*Algebraic &* *Geometric* *Topology*

**A** **T** ^{G}

^{G}

Volume 3 (2003) 623{675 Published: 25 June 2003

**On the rho invariant for manifolds with boundary**

Paul Kirk Matthias Lesch

**Abstract** This article is a follow up of the previous article of the authors
on the analytic surgery of{ and*{invariants. We investigate in detail the*
(Atiyah{Patodi{Singer){{invariant for manifolds with boundary. First we
generalize the cut{and{paste formula to arbitrary boundary conditions. A
priori the *{invariant is an invariant of the Riemannian structure and a*
representation of the fundamental group. We show, however, that the de-
pendence on the metric is only very mild: it is independent of the metric
in the interior and the dependence on the metric on the boundary is only
up to its pseudo{isotopy class. Furthermore, we show that this cannot be
improved: we give explicit examples and a theoretical argument that dier-
ent metrics on the boundary in general give rise to dierent *{invariants.*

Theoretically, this follows from an interpretation of the exponentiated *{*
invariant as a covariantly constant section of a determinant bundle over a
certain moduli space of flat connections and Riemannian metrics on the
boundary. Finally we extend to manifolds with boundary the results of
Farber{Levine{Weinberger concerning the homotopy invariance of the *{*
invariant and spectral flow of the odd signature operator.

**AMS Classication** 58J28; 57M27, 58J32, 58J30
**Keywords** rho-invariant, eta-invariant

**1** **Introduction**

The *{invariant of a closed odd-dimensional manifold was dened in [2] as a*
dierence of two spectral invariants. To a closed Riemannian manifold *M* and
a unitary representation of its fundamental group : _{1}(M) *!* *U*(n) Atiyah,
Patodi, and Singer assigned the real number

*(M; ) =(D*_{}*; M)−(D*_{}*; M*)

where*D* denotes the odd signature operator with coecients in the flat bundle
determined by , and *D** _{}* is similar with respect to the trivial representation
, and

*(D) denotes the regularized signature of a self-adjoint Dirac operator*

*D, introduced in [2]. As a consequence of their index theorem they showed that*

(1) *(M; ) is independent of the choice of Riemannian metric on* *M*.
(2) *(M; ) extends the signature defect, that is, if (M; ) =* *@(W; ) for*

some manifold *W* with unitary representation : 1(W)*!U*(n), then
*(M; ) =n*Sign(W)*−*Sign* _{}*(W)

where Sign* _{}*(W) denotes the signature of

*W*with local coecients in the flat bundle determined by .

Thus *(M; ) is a fundamental smooth invariant, but it remains largely myste-*
rious since its general denition depends on the spectra of dierential operators
on *M*.

In [26] we dened the *{invariant* *(M; ; g) in the case when the boundary*
of *M* is non-empty and proved a non-additivity formula as a consequence of
our cut-and-paste formula for {invariants of Dirac operators: if *M* is a closed
manifold split into two parts *X* and *Y* along a hypersurface,

*(M; ) =(X; ; g) +(Y; ; g) +m(V*_{X;}*; V** _{Y;}*)

_{(g;)}

*−m(V*

_{X;}*; V*

*)*

_{Y;}_{(g;)}

*:*(For the denitions of the terms see (2.4) and (2.8).)

It is the purpose of this article to explore the properties of *(M; ; g), particu-*
larly those which flow from this formula. We will describe the behavior of this
invariant with respect to variations in Atiyah-Patodi-Singer (APS) boundary
conditions, bordisms, and variations of and *g*. As applications we prove
generalizations of the main results of Farber-Levine-Weinberger [14] concerning
the homotopy invariance of to manifolds with boundary. Special attention is
given to the construction of explicit examples.

The invariant *(M; ; g) is dened as a dierence of* *{invariants for manifolds*
with boundary and as such is also a spectral invariant. This has the happy
consequence that it is gauge and isometry invariant. But in contrast to the
closed case, when the boundary of *M* is non-empty the resulting invariant
depends on the choice of Riemannian metric, *g, on the boundary.*

Hidden from the notation is the fact that elliptic boundary conditions are re- quired to dene {invariants on manifolds with boundary. Our choice in [26]

is to use APS boundary conditions with respect to the Lagrangian subspace
of *limiting values of extended* *L*^{2} *solutions* in the sense of [2]. This choice is
intrinsic, homotopy invariant, and natural in a sense we will describe with re-
spect to bordisms, but is not continuous in families. This fact is apparent when
one considers families for which the dimension of the kernel of the tangential
operator is not constant, but discontinuities can also occur in families for which
the kernel is constant dimensional.

For this reason it turns out to be useful to allow more general Lagrangian subspaces; we describe this generalization and derive the corresponding cut-and- paste formula for - and {invariants with respect to arbitrary APS boundary conditions in Theorem 3.2. Among other things, Theorem 3.2 says:

**Theorem** *Suppose that* *M* =*Y[**X; :* _{1}(M)*!U*(n) *is a representation,*
*W*^{X}*; W*^{Y}*H** ^{}*(;C

*)*

^{n}*are Lagrangian subspaces, and let*

*B*

*be a flat connec-*

*tion on*

*M*

*with holonomy*

*in cylindrical form near*. Then the dierence

*(D*

_{B}*; M*)

*−(D*

_{B;W}*X*

*; X)−(D*

_{B;W}*Y*

*; Y*)

*−m(W*

^{X}*; W*

*)*

^{Y}_{(;g)}

*equals the integer*

~

*(V**X;**; V**Y;**; γ(W** ^{Y}*))

*−(γ(V*~

*X;*); W

^{X}*; W*

*):*

^{Y}In this statement *(D*_{B;W}*; X) denotes the* {invariant of the odd signature
operator coupled to a flat connection *B* on the manifold *X* with respect to
APS boundary conditions determined by the Lagrangian subspace *W* of the
kernel of the tangential operator. Moreover, *m(V; W*)_{(;g)} is an explicit real-
valued invariant of pairs of Lagrangian subspaces of the Hermitian symplectic
space *H** ^{}*(;C

^{n}*) with its induced*

_{}*L*

^{2}metric (this is dened in Section 2),

*γ*is the associated complex structure, ~ is the Maslov triple index which appears in Wall’s non-additivity theorem [36],

*V*

*= image*

_{X;}*H*

*(X;C*

^{}

^{n}*)*

_{}*!*

*H*

*(;C*

^{}

^{n}*)*

_{},
and similarly for *V**Y;*.

