each W r,δ,l i gives invertibility on H r,δ,l . This proves Lemma 2.10. Theorem 2.9 then follows from [37] as explained in [3, Sect. 2] using the invertibility of the normal operator from Lemma 2.10. In the notation of those papers, one has the numbers δ := inf δ : L(y, η) : σ δ L 2 (dσdV ol z ; S y ) −→ L 2 (dσdV ol z ; S y ) is injective for all y. δ := sup δ : L(y, η) : σ δ L 2 (dσdV ol z ; S y ) −→ L 2 (dσdV ol z ; S y ) is surjective for all y.

Proof of Theorem 1 and the generalized inverse of ð.
In this section we will prove Theorem 1 and describe the properties of the integral kernel of the generalized inverse of ð. We start by recalling the statement for the convenience of the reader: Proof. The proof of Theorem 1 will follow from combining various elements of [2,3]. The first and main step is the construction of a left parametrix for the map ð : D max −→ L 2 (M ; S), where D max is the maximal domain defined in (6).
Consider Q 1 from (2.16) and set Q 1 x = Q 1 . Then by (2.16) where both sides of this equation are thought of as maps of x δ L 2 e (M ; S). We claim that in fact equation (2.30) holds not only on x δ L 2 e (M ; S), but on the maximal domain D max defined in (6). This follows from [3, Lemma 2.7] as follows. In the notation of that paper, L = ð and P = xð. Taking (again, in the notation of that paper) E(L) to be D max , by [ Thus Id = Q 1 ð + Π ker,δ on D max , and since the right hand side is bounded L 2 (M ; S) to x δ L 2 , for any δ ∈ (0, 1), we have in particular for any δ < 1, D max ⊂ H 1 loc ∩ x δ L 2 (M, S) which is a compact subset of L 2 (M ; S). It then follows from Gil-Mendoza [29] (see [2,Prop. 5.11]) that D max ⊂ D min , i.e. that ð is essentially self-adjoint. By a standard argument, e.g. [43,Lemma 4.2], the compactness of D max implies that ð has finite dimensional kernel and closed range. But the self-adjointness of ð on D now implies that ð has finite dimensional cokernel, so ð is self-adjoint and Fredholm. where π ker is L 2 orthogonal projection onto the kernel of ð in D with respect to the pairing induced by the Hermitian inner product on S. To be precise, if {φ i }, i = 1, . . . , N is an orthonormal basis for the kernel of ð on D, then π ker has Schwartz kernel From (2.31), we see that π ker ∈ Ψ −∞ e (M ; S; a, b). The properties of the integral kernel of Q can be deduced from those of the parametrices Q i in (2.16). Indeed, setting Q = Qx −1 , we see that Q(xð) = Id − π ker and (xð) Q = Id − x · π ker · x −1 . Applying the argument from [37,Sect. 4], specifically equations (4.24) and (4.25) there, shows that Q ∈ Ψ −1 e (M ; S; a, b) for the same a, b as in (2.16), and in particular that N ( Q) = N ( Q i ). In particular, by Theorem 2.9, we have the bounds K Q (p) = O(ρ a lf ) as p → lf and K Q (p) = O(ρ b rf ) as p → rf where a > δ − f /2 − 1/2 and b > −δ − f /2 + 1/2, 0 < δ < 1, and again the bounds hold for K Q as a section of End(S) over M × M . Finally, by self-adjointness of Q, we have that By (2.32), the bound at rf, which one approaches in particular if w remains fixed in the interior of M and w goes to the boundary, gives a bound at lf. Thus we obtain the following.
where a > −δ − f /2 + 1/2 for any δ > 0 and c is an arbitrary small positive number. Essentially, D ε consists of sections over M ε whose boundary values correspond with the boundary values of an L 2 harmonic section over M − M ε . We also have the chirality spaces

Boundary values and boundary value projectors
where S ± are the chirality subbundles of even and odd spinors. In this section we will prove the following. ). In the process of proving Theorem 3.1, we will construct a family of boundary value projectors π ε which define D ε in the sense of Claim 3.2 below, and whose microlocal structure we will use in Section 4 to relate the index of ð on M ε with domain D ε to the index of ð on M ε with the APS boundary condition, see Theorem 4.1.