This theorem generalizes [26, Theorem 8.8] to arbitrary APS boundary condi-
tions. Taking dierences gives a corresponding formula for *{invariants.*

We next give a topological description of how the spaces *V**X;* propagate across
a bordism; the result is given as Theorem 4.1 which gives a functorial framework
to keep track of APS boundary conditions and a companion additivity formula
for and .

With these technical results in place, we can then begin a careful investigation of how the {invariants for manifolds with boundary depend on the choice of metric on the boundary and the representation. For example, we show:

**Corollary 5.2** *The* *{invariant for a manifold with boundary depends on*
*the Riemannian metric on the boundary only up to its pseudo-isotopy class.*

*Precisely, if* *f*_{0}*; f*_{1}: *@X* *!@X* *are pseudo-isotopic dieomorphisms, then*
*(X; ; f*_{0}* ^{}*(g)) =

*(X; ; f*

_{1}

*(g)):*

^{}In Section 6 we give explicit examples which show that*(X; ; g) and* *m(V; W*)
depend on the choice of Riemannian metric:

**Theorem 6.1**

(1) *There exists a 3-manifold* *Y* *with non-empty boundary, Riemannian met-*
*rics* *g*0*,* *g*1 *on* *@Y, and a representation* : 1*Y* *!U*(2) *so that*

*(Y; ; g*_{0})*6*=*(Y; ; g*_{1}):

*Examples exist with vanishing kernel of the tangential operator, i.e.*

ker*A**b* =*H** ^{}*(@Y;C

^{2}) = 0:

(2) *There exist metrics* *g*_{0} *and* *g*_{1} *on the torus* *T* *and 3-manifolds* *X* *and* *Y*
*with boundary* *T* *such that setting* *V** _{X}* =

*image*

*H*

*(X;C)*

^{}*!H*

*(T;C)*

^{}*and*

*V*

*=*

_{Y}*image*

*H*

*(Y;C)*

^{}*!H*

*(T;C)*

^{}*(with* *the trivial conection),*
*m(V**X**; V**Y*)_{(;g}_{0}_{)}*6*=*m(V**X**; V**Y*)_{(;g}_{1}_{)}*:*

This theorem drives home the point that the choice of Riemannian metric on the boundary is an essential ingredient of the {invariant on a manifold with non-empty boundary.

In Section 7 we extend to manifolds with boundary the results of Farber-Levine- Weinberger concerning the homotopy invariance of the {invariants and spec- tral flow of the odd signature operator. Let

*(*_{1}*X; U*(n)) = Hom(_{1}*X; U*(n))=conjugation

and let ^{M}*@X* denote the space of Riemannian metrics on *@X*. Notice that
a map *F*: *X* *!X** ^{0}* which restricts to a dieomorphism on the boundary and
which induces an isomorphism on fundamental groups provides an identication
of

^{M}

*with*

_{@X}^{M}

_{@X}*0*and

*(*

_{1}

*X; U*(n)) with

*(*

_{1}

*X*

^{0}*; U*(n)).

**Theorem 7.2** *Let* *F*: *X* *!X*^{0}*be a homotopy equivalence of compact mani-*
*folds which restricts to a dieomorphism on the boundary. Then the dierence*

*(X)−(X** ^{0}*) :

*(*1

*X; U*(n))

^{M}

*@X*

*!*R

*factors through*_{0}((_{1}*X; U*(n)))(^{M}_{@X}*=*^{D}_{@X}^{0} )*(where*^{D}_{@X}^{0} *denotes the group*
*of dieomorphisms of* *@X* *pseudo-isotopic to the identity) and takes values in*
*the rational numbers.*

*In other words there is a commutative diagram*

*(*1*X; U*(n))^{M}*@X* R

_{0}((_{1}*X; U*(n)))(^{M}_{@X}*=*^{D}_{@X}^{0} ) Q

*(X*)*−**(X** ^{0}*-)

? -

6

*Moreover the dierence* *(X; ; g)−(X*^{0}*; ; g)* *vanishes for* *in the path com-*
*ponent of the trivial representation.*

We also show that the spectral flow of the odd signature operator coupled to a
path of flat connections is a homotopy invariant. In the following statement*P*(t)
denotes a smooth path of self-adjoint APS boundary conditions with prescribed
endpoints. (We show how to construct such a path in Lemma 7.3.)

**Theorem 7.4** *Suppose that* *F*: *X*^{0}*!* *X* *is a homotopy equivalence which*
*restricts to a dieomorphism* *f* = *Fj**@X** ^{0}*:

*@X*

^{0}*!*

*@X. Assume that*

*B*

_{t}*is a*

*continuous, piecewise smooth path of flat*

*U*(n)

*connections on*

*E*

*!*

*X. Use*

*F*

*to pull back the path*

*B*

_{t}*to a path of flat connections*

*B*

_{t}

^{0}*on*

*X*

^{0}*and to*

*identify*

*@X*

*with*

*@X*

^{0}*, and choose a path*

*P(t)*

*of APS boundary conditions as*

*in Lemma 7.3.*

*Then*

SF(X; D_{B}_{t}* _{;P(t)}*)

_{t}

_{2}_{[0;1]}= SF(X

^{0}*; D*

_{B}*0*

*t**;P*(t))_{t}_{2}_{[0;1]}*:*

In Section 8 we use the machinery of determinant bundles, especially the Dai-
Freed theorem, to study the variation of *(X; ; g) modulo* Z. By working
modulo Z one loses geometric information but the discontinuities of as a
function of are eliminated. In particular variational techniques can be ap-
plied.

Theorem 8.4 implies the following. In this theorem *r** ^{Q}* denotes the connection
on the determinant bundle as introduced by Quillen in [32]. (See [5] for the
construction of

*r*

*in general.)*

^{Q}**Theorem** *The assignment of the exponentiated* *{invariant to a flat* *SU*(n)
*connection* *B* *on a manifold with boundary* *X* *and a choice of Riemannian*
*metric* *g* *on* *@X,*

(B; g)*7!*exp(i (X; ; g));

*(where* *is the holonomy of* *B) denes a smooth horizontal (with respect to*
*the connection* *r*^{Q}*) cross section of the determinant bundle of the family of*
*tangential operators to the odd signature operators.*

This theorem allows one to relate the mod Z reduction of the {invariant on
manifolds with isomorphic fundamental groups and dieomorphic boundaries,
and also shows that the manner in which *(X; ; g) depends on the choice of*
metric *g* on *@X* is intimately tied to the connection *r** ^{Q}*.

For example, the following is a consequence of Theorem 8.5. We view *(X) as*
a function of the conjugacy class of the representation and the metric *g*.

**Theorem** *LetX* *and* *X*^{0}*be two odd dimensional manifolds and suppose that*
*F*: *X*^{0}*!* *X* *is a smooth map which induces an isomorphism on fundamental*
*groups and such that the restriction* *f* = *Fj**@X** ^{0}*:

*@X*

^{0}*!*

*@X*

*is a dieomor-*

*phism.*

*Then there is a factorization*

*(*_{1}(X); SU(n))^{M} R*=*Z

0((1(X); SU(n)))^{M}
*(X)**−**(X** ^{0}*) -

HHHHHj p p p pp p p pp p p p*p

*and* *(X)−(X** ^{0}*)

*is zero on the path component of the trivial representation.*

*The result holds for* *U*(n) *replacing* *SU*(n) *if* dim*X*= 4‘*−*1*.*

These results, together with the cut{and{paste formula for{invariants (Theo- rem 3.2) are a step in the program of determining what the homotopy properties of the {invariant are. A discussion of problems in this topic is given in Section 9, including the following consequence of Theorem 9.2 concerning the homotopy invariance of the {invariant for closed manifolds.

**Theorem** *LetM* *and* *M*^{0}*be closed manifolds, and suppose there exists a sep-*
*arating hypersurfaceM* *and a smooth homotopy equivalence* *F*: *M*^{0}*!M*
*so that the restriction ofF* *to* *F*^{−}^{1}()*is a dieomorphism. Write* *M* =*X[**Y*
*and* *M** ^{0}* =

*X*

^{0}*[*

*Y*

^{0}*and suppose that*

*F*

*restricts to homotopy equivalences*

*X*

^{0}*!X*

*and*

*Y*

^{0}*!Y. Let*

*2(*

_{1}

*M; U*(n)).

*If the restriction* *j**X* *(resp.* *j**Y**) of* *to* _{1}*X* *(resp.* _{1}*Y) lies in the path*
*component of the trivial representation of* *(*_{1}*X; U*(n)) *(resp.* *(*_{1}*Y; U*(n)))
*then* *(M; ) =(M*^{0}*; ).*

We nish the article with a brief discussion of the relation of our investigations to one of the approaches to the program of constructing topological quantum eld theories proposed in [1].

**Acknowledgements**

The authors thank J.F. Davis, C. Livingston, S. Paycha, and K.P. Wojciechowski for very helpful discussions.

The rst named author gratefully acknowledges the support of the National Science Foundation under grant no. DMS-0202148.

**2** **The** *{invariant on manifolds with boundary*

We begin by recalling the context and the denition of the {invariant for a manifold with boundary. More details can be found in [26].

Let *X* be a (2k + 1)-dimensional smooth, oriented, compact manifold with
(possibly empty) boundary . Fix a Riemannian metric ~*g* on *X* in product
form near the boundary . To keep track of signs it is crucial to x a convention
for the orientation of a collar of the boundary. In this paper we will use the
convention of [26]: if not indicated otherwise a collar of the boundary will be
written as [0; ), i.e. the manifold *X* is \on the right" of the boundary. The
choice of the sign convention has consequences for the denition of *A** _{b}* and

*γ*(and hence the Hermitian symplectic structure on

*H*

*(;C*

^{}*)) below.*

^{n}Let *B* be a flat *U*(n) connection on *X* in product form near the boundary, i.e.

*Bj*[0;) =* ^{}*(b) for some flat connection

*b*on ; here : [0; )

*!*de- notes the projection. Denote by :

_{1}

*X*

*!U*(n) the holonomy representation of

*B*. Since it will be central in what follows, denote the restriction of ~

*g*to the boundary by

*g.*

The*odd signature operator coupled to the flat connection* *B*
*D**B*: *p*Ω^{2p}(X;*E)! **p*Ω^{2p}(X;*E)*
is dened by

*D** _{B}*() =

*i*

*(*

^{k+1}*−*1)

^{p}

^{−}^{1}(

*d*

_{B}*−d*

*)() for*

_{B}*2*Ω

^{2p}(X;

*E):*

Here, : Ω* ^{‘}*(X;

*E)*

*!*Ω

^{2k+1}

^{−}*(X;*

^{‘}*E) denotes the Hodge * operator (which is*determined by the Riemannian metric ~

*g*on

*X*),

*d*

*B*: Ω

*(X;*

^{‘}*E)!*Ω

*(X;*

^{‘+1}*E)*denotes the covariant derivative associated to the flat connection

*B*, and

*E*

*!*

*X*denotes the associated Hermitian C

*vector bundle.*

^{n}On the collar [0; ), *D** _{B}* takes the form (after conjugating with a certain
unitary transformation, see [26, (8.1)] for details)

*D** _{B}* =

*γ(*

*@*

*@x* +*A** _{b}*);

where the*de Rham operator*

*A** _{b}*:

*k*Ω

*(;*

^{k}*Ej*)

*!*

*k*Ω

*(;*

^{k}*Ej*) is dened by

*A** _{b}*() =

(*−*(d* _{b}*^+ ^

*d*

*); if*

_{b}*2*

*k*Ω

^{2k}(;

*Ej*);

(d* _{b}*^+ ^

*d*

*); if*

_{b}*2*

*k*Ω

^{2k+1}(;

*Ej*):

In these formulas ^ denotes the Hodge operator on and
*γ*: *p*Ω* ^{p}*(;

*Ej*)

*!*

*p*Ω

*(;*

^{p}*Ej*) coincides with ^ up to a constant:

*γ*() =
(

*i** ^{k+1}*(

*−*1)

^{p}

^{−}^{1}^

*;*if

*2*Ω

^{2p}(;

*Ej*);

*i** ^{k+1}*(

*−*1)

^{k}

^{−}*^*

^{q}*;*if

*2*Ω

^{2q+1}(;

*Ej*):