3.1. Boundary value projector for D ε . As already mentioned, the main tool for proving Theorem 3.1 and also for proving Theorem 4.1 below will be to express the boundary condition in the definition of D ε in (3.1) in terms of a pseudodifferential projection over ∂M ε . We discuss the construction of this projection now.
First we claim that the invertible double construction of [16,Chapter 9]   We describe the construction of this "invertible double" for the convenience of the reader, though it is essentially identical to that in [16,Chapter 9], the only difference being that we must introduce a product type boundary while they have one to begin with. Choosing any point p ∈ M ε0 , let D 1 , D 2 denote open discs around p with p ∈ D 1 D 2 and D 2 ∩ (M − M ε0 ) = ∅. We can identify the annulus D 2 − D 1 with [1, 2) s × S d−1 by a diffeomorphism and the metric g is homotopic to a product metric ds 2 + |dx| 2 where x is the standard coordinate on B d−1 . Furthermore, the connection can be deformed so that the induced Dirac operator ð is of product type on the annulus (see equation 9.4 in [16]). Call the bundle over N 1 := M − D 1 thus obtained S. Letting N 2 := −N 1 , the same incomplete edge space with the opposite orientation, let M = N 1 N 2 / {s = 1} and consider the vector bundle S over M obtained by taking S + over N 1 and S − over N 2 and identifying the two bundles over D 2 using Clifford multiplication by ∂ s . The resulting Dirac operator, which we still denote by ð , is seen to be invertible on M by the symmetry and unique continuation argument in Lemma 9.2 of [16].
We will now work on a neighborhood in M − M ε of ∂M (or equivalently of the singular stratum Y ), so we drop the distinction between M and M . Using notation as in (3.2), and given f ∈ C ∞ (∂M ε ; S), define the harmonic extension where K Q is the Schwartz kernel of Q (see (2.9)), and c ν = c(∂ x ). Since ð Ext ε f (w) = 0 for w ∈ ∂M ε . Recall Green's formula for Dirac operators; specifically, for a smoothly bounded region Ω with normal vector ∂ ν , Green's formula for sections u satisfying ðu ≡ 0 in M − M ε gives that for u ∈ H loc,ε , The identity in (3.6) is obtained by integrating by parts in and using (3.4). In fact, as we will see in the proof of Claim 3.2 below, (3.7), and thus (3.6), hold for all u ∈ H loc,ε . It follows from (3.3) and (3.6) that, for ðu = 0 satisfying (3.7), We will show that the E ε define the domains D ε as follows.
Assuming Claim 3.2 for the moment, we prove Theorem 3.1.
Proof of Theorem 3.1 assuming Claim 3.2. The main use of Claim 3.2 in this context (it will be used again in Theorem 4.1 ) is to show that the map is self-adjoint on L 2 (M ε ; S) and Fredholm. The Fredholm property follows from the principal symbol equality (3.11), since from [16] any projection in Ψ 0 (∂M ε , S) with principal symbol equal to that of the Atiyah-Patodi-Singer boundary projection defines a Fredholm problem. To see that it is self-adjoint, note that from (3.5) the adjoint boundary condition is D * ε = {φ : g, cl ν φ| ∂Mε ∂Mε = 0 for all g with (Id − E ε )g = 0} . Again by (3.5), for any v ∈ H loc,ε , v| ∂Mε ∈ D * ε . Thus D ε ⊂ D * ε , and it remains to show that D * ε ⊂ D ε . Let φ ∈ D * ε , and set f := φ| ∂Mε . We want to show that (I − E ε )f = 0, or equivalently Since f, c ν g = − c ν f, g , by (3.5) we have f, c ν g = 0 for every g ∈ Ran E, and thus (3.13) will hold if (I − E * ε )g ∈ Ran c ν E. In fact, we claim that I − E * ε = −c ν Ec ν . To see that his holds, note that by Claim 3.2 and self-adjointness of Q , specifically (2.32), which proves self-adjointness. Now that we know that (3.12) is self-adjoint, we proceed as follows. We claim that for ε > 0 sufficiently small, the map is well defined and an isomorphism. It is well defined since by definition any section φ ∈ ker(ð : D −→ L 2 (M ; S)) satisfies that φ = φ| Mε ∈ D ε . It is injective by unique continuation. For surjectivity, note that for any element φ ∈ ker(ð : D ε −→ L 2 (M ε ; S)), by definition there is a u ∈ H loc,ε such that u| ∂Mε = φ| ∂Mε . It follows that is in H 1 and satisfies ð φ = 0 on all of M , i.e. φ ∈ ker(ð : D −→ L 2 (M ; S)). Since the full operator ð on D is self-adjoint, and since the operator in (3.12) is also, the cokernels of both maps are equal to the respective kernels. Restricting ð to a map from sections of S + to sections of S − gives the theorem.