One calculates that*γ*^{2}=*−Id; γA** _{b}* =

*−A*

_{b}*γ*, and that

*γ*is unitary with respect to the

*L*

^{2}inner product on Ω

*(;*

^{}*Ej*) dened by

*h*1*; *2*i*=
Z

1*^*^2*:*

(The Riemannian metric on is used to dene the Hodge -operator ^, and
we have suppressed the notation for the inner product in the bundle *E*.) The
operator *A** _{b}* is elliptic and self-adjoint and hence one has an orthogonal decom-
position

*L*^{2}(Ω* ^{}*(;

*Ej*)) =

*F*

_{b}*ker*

^{−}*A*

_{b}*F*

_{b}^{+}(2.1) into the negative eigenspan, kernel, and positive eigenspan of

*A*

*. The relation*

_{b}*γA*

*=*

_{b}*−A*

_{b}*γ*implies that ker

*A*

*is preserved by*

_{b}*γ*and that

*γ*maps

*F*

_{b}^{+}unitarily onto

*F*

_{b}*.*

^{−}The kernel of *A** _{b}* is identied by the Hodge theorem with the twisted de Rham
cohomology of the complex (Ω

*(;*

^{}*Ej*); d

*); indeed the elements of ker*

_{b}*A*

*are just the*

_{b}*d*

*b*-harmonic forms and so the composite

ker*A** _{b}* = ker

*d*

_{b}*\*ker

*d*

^{}*ker*

_{b}*d*

_{b}*!*ker

*d*

*image*

_{b}*d*

_{b}is an isomorphism. The de Rham theorem then identies the cohomology of
(Ω* ^{}*(;

*Ej*); d

*) and the (singular or cellular) cohomology*

_{b}*H*

*(;C*

^{}*) with local coecients given by the holonomy representation .*

^{n}The triple (ker*A**b**;h* *;* *i; γ) gives kerA**b* the structure of a Hermitian sym-
plectic space. In general a *Hermitian symplectic space* (H;*h* *;* *i; γ) is a nite*
dimensional complex vector space *H* with a positive denite Hermitian inner
product *h;* *i*: *HH!* C and an isomorphism *γ*: *H* *!H* which is unitary,
i.e. *hγ*(x); γ(y)*i* =*hx; yi*, satisfying *γ*^{2} =*−I* such that the signature of *iγ* is
zero. The *underlying symplectic structure* is the pair (H; !), where *!* is the
non-degenerate skew-Hermitian form

*!(x; y) =hx; γ(y)i:*

The signature of *iγ* on ker*A** _{b}* =

*H*

*(;C*

^{}*) is zero. This is a consequence of the fact that (; j) bounds (X; ), and is not true for a general pair (; ).*

^{n}However, it is true in many important cases, for example if is a 4‘*−*2
dimensional manifold and : _{1}*!U*(n) factors through *O(n).*

In contrast to the Hermitian inner product *h* *;* *i* and the unitary map *γ* on
ker*A** _{b}*, the symplectic form

*!*does not depend on the Riemannian metric and in fact is given by the cup product:

*!(*1*; *2) =*i** ^{r}*
Z

1*^*2 =*i** ^{r}*([1]

*[*[2])

*\*[];

where *i** ^{r}* is a constant depending on the degrees of the

*i*.

A subspace *W* of a Hermitian symplectic space (H; !) is called *Lagrangian* if

*!* vanishes on *W* and *W* is maximal with this property. This is equivalent to
*γ(W*) =*W** ^{?}*, but being a Lagrangian subspace is a property of the underlying
symplectic structure. Note that dim

*W*=

^{1}

_{2}dim

*H*. Denote the Grassmannian of all Lagrangian subspaces of

*H*by

^{L}(H).

We summarize: The symplectic structure on *H** ^{}*(;C

*), and hence the Grass- mannian*

^{n}^{L}(H

*(;C*

^{}*)), depends only on the cohomology and cup product, and therefore is a homotopy invariant of (; ). On the other hand, the Her- mitian symplectic structure on*

^{n}*H*

*(;C*

^{}*) depends on its identication with ker*

^{n}*A*

*via the Hodge and de Rham theorems, since the inner product*

_{b}*h*

*;*

*i*is the restriction of the

*L*

^{2}inner product (which depends on the Riemannian metric on ) to ker

*A*

*.*

_{b}The following lemma is well{known; it follows by a standard argument using Poincare duality (cf. also [26, Cor. 8.4]).

**Lemma 2.1** *The image of the restriction*

*H** ^{}*(X;C

*)*

^{n}*!H*

*(;C*

^{}*) (2.2)*

^{n}*is a Lagrangian subspace.*

We will denote this subspace by *V**X;*, and, by slight abuse of notation, its
preimage in ker*A** _{b}* via the isomorphism ker

*A*

*=*

_{b}*H*

*(;C*

^{}*) will also be de- noted by*

^{n}*V*

*. We emphasize that the Lagrangian*

_{X;}*V*

*is a homotopy invari- ant of (X; ). Moreover it gives a distinguished element in the Grassmannian*

_{X;}L(H* ^{}*(;C

*)). Considered as a subspace of ker*

^{n}*A*

*,*

_{b}*V*

*X;*coincides with the

*limiting values of extended*

*L*

^{2}

*solutions of*

*D*

*= 0 on ((*

_{B}*−1;*0])

*[X*in the sense of [2].

Lagrangian subspaces of *H** ^{}*(;C

*) are used to produce elliptic self-adjoint Atiyah-Patodi-Singer (APS) boundary conditions for the odd signature opera- tor*

^{n}*D*

*B*as follows. Given a Lagrangian subspace

*W*

*H*

*(;C*

^{}

^{n}*) we consider*

_{}the orthogonal projection in *L*^{2}(Ω* ^{}*(;

*Ej*)) onto

*F*

_{b}^{+}

*W*. This orthogonal projection denes a well{posed boundary condition for

*D*

*(see e.g. [9]).*

_{B}Restricting *D**B* to the subspace of sections whose restriction to the boundary
lies in the kernel of this projection makes *D** _{B}* a discrete self-adjoint operator
which we denote by

*D*

*B;W*. The following properties of this operator are the starting point of the investigations of this article and go back to Atiyah, Patodi, and Singer’s fundamental articles [2, 3, 4]. In this context the following facts are explained in [26].