This completes the proof.
Thus to prove Theorem 3.1 it remains to prove Claim 3.2, which we proceed to do Proof of Claim 3.2. We begin by proving (3.10). It is a standard fact (see [51,Sect. 7.11]) that E ε ∈ Ψ 0 (∂M ε ; S). Obviously, where the last containment refers to the Schwartz kernel of A. We claim that A = 1 2 id and B ε ∈ Ψ 0 (∂M ε ; S).
Using that Q has principal symbol σ(Q ) = σ(ð) −1 we can write Q in local coordinates w as Given a bump function χ supported near w 0 ∈ ∂M ε , let Q χ := χQ χ and define the distributions where, as in (2.9), K Q χ denotes the Schwartz kernel of Q χ . The distribution K 2 is that of a pseudodifferential operator of order −2 on M , and it follows from the theory of homogeneous distributions (see [51,Chapter 7]) that the distribution K 2 restricts to ∂M ε to be the Schwartz kernel of a pseudodifferential operator of order −1. The distribution K 1 is that of a pseudodifferential operator on M of order −1. It is smooth in x with values in homogeneous distributions in x− x of order −n+1, and it follows that the restriction of the Schwartz kernel K 1 (w, w ) to ∂M ε gives a pseudodifferential operator of order zero. Letting B ε in (3.14) be the operator defined by the restriction of K 1 to ∂M ε , we have that B ε is in Ψ 0 (∂M ε ; S) and it remains to calculate A. Choosing coordinates of the form w = (x, x ) and w = ( x, x ) of the form in (1.19) and such that at the fixed value w 0 = (ε, x 0 ) ∈ ∂M ε the metric satisfies g(x) = id, it follows (see [4]) that the Schwartz kernel of B in (3.15) satisfies where ω n−1 is the volume of the unit sphere S n−1 . If we let B(x , x ) = B(0, x , 0, x ), then near x 0 This proves that A = 1/2. The principal symbol of E ε (again see [51,Sect. 7.11]) is given by the integral where in the third line we used the residue theorem. Now recall that the term −ic(∂ x )c( ξ · ∂ x )) is precisely the endomorphism appearing in (1.20), so where the projections are those in (1.20). Thus, Theorem 1.2 implies the desired formula for the principal symbl of E ε , (3.11).