(1) The function of the operator *D** _{B;W}*,

*(s) =* X

*2*^{Spec}(D*B;W*)*nf*0*g*

sign*jj*^{−}^{s}*;*

converges for Re(s) *>>* 0 and has a meromorphic continuation to the
entire complex plane with no pole at *s*= 0. Denote its value at *s*= 0 by

*(D*_{B;W}*; X) :=(0):*

(2) The kernel of *D** _{B;W}* ts into an exact sequence
0

*!*image

*H*

*(X;;C*

^{}*)*

^{n}*!H*

*(X;C*

^{}*)*

^{n}*!: : :*

*: : :!*ker*D*_{B;W}*!W* *\γ*(V* _{X;}*)

*!*0: (2.3) In particular, taking

*W*=

*V*

*we see*

_{X;}ker*D**B;V**X;* = image*H** ^{}*(X;;C

*)*

^{n}*!H*

*(X;C*

^{}*):*

^{n}We next recall the denition of the {invariant for manifolds with boundary
from [26]. Let denote the trivial connection in the product bundle C^{n}*X*
in the form = * ^{}*() in the collar of

*@X*, and :

_{1}

*X*

*!*

*U*(n) the trivial representation. Then dene

*(X; ; g) =(D*_{B;V}_{X;}*; X)−(D*_{;V}_{X;}*; X):* (2.4)
It is shown in [26, Sec. 8] that *(X; ; g) depends only on the smooth structure*
on *X*, the conjugacy class of : _{1}*X* *!* *U*(n), and the Riemannian metric *g*
on =*@X*. In particular, it is independent of the choice of flat connection *B*
with holonomy conjugate to and also independent of the Riemannian metric

~

*g* on *X* extending *g*.

When *@X* is empty, then the dieomorphism invariance of *(X; ) was estab-*
lished by Atiyah, Patodi, and Singer in [3] and follows straightforwardly from
their index theorem. The cut{and{paste formulae

*(D**B**; M*) =*(D**B;V**X;**; X) +(D**B;V**Y;**; Y*) +*m(V**X;**; V**Y;*)_{(b;g)} (2.5)

and

*(M; ) =(X; ; g) +(Y; ; g) +m(V**X;**; V**Y;*)_{(b;g)}*−m(V**X;**; V**Y;*)_{(;g)} (2.6)
when *M* = *Y* *[* *X* were proven in [26, Sec. 8] and are the basis for our
investigations in the present article.

In Equations (2.5) and (2.6) the correction term *m(V; W*)_{(b;g)} is a real valued
invariant of pairs of Lagrangians in *H** ^{}*(;C

*); it depends on the identication of*

^{n}*H*

*(;C*

^{}*) with the kernel of*

^{n}*A*

*and hence may*

_{b}*a priori*(and

*a posteriori*as well, see Section 6) depend on the Riemannian metric

*g*on . It is dened as follows.

Let ker*A*^{+}* _{b}* denote the +i-eigenspace of

*γ*acting on ker

*A*

*and let ker*

_{b}*A*

^{−}*denote the*

_{b}*−i-eigenspace. Then every Lagrangian subspace*

*W*of ker

*A*

*b*=

*H*

*(;C*

^{}*) can be written uniquely as a graph*

^{n}*W* =*fx*+*(W*)(x)jx *2*ker*A*^{+}_{b}*g;* (2.7)
where *(W*) : ker*A*^{+}_{b}*!* ker*A*^{−}* _{b}* is a unitary isomorphism. The map

*W*

*7!*

*(W*) determines a dieomorphism between the space ^{L}(ker*A** _{b}*) of La-
grangians in ker

*A*

*to the space of unitary operators*

_{b}*U*(ker

*A*

^{+}

_{b}*;*ker

*A*

^{−}*). We take the branch log(re*

_{b}*) = ln*

^{it}*r*+

*it; r >*0;

*− < t*and use it to dene tr log :

*U*(ker

*A*

^{+}

*)*

_{b}*!i*R via tr log(U) =P

log(* _{i}*);

_{i}*2*Spec

*U*. Then dene

*m(V; W*)

_{(b;g)}=

*−*

_{i}^{1}tr log(

*−(V*)(W)

*) + dim(V*

^{}*\W*)

=*−*_{i}^{1} X

*2*Spec(*−**(V*)(W)* ^{}*)

*6*=

*−*1

log*:*

(2.8)

We will abbreviate this to *m(V; W*) when (b; g) is clear from context. Since

*−(V*)(W)* ^{}* is unitary, its eigenvalues are unit complex numbers, and hence

*m(V; W*) is a real number. The term dim(V

*\W*) is added to match conventions and to simplify formulas; notice that its eect is to remove the contribution of the

*−*1 eigenspace of

*−(V*)(W)

*to tr log(*

^{}*−(V*)(W)

*). Thus*

^{}*m*is not in general a continuous function of

*V*and

*W*. The function

*m*has been investigated before, the notation is taken from [10].

**3** **Cutting and pasting formulas with arbitrary** **boundary conditions**

The {invariants appearing in the denition of of Equation (2.4) are taken
with respect to the boundary conditions *V**X;* *H** ^{}*(;C

^{n}*) and*

_{}*V*

*X;*

*H** ^{}*(;C

*). More precisely, the Lagrangian*

^{n}*V*

_{X;}*H*

*(;C*

^{}*) determines a subspace (still denoted*

^{n}*V*

*) of ker*

_{X;}*A*

*, and this in turn determines the orthog- onal projection to*

_{b}*F*

_{b}^{+}

*V*

*X;*, (recall that

*F*

_{b}^{+}is shorthand for the positive eigenspan of

*A*

*). A similar comment applies to*

_{b}*V*

*. Since these Lagrangians are canonically determined by the homotopy type of the pair (X; ) and the Riemannian metric on , they present a natural choice for the boundary con- ditions. Nevertheless it is useful to use other Lagrangians in*

_{X;}*H*

*(;C*

^{}*) to dene boundary conditions. One important reason is that the*

^{n}*V*

*do not vary continuously in families, even if ker*

_{X;}*A*

*b*does.

**Denition 3.1** Let *X* have boundary and let : _{1}*X* *!* *U*(n) be a
representation. Given Lagrangian subspaces *W* *H** ^{}*(;C

^{n}*) and*

_{}*W*

*H*

*(;C*

^{}*), dene*

^{n}*(X; ; g; W*

_{}*; W*

*) by*

_{}*(X; ; g; W*_{}*; W** _{}*) :=

*(D*

_{B;W}

_{}*; X)−(D*

_{;W}

_{}*; X):*

Thus *(X; ; g) is shorthand for* *(X; ; g; V*_{X;}*; V** _{X;}*).