To finish the claim, we must show the equivalence of domains in (3.9). We first show that for any u ∈ H loc,ε , the formula in (3.6) holds. This will show that any f ∈ H 1/2 (∂M ε ; S) with f = u| ∂Mε for some u ∈ H loc,ε satisfies (Id − E ε )f = 0, i.e., that Thus we must show that (3.7) holds for u ∈ H loc,ε . For such u, we claim that for some δ > 0, as ε → 0 To see this, note first that u ∈ x 1−δ H 1 e (M − M ε , S) for every δ > 0, which follows since u has an extension to a section in D max ⊂ H 1 loc ∩ δ>0 x 1−δ L 2 (M ; S). In particular, x δ+f /2 u ∈ H 1 (M, dxdydz), the standard Sobolev space of order 1 on the manifold with boundary M . Using the restriction theorem [50, Prop 4.5, Chap 4], x δ+f /2 u = ε δ+f /2 u ∈ H 1/2 (∂M ) uniformly in ε, so (3.16) holds. Thus, for fixed w ∈ M − M ε , writing dV ol g = x f adxdydz for some a = a(x, y, z) with a(0, y, z) = 0, we can use the bound for K Q in (2.33) with x fixed and x = ε to conclude To prove the other direction of containment in (3.9), we need to know that for f ∈ H 1/2 (∂M ε ) satisfying (Id − E ε )f = 0, the section u := Ext ε f | M −Mε ∈ H loc,ε , where Ext ε is the extension operator in (3.3). This is true since for any H 1/2 section h over ∂M ε , there is an H 1 extension v to the manifold M defined above, that can be taken with support away from the singular locus. If 1 M ε is the indicator function of M ε , then ð ( This completes the proof of Claim 3.2.

Equivalence of indices
In the previous section we have shown , in this section we will prove the following.  The main tool for proving Theorem 4.1 is the following theorem from [16]. We define the 'pseudodifferential Grassmanians' Gr AP S,ε = π ∈ Ψ 0 (∂M ε ; S) : π 2 = π and σ(π) = σ(π AP S,ε ) . (4.1) We endow Gr AP S,ε with the norm topology. If π ∈ Gr AP S,ε , then defining the domain D π,ε = u ∈ H 1 (M ε ; S) : (Id − π)(u| ∂Mε ) = 0 , the map To apply Theorem 4.2 in our case, we will study the two families of boundary values projectors π AP S,ε and E ε using the adiabatic calculus of Mazzeo and Melrose, [38].

4.1.
Review of the adiabatic calculus. Consider a fiber bundle Z → X π −−→ Y . The adiabatic double space X 2 ad is formed by radial blow up of X 2 × [0, ε 0 ) ε along the fiber diagonal, diag f ib ( X) (see (2.7)) at ε = 0. That is, Thus, X 2 ad is a manifold with corners with two boundary hypersurfaces: the lift of {ε = 0}, which we continue to denote by {ε = 0}, and the one introduced by the blowup, which we call ff. Similar to the edge front face above, ff is a bundle over Y whose fibers are isomorphic to Z 2 × R b where b = dim Y , and in fact this is the fiber product of π * T * Y and X.
We define ff y to be the fiber of ff lying above y.
The adiabatic vector fields on the fibration X are families of vector fields V ε parametrized smoothly in ε ∈ [0, ε), such that V 0 is a vertical vector field, i.e., a section of T X/Y. Locally these are C ∞ ( X × [0, ε 0 ) ε ) linear combinations of the vector fields ∂ z , ε∂ y .

Such families of vector fields are in fact sections of a vector bundle
(4.2) We will now define adiabatic differential operators on sections of S. The space of m th order adiabatic differential operators Diff m ad ( X; S) is the space of differential operators obtained by taking C ∞ ( X; End(S)) combinations of powers (up to order m) of adiabatic vector fields. An adiabatic differential operator P admits a normal operator N (P ), obtained by letting P act on X × X × [0, ε 0 ) ε , pulling back P to X 2 ad , and restricing it to ff. The normal operator acts tangentially along the fibers of ff over Y , and N (P ) y will denote the operator on sections of S restricted to over ff y . More concretely, if P is an adiabatic operator of order m, then near a point y 0 in Y , we can write for y near y 0 , where a α,β (z, y, ε) is a smooth family of endomorphisms of S. The normal operator is given by Returning to the case that X = ∂M for M a compact manifold with boundary, we take a collar neighborhood U ∂M × [0, ε 0 ) x as in (2.1), and treating the boundary defining function, x, as the parameter ε in the previous paragraph, identify the adiabatic double space (∂M ) 2 ad with a blow up of {x = x } ⊂ U × U.  .3), and ð Y is the standard Dirac operator on T y Y .