We next recall the denition of ~ from [26, Sec. 8]. Given Lagrangian subspaces
*U; V; W* of a Hermitian symplectic space, dene

~

*(U; V; W*) :=*m(U; V*) +*m(V; W*) +*m(W; U*):

Then ~ is integer-valued, depends only on the symplectic form *!*, and coincides
with Wall’s correction term for the non-additivity of the signature [36] as well
as the Maslov triple index of [11].

The following theorem gives a complete formulation of the dependence of the - and{invariants for a manifold with boundary on the choice of Lagrangians used for APS boundary conditions.

**Theorem 3.2** *Suppose that* *M* = *Y* *[* *X,* : _{1}(M) *!* *U*(n) *is a repre-*
*sentation,* *W*_{}^{X}*; W*_{}^{Y}*H** ^{}*(;C

^{n}*)*

_{}*and*

*W*

_{}

^{X}*; W*

_{}

^{Y}*H*

*(;C*

^{}*)*

^{n}*are Lagrangian*

*subspaces, and let*

*B*

*be a flat connection on*

*M*

*with holonomy*

*in cylindri-*

*cal form near*. Orientation dependent quantities like

*γ*

*etc. are taken with*

*respect to*

*X*

*according to the convention explained on page 629.*

*Then:*

(1) *(D*_{B;W}*X*

*; X)−(D*_{B;V}_{X;}*; X) =m(γ(V** _{X;}*); W

_{}*):*

^{X}(2) *(X; ; g; W*_{}^{X}*; W*_{}* ^{X}*)

*depends only on the dieomorphism type of*

*X, the*

*representation*

*, the Lagrangian subspaces*

*W*

_{}

^{X}*; W*

_{}

^{X}*and the Rieman-*

*nian metric*

*g*

*on*=

*@X.*

(3) *The dierence* *(D*_{B}*; M)−(D*_{B;W}*X*

*; X)−(D*_{B;W}*Y*

*; Y*)*−m(W*_{}^{X}*; W*_{}* ^{Y}*)

*is an integer. In fact it equals*

~

*(V**X;**; V**Y;**; γ(W*_{}* ^{Y}*))

*−*~

*(γ(V*

*X;*); W

_{}

^{X}*; W*

_{}*):*

^{Y}(4) *(D*_{B}*; M*) =*(D*_{B;V}_{X;}*; X) +(D*_{B;γ(V}_{X;}_{)}*; Y*) *and so*

*(M; ) =(X; ; V*_{X;}*; V** _{X;}*) +

*(Y; ; γ(V*

*); γ(V*

_{X;}*)):*

_{X;}**Proof** We use the results of [26]. Recall the notation ~*(D) =* ^{1}_{2}((D) +
dim ker*D). For the proof of (1) we omit the sub- and superscripts of* *W*:
By [26, Theorem 4.4] we have

~

*(D*_{B;W}*; X)−(D*~ _{B;V}_{X;}*; X)*

= _{2i}^{1} tr log((P^{+}(W))(P*X*)* ^{}*)

*−*tr log((P

^{+}(V

*X;*))(P

*X*)

*)*

^{}*:* (3.1)
Here, *P*^{+}(W) denotes the orthogonal projection onto *W* *F*_{b}^{+}, *P**X* denotes
the Calderon projector for *D**B* acting on *X*, and is the innite{dimensional
version of *: it denotes the dieomorphism from the (innite{dimensional)*
Lagrangian Grassmannian onto ^{U}(ker(γ*−i);*ker(γ+i)) (cf. [26, Sec. 2]). Using
[26, Lemma 6.9] we identify the right side of (3.1) with

* _{}*(P

^{+}(V

*); P*

_{X;}^{+}(W); P

*)*

_{X}*−*

_{2i}

^{1}tr log (P

^{+}(V

*))(P*

_{X;}^{+}(W))

^{}*;* (3.2)
where is the Maslov triple index dened in [26, Sec. 6].

In view of [26, Lemma 8.10] the quantity * _{}*(P

^{+}(V

*); P*

_{X;}^{+}(W); P

*) is invariant under adiabatic stretching and equals*

_{X}(V*X;**; W; V**X;*) = dim *V**X;**\γ*(W)

*;* (3.3)

where the last equality follows from [26, Prop. 6.11].

As in the proof of [26, Theorem 8.12] one calculates
tr log (P^{+}(V*X;*))(P^{+}(W)* ^{}*)

= tr log *(V**X;*)(W)^{}

*:* (3.4)
The identity *γ*^{2} =*−I* shows that dim(V_{X;}*\γ*(W)) = dim(γ(V* _{X;}*)

*\W*) and clearly

*(γ(W*)) =

*−(W*). These facts together with the denition of

*m(V; W*) and Equation (3.4) imply

~

*(D*_{B;W}*; X)−(D*~ _{B;V}_{X;}*; X)*

= dim(γ(V* _{X;}*)

*\W*)

*−*

_{2i}

^{1}tr log((V

*)(W)*

_{X;}*)*

^{}= ^{1}_{2} *m(γ(V**X;*); W) + dim(γ(V*X;*)*\W*)
*:*

(3.5)

Using the denition ~*(D) =* ^{1}_{2}((D) + dim ker*D) we see that* *(D*_{B;W}*; X)−*
*(D*_{B;V}_{X;}*; X)−m(γ(V** _{X;}*); W) equals

*−*dim ker*D** _{B;W}* + dim ker

*D*

_{B;V}*+ dim(γ(V*

_{X;}*)*

_{X;}*\W*): (3.6) But (3.6) vanishes, as one sees by using the exact sequence (2.3). This proves the rst assertion of Theorem 3.2.

The second assertion follows from the rst part and [26, Lemma 8.15].