The space of adiabatic pseudodifferential operators with bounds on X of order m acting on sections of S, denoted Ψ m ad,bnd ( X; S), is the space of families of pseudodifferential operators {A ε } 0<ε<ε0 , where A ε is a (standard) Ψ of order m for each ε, and whose integral kernel of A ε is conormal to the lifted diagonal ∆ ad := diag X ×(0, ε 0 ) ε , smoothly up to ff. To be precise, the Schwartz kernel of an operator A ∈ Ψ m (∂M ; S) is given by a family of Schwartz kernels K Aε = K 1,ε + K 2,ε where K 1,ε is conormal of order m at ∆ ad smoothly down to ff and supported near ∆ ad , and K 2,ε is smooth on the interior and bounded at the boundary hypersurfaces.
An adiabatic pseudodifferential operator A ∈ Ψ m (∂M ; S) with bounds comes with two crucial pieces of data: a principal symbol and a normal operator. The principal symbol σ(A)(ε) is the standard one defined for a conormal distribution, i.e. as a homogeneous section of N * (∆; End(S)) ⊗ Ω 1/2 , the conormal bundle to the lifted diagonal (with coefficients in half-densities). In our case N * (∆) is canonically isomorphic to T * ad (∂M ), the dual bundle to T ad (∂M ) defined in (4.2); in particular, the symbol of A is a map σ(A) : T * ad (∂M ) −→ C ∞ (M ; End(S)), well defined only to leading order, and smooth down to ff. The normal operator is the restriction of the Schwartz kernel of A to the front face We thus have maps We have the following.

APS projections as an adiabatic family.
To study the integral kernel of the projector π AP S,ε we will make use of the fact that the boundary Dirac operator ð ε from (1.18) is invertible for small ε. This is a general fact about adiabatic pseudodifferential operators: invertibility at ε = 0 implies invertibility for small epsilon, or formally Theorem 4.5. Let A ε ∈ Ψ m ad (M ; S) and assume that on each fiber ff y the Fourier transform of the normal operator N (A ε ) y is invertible on L 2 (Z; S y , k y ), with S y the restriction of the spinor bundle to the fiber over y and k y = g N/Y y . Then A ε is invertible for small ε.
It is well known [7] that for each fixed ε > 0, π AP S,ε is a pseudodifferential operator of order 0. As ε varies, these operators form an adiabatic family: Lemma 4.6. The family π AP S,ε lies in Ψ 0 ad (∂M ; S). Its normal symbol N (π AP S,ε ) satisfies Proof. By Assumption 2.1, ð ε is invertible for small ε. Indeed, by (4.8), N ( ð ε ) y does not have zero as an eigenvalue. The projectors π AP S,ε can be expressed in terms of functions of the tangential operators ð ε [7] via the formula Following [49], the operator ð −1 ε | ð ε | is in Ψ 1 ad (∂X; S) and has the expected normal operator, namely the one obtained by applying the appropriate functions to the normal operator of N ( ð ε ) y and composing them.
Proof. The proof proceeds in two main steps. First, we construct a homotopy from the normal operators N (E ε ) to N (π AP S,ε ). Then we extend this homotopy to a homotopy of the adiabatic families as claimed in the theorem. For the homotopy of the normal operators, the main lemma will be the following Claim 4.8. For each y ∈ Y , the normal operators N (E ε ) y and N (π AP S,ε ), acting on for some δ > 0 independent of y.
Assuming the claim for the moment, the following argument from [16, Chap 15] furnishes a homotopy. In general, let P and Q be projections on a separable Hilbert space. Define T t = Id + t(Q − P )(2P − Id), and note that T 1 P = QT 1 . Now assume that T t is invertible for all t. Then the operator F t = T −1 t P T t is a homotopy from P to Q, i.e. F 0 = P , F 1 = Q. This holds in particular if P − Q < 1, in which case T t is invertible by Neumann series for t ∈ [0, 1].
To apply this in our context, we first take P = N (E ε ) y and Q = N (π AP S,ε ) y , and see that the corresponding operator T t is invertible by Claim 4.8. Now taking P = E ε and Q = π AP S,ε (so P , Q, and T t depend on ε) by Theorem 4.5, T t is invertible for small ε. Thus the homotopy F t = F t (ε) =: π ε,t is well defined for small ε. In fact, π ε,t is a smooth family of adiabatic pseudodifferential projections with principal symbol equal to that of π AP S,ε for all ε.
Thus it remains to prove Claim 4.8. By the formulas for the normal operators given in (4.4) and (4.6) and Plancherel, the claim will follow if we can show that for each µ with |µ| > 1/2, and all |η|, that for some δ independent of µ ≥ 1/2 and |η|. Here the norm is as a map of R 2 with the standard Euclidean norm. We prove the bound in (4.9) using standard bounds on modified Bessel functions in the Appendix, §7.

Proof of Main Theorem: limit of the index formula
Recall (e.g., [46, §2.14]) that if E −→ M is a real vector bundle of rank k connection ∇ E and curvature tensor R E then every smooth function (or formal power series) that is invariant under the adjoint action of SO(k), determines a closed differential form P (R E ) ∈ C ∞ (M ; Λ * T * M ). If ∇ E 1 is another connection on E, with curvature tensor R E 1 then P (R E ) and P (R E 1 ) differ by an exact form. Indeed, define a family of connections on E by The differential form Now consider for ε < 1 the truncated manifold M ε = {x ≥ ε} and the corresponding truncated collar neighborhood C ε = [ε, 1] × N. Let ∇ pt be the Levi-Civita connection of the metric The Atiyah-Patodi-Singer index theorem on M ε has the form [30,31], cf. [25] Mε where AS is a characteristic form associated to a connection ∇ and T AS(∇, ∇ pt ) is its transgression form with respect to the connection ∇ pt . The Levi-Civita connection of g pt induces a connection on T ie M ε , which we continue to denote ∇ pt . Let θ ε = ∇ − ∇ pt . Since g ie and g pt coincide on {x = ε} we have On the other hand, if A, B, C ∈ {∂ x , 1 x V, U }, we have Note that, analogously to (1.9), we have In particular note that j * ε θ ε is independent of ε and is equal to Next we need to compute the restriction to x = ε of the curvature Ω t of the connection (1 − t)∇ + t∇ pt = ∇ + tθ ε . Locally, with ω the local connection one-form of ∇ (1.10), the curvature Ω t is given by where [·, ·] s denotes the supercommutator with respect to form parity, so that [ω, θ ε ] s = ω ∧ θ ε + θ ε ∧ ω. In terms of the splitting (1.7) we have In particular, if we denote Ω v+,t the curvature of the connection (1 − t)∇ v+ + t∇ v on the bundle ∂ x + T N/Y, we have It follows that and similarly for any multiplicative characteristic class.
We can now prove the main theorem, whose statement we recall for the reader's convenience.
Theorem 5.1. Let X be stratified space with a single singular stratum endowed with an incomplete edge metric g and let M be its resolution. If ð is a Dirac operator associated to a spin bundle S −→ M and ð satisfies Assumption 2.1, then where A denotes the A-genus, T A(∇ v+ , ∇ pt ) denotes the transgression form of the A genus associated to the connections ∇ v+ and ∇ pt above, and η the η-form of Bismut-Cheeger [11].
Proof of Main Theorem. Combining Theorems 3.1 and 4.1 we know that, for ε small enough,

5.1.
Four-dimensions with circle fibers. An incomplete edge space whose link is a sphere is topologically a smooth space. So let us consider a four-dimensional manifold X with a submanifold Y and a Riemannian metric on X \ Y that in a tubular neighborhood of Y takes the form Here β is a constant and 2πβ is the 'cone angle' along the edge.