Using (2.5) and the rst assertion one sees that
*(D**B**; M)−(D*_{B;W}*X*

*; X)−(D*_{B;W}*Y*

*; Y*)*−m(W*_{}^{X}*; W*_{}* ^{Y}*)
equals

*m(V*_{X;}*; V** _{Y;}*)

*−m(γ(V*

*); W*

_{X;}

_{}*) +*

^{X}*m(γ*(V

*); W*

_{Y;}

_{}*)*

^{Y}*−m(W*

_{}

^{X}*; W*

_{}*): (3.7) (There is one subtlety: the sign change of the term*

^{Y}*m(γ*(V

*Y;*); W

_{}*) occurs because viewed from the \Y" side, the Hermitian symplectic structure changes sign.)*

^{Y}Using the identities *m(V; W*) = *−m(W; V*) and *(γ(W*)) = *−(W*), so that
*m(γ*(V); γ(W)) =*m(V; W*), we can rewrite (3.7) as

*−m(γ*(V* _{X;}*); W

_{}*)*

^{X}*−m(W*

_{}

^{X}*; W*

_{}*) +*

^{Y}*m(V*

_{X;}*; V*

*) +*

_{Y;}*m(V*

_{Y;}*; γ(W*

_{}*));*

^{Y}which equals

~

(V*X;**; V**Y;**; γ(W*_{}* ^{Y}*))

*−m(γ(W*

_{}*); V*

^{Y}*X;*)

*−(γ*~ (V*X;*); W_{}^{X}*; W*_{}* ^{Y}*) +

*m(W*

_{}

^{Y}*; γ(V*

*X;*))

=~(V*X;**; V**Y;**; γ(W*_{}* ^{Y}*))

*−(γ*~ (V

*X;*); W

_{}

^{X}*; W*

_{}*) as desired. This proves the third assertion.*

^{Y}The last statement follows straightforwardly from the previous or, alternatively, can be immediately recovered from [26, Theorem 8.8].

**4** **Lagrangians induced by bordisms**

Theorem 3.2 gives splitting formulas for the and *{invariants of* *D** _{B}* in the
situation when a manifold

*M*is decomposed into two pieces

*X*and

*Y*along a hypersurface . To develop this into a useful cut-and-paste machinery for the {invariant requires keeping track of the Lagrangian subspaces

*V*

*= image*

_{X;}*H*

*(X;C*

^{}

^{n}*)*

_{}*!H*

*(;C*

^{}

^{n}*) and their generalizations. It is clearest to give*

_{}an exposition based on the eect of a bordism on Lagrangian subspaces and we do this next.

Let *X* be a Riemannian manifold with boundary *−*_{0} *q*_{1} (we allow _{0} or
1 empty). Let : 1*X* *!U*(n) be a representation. Fix a flat connection *B*
on *X* with holonomy in cylindrical form near 0 and 1. The tangential
operator *A** _{b}* of

*D*

*acting on*

_{B}*X*decomposes as a direct sum

*A*

*=*

_{b}*A*

_{b;0}*A*

*since*

_{b;1}*L*

^{2}(@X) =

*L*

^{2}(0)

*L*

^{2}(1). In particular

ker*A** _{b}* = ker

*A*

_{b;0}*A*

*=*

_{b;1}*H*

*(*

^{}_{0};C

^{n}*)*

_{}*H*

*(*

^{}_{1};C

^{n}*):*

_{}We view *X* as a bordism from _{0} to _{1}.

We explained in the previous section that ker*A** _{b}* =

*H*

*(@X;C*

^{}*) is a Hermitian symplectic space. At this point we add the hypothesis that both ker*

^{n}*A*

*b;0*=

*H*

*(0;C*

^{}*) and ker*

^{n}*A*

*=*

_{b;1}*H*

*(1;C*

^{}*) be Hermitian symplectic spaces. This is not automatic, but follows for example if there exists a manifold*

^{n}*Y*with boundary 0 over which

*j*0: 10

*!*

*U*(n) extends. It is in this context that we will usually work.

We use *X* to dene a function *L**X;* from the set of subspaces of *H** ^{}*(0;C

*) to the set of subspaces of*

^{n}*H*

*(*

^{}_{1};C

*) by*

^{n}*L** _{X;}*(W) =

*P*

_{1}

*V*

_{X;}*\*(W

*H*

*(*

^{}_{1};C

*))*

^{n}*;* (4.1)

where *P*1: *H** ^{}*(@X;C

*)*

^{n}*!H*

*(1;C*

^{}*) denotes the projection onto the second factor:*

^{n}*P*_{1}: *H** ^{}*(@X;C

*) =*

^{n}*H*

*(*

^{}_{0};C

*)*

^{n}*H*

*(*

^{}_{1};C

*)*

^{n}*!H*

*(*

^{}_{1};C

*):*

^{n}In the following theorem, let *Y* be a Riemannian manifold with boundary 0

with a product metric *g*_{0}+*du*^{2} near the collar. Write
*Z* =*Y* *[*0 *X*

and assume that extends over *Z*. Let *γ*_{0} be the restriction of *γ* to ker*A** _{b;0}*.
Notice that

*γ*

_{0}(V

*)*

_{Y;}*L*

*(V*

_{X;}*) is a Lagrangian subspace of ker*

_{Y;}*A*

*.*

_{b}**Theorem 4.1** *The function of Equation* (4.1) *takes Lagrangian subspaces to*
*Lagrangian subspaces, i.e. it induces a function*

*L**X;*: ^{L}(H* ^{}*(0;C

*))*

^{n}*!*

^{L}(H

*(1;C*

^{}*)):*

^{n}*This function has the properties:*

(1) *If* *Y,* *Z* =*Y* *[*0*X* *are as above then*

*V** _{Z;}*=

*L*

*(V*

_{X;}*):*

_{Y;}*In short, the bordism propagates the distinguished Lagrangian. Moreover*
*(D**B;V**Z;**; Z*) =*(D**B;V**Y;**; Y*) +*(D*_{B;γ}_{0}_{(V}_{Y;}_{)}_{}_{L}_{X;}_{(V}_{Y;}_{)}*; X)*
*and hence* *(Z; ; g*1) *equals*

*(Y; ; g*0) +*(X; ; g*0*qg*1*; γ*0(V*Y;*)*L**X;*(V*Y;*); γ0(V*Y;*)*L**X;*(V*Y;*)):

*where* *g*_{i}*is a metric on* _{i}*.*

(2) *If* *X*_{1} *is a bordism from* _{0} *to* _{1} *and* *X*_{2} *is a bordism from* _{1} *to* _{2}
*and* : 1(X1*[*1*X*2)*!U*(n) *then*

*L**X*1*[*_{1}*X*2*;*=*L**X*2*;**L**X*1*;**:*

**Proof** The map of (4.1) is just the map taking*V**X;* to its symplectic reduction
with respect to the subspace *W* *H** ^{}*(

_{1};C

*)*

^{n}*H*

*(*

^{}_{0};C

*)*

^{n}*H*

*(*

^{}_{1};C

*) (cf.*

^{n}[26, Sec. 6.3]). Symplectic reduction takes Lagrangians to Lagrangians.