Recall that the circle has two distinct spin structures, and with the round metric the corresponding Dirac operators have spectra equal to either the even or odd integer multiples of π. The non-trivial spin structure on the circle is the one that extends to the disk, and so any spin structure on X will induce non-trivial spin structures on its link circles. Thus, cf. [23, Proposition 2.1], the generalized Witt assumption will be satisfied as long as β ≤ 1.
Next let us consider θ in more detail. From (5.1), with respect to the splitting (1.6), we have where α is a vertical one-form of g Z length one. This form is closely related to the 'global angular form' described in [17, pg. 70]. Indeed, α restricts to each fiber to be βdθ which integrates out to 2πβ. It follows that dα = −2πβφ * e, where e ∈ C ∞ (Y ; T * Y ) is the Euler class of Y as a submanifold of X, and hence Thus we find Theorem 5.2. Let X be an oriented four dimensional manifold, Y a smooth compact oriented embedded surface, and g an incomplete edge metric on X \ Y with cone angle 2πβ along Y.
1)[Atiyah-LeBrun [6]] The signature of X is given by 2)If X is spin and β ∈ Proof. 1) Since X is a smooth manifold we can use Novikov additivity of the signature to decompose the signature as sgn(X) = sgn(X \ M ε ) + sgn(M ε ).
Identifying X \ M ε with a disk bundle over Y we have from [24,Pg. 314] that sgn(X \ M ε ) = sgn Y e , i.e., the signature is the sign of the self-intersection number of Y in X. In fact this is a simple exercise using the Thom isomorphism theorem. where B e is the bilinear form on H 0 (Y ) given by H 0 (Y ) c, c → cc e, Y ∈ R, i.e., it is again the sign of the self-intersection of Y. (In comparing with [26] note that the orientation of ∂M ε is the opposite of the orientation of the spherical normal bundle of Y in X, and so sgn(B e ) = − sgn(X \ M ε ).) The only term in L(T Y )(coth e − e −1 ) of degree two is 1 3 e, and hence sgn(X) = 1 as required.
2) As mentioned above, the fact that the spin structure extends to all of X and β ∈ (0, 1] implies that the generalized Witt assumption for ð is satisfied. The degree four term of the A genus is −p 1 /24, the adiabatic limit of the eta-invariant for the spin Dirac operator is [26,

Positive scalar curvature metrics
In this short section, we prove Theorem 3 following [22]. We recall the statement of the theorem for the benefit of the reader: Proof. a) Taking traces in (1.11), the scalar curvature R g satisfies where R cone is the scalar curvature of the cone with metric dx 2 + x 2 g N/Y | ∂x ⊕ 1 x T N/Y , as in (1.7). On the other hand, by [22,Sect. 4], the scalar curvature of an exact cone C(Z) is equal to x −2 (R Z − dim(Z)(dim(Z) − 1)), where R Z is the scalar curvature of Z. Thus R g ≥ 0 implies that R Z ≥ 0, which by [22,Lemma 3.5] shows that Assumption 2.1 holds.
b) First off, by Theorem 1, ð is essentially self-adjoint. That is, the graph closure of ð on C ∞ comp (M ) is self-adjoint, with domain D from Theorem 1, and furthermore by the Main Theorem its index satisfies (4).

Appendix
In this appendix we prove Claim 4.8 by using standard bounds on modified Bessel functions to prove the sup norm bound (4.9): for each µ with |µ| > 1/2, and all |η|, Among references for modified Bessel functions we recall [5,8,9,47].
To begin with, using the Wronskian equation (2.27), note that Tr N µ,z = Tr N AP S µ,z = 1.
The limit as z → ∞ can be seen using the large argument asymptotic formulas from [1, Sect. 9.7], while the limit as z → 0 follows from the recurrence relations (2.27) and the small argument asymptotics in [1, Sect. 9.6]. Thus zK ν (z)I ν (z) is monotone on the region under consideration. Using the asymptotic formulas again shows that zK ν (z)I ν (z) → 0 as z → 0 → 1/2 as z → ∞. so (7.3) holds.