To prove the the rst part of (1) consider a *2* *V*_{Y}_{[}_{0}* _{X;}*. Then there is a

*w*

*2H*

*(Y*

^{}*[*0

*X;*C

*) with*

^{n}*i*

^{}_{}

_{1}

*w*= . We put 0 :=

*−i*

^{}_{}

_{0}

*w. Since certainly*

*w*

_{j}

_{X}*2*

*H*

*(X;C*

^{}*) we infer*

^{n}_{0}=

*i*

^{}

_{@X}*w*

*2*

*V*

*. Thus =*

_{X;}*P*

_{1}(

_{0}

*)*

*2*

*P*

_{1}(V

_{X;}*\*(V

_{Y;}*H*

*(*

^{}_{1};C

^{n}*))).*

_{}Conversely, let *2* *P*1(V*X;**\*(V*Y;**H** ^{}*(1;C

^{n}*))) be given. Then there is*

_{}_{0}

*2V*

*such that*

_{Y;}_{0}

*2V*

*. Thus we may choose*

_{X;}*w*

_{X}*2H*

*(X;C*

^{}*) with*

^{n}*i*

^{}

_{@X}*w*

*=*

_{X}_{0}and

*w*

_{Y}*2H*

*(Y;C*

^{}*) with*

^{n}*i*

^{}_{}

0*w** _{Y}* =

_{0}.

From the Mayer{Vietoris sequence of *Y* *[*0 *X* we obtain an *w* *2* *H** ^{}*(Y

*[*0

*X;*C* ^{n}*) with

*w*

_{j}*=*

_{Y}*w*

*Y*and

*w*

_{j}*=*

_{X}*w*

*X*. Then =

*i*

^{}_{}

_{1}

*w*

*X*=

*i*

^{}_{}

_{1}

*w2V*

*Y*

*[*

_{0}

*X;*

and we reach the conclusion.

Consider now the second part of (1). We have explained in [26, Sec. 7] that
the gluing formula for {invariants remain true if one glues (a nite union of)
components of the boundary and xes a boundary condition at the remaining
components. The result now follows from *V** _{Z;}*=

*L*

*(V*

_{X;}*) and Theorem 3.2.*

_{Y;}The proof of (2) proceeds along the same lines as the proof of the rst part
of (1). Consider *W* *H** ^{}*(0;C

^{n}*) and a*

_{}*2*

*L*

*X*1

*[*

_{1}

*X*2

*;*(W). Then there is a

*w*

*2*

*H*

*(X*

^{}_{1}

*[*1

*X*

_{2};C

*) with*

^{n}*i*

^{}_{}

2*w* = and _{0} := *−i*^{}_{}

0*w* *2* *W*. Put
_{1}:=*i*^{}_{}

1*w. Then it is immediate that* _{0}_{1}*2V*_{X}_{1}_{;}*\*(W*H** ^{}*(

_{1};C

*)) and*

^{n}*−*1*2V**X*2*;**\*(L*X*1*;*(W)*H** ^{}*(2;C

^{n}*)). This proves*

_{}*2L*

*X*2

*;*

*L*

*X*1

*;*(W).

Conversely, let *2* *L*_{X}_{2}_{;}*L*_{X}_{1}* _{;}*(W) be given. Then there exists a

_{1}

*2*

*H*

*(*

^{}_{1};C

^{n}*) such that*

_{}*−*

_{1}

*2V*

_{X}_{2}

_{;}*\*(L

_{X}_{1}

*(W)*

_{;}*H*

*(*

^{}_{2};C

^{n}*)) and a*

_{}_{0}

*2*

*H*

*(0;C*

^{}*) such that*

^{n}*−*01

*2V*

*X*1

*;*

*\*(W

*H*

*(1*

^{}*;*C

*)).*

^{n}A Mayer{Vietoris argument as in the proof of the rst part of (1) shows the
existence of a *w* *2* *H** ^{}*(X

_{1}

*[*1

*X*

_{2};C

*) such that*

^{n}*i*

^{}_{}

2*w* = and *i*^{}_{}

0*w* = *−*_{0}.
This proves *−*_{0} *2* *V*_{X}_{1}_{[}_{1}_{X}_{2}_{;}*\*(W *H** ^{}*(

_{2};C

^{n}*)), and hence*

_{}*L*

_{X}_{2}

_{;}*L*

*X*1

*;*(W)

*L*

*X*1

*[*

_{1}

*X*2

*;*(W).

Theorem 4.1 easily extends to the situation

*Z* =*Y* *[*0 *X*_{1}*[*1* [**n**X*_{n+1}*:*

This gives a useful strategy for computing{invariants by decomposing a closed manifold into a sequence of bordisms, e.g. by cutting along level sets of a Morse function.

We use Theorem 4.1 and the denitions to write down a formula which expresses
the dependence of *(Y; ; g) on the metric* *g* on *@Y*.

**Corollary 4.2** *LetY* *be a compact manifold with boundary*. Let: _{1}*Y* *!*
*U*(n) *be a representation. Suppose that* *g*_{0}*,* *g*_{1} *are two Riemannian metrics on*
. Choose a path of metrics from *g*0 *to* *g*1 *and view this path as a metric on*
[0;1].

*Then*

*(Y; ; g*_{1})*−(Y; ; g*_{0})

=*(D*_{B;γ}_{0}_{(V}_{Y;}_{)}_{}_{V}_{Y;}*;*[0;1])*−(D*_{;γ}_{0}_{(V}_{Y;}_{)}_{}_{V}_{Y;}*;*[0;1])

=*(*[0;1]; g0*qg*1*; γ*0(V*Y;*)*V**Y;**; γ*0(V*Y;*)*V**Y;*):

Here, is oriented such that a collar of the boundary takes the form (*−;*0].

**Proof** Apply Theorem 4.1 with *X* = [0;1] and note that for the cylinder
*X*= [0;1] the map *L** _{X;}* is the identity.

In Section 6 we will use Corollary 4.2 to give examples that show that*(Y; ; g)*
depends in general on the choice of*g, in contrast with the* *{invariant for closed*
manifolds.