On localized signature and higher rho invariant of fibered manifolds

Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in"On the index of a fibered manifold". We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in"Positive scalar curvature, higher rho invariants and localization algebras"and Zeidler in"Positive scalar curvature and product formulas for secondary index invariants".


Introduction
The signature of a 4k-dimensional manifold is defined to be the signature of the cup product as a non-degenerate symmetric bilinear form on the vector space of 2k-cohomology classes. In [1], Chern, Hirzebruch and Serre established a product formula of signature for fibered manifold. More precisely, let F Ñ E Ñ B be a fibered manifold with base manifold B and fiber manifold F , if π 1 pBq acts trivially on Hd R pF q, the de Rham cohomology of F . We have the following product formula sgnpBqˆsgnpF q " sgnpEq. (1.1) The signature of a manifold is also equal to the Fredholm index of the signature operator. When taking into account of the fundamental group of the manifold, one can introduce higher invariants of the signature operator, which lie in the K-theory of certain geometric C˚-algebras. Let M be an m-dimensional manifold with fundamental group π 1 pM q " G and universal covering Ă M . Let D sgn M be the signature operator on M . The higher index of D sgn M , IndpD sgn M q is a generalization of the Fredholm index, and is defined to be an element in K m pC˚p Ă M q G q, where C˚p Ă M q G is the equivariant Roe algebra of Ă M and is Morita equivalent to the reduced group C˚-algebra Cr pGq. The higher index of signature operator is invariant under homotopy equivalence and oriented cobordism, and plays a fundamental role in the study of classification of manifolds. On the other hand, D sgn M defines a K-homology class rD sgn M s in K m pCLp Ă M q G q, the K-theory of the equivariant localization algebra. See Section 2 and 3 for the explicit definitions of equivariant geometric C˚-algebras, higher index and K-homology class of signature operator.
Furthermore, if f : M 1 Ñ M is an orientation-preserving homotopy equivalence of closed manifolds, then there exists a concrete homotopy path that realizes the equality where m is the dimension of M and M 1 . This homotopy path allows one to define a secondary invariant of signature operator associated to the homotopy equivalence f , called higher rho invariant, in the K-theory of the equivariant obstruction algebra CL ,0 p Ă M q G . The higher rho invariant of signature operator associated to homotopy equivalence plays a central role in estimating the topological nonrigidity of a manifold (cf: [4,7,8,13,18]).
Inspired by Chern, Hirzebruch and Serre's product formula, we prove a product formula for higher index and higher rho invariant of signature operator on fibered manifold. More precisely, consider a closed fibered manifold F Ñ E Ñ B with base space B and fiber F . Denote the fundamental group of E by G, and the fundamental group of B by H. Let r E and r B be the universal covering spaces of E and B respectively. Set n " dim F and m " dim B.
We first define equivariant family localization algebra CLp r E; r Bq G , and family obstruction algebra CL ,0 p r E; r Bq G . We show that there are naturally defined product maps: φ : K m pCLp r Bq H q b K n pCLp r E, r Bq G q Ñ K m`n pCLp r Eq G q, φ 0 : K m pCLp r Bq H q b K n pCL ,0 p r E, r Bq G q Ñ K m`n pCL ,0 p r Eq G q.
Taking advantage of the fiberwise signature operator, we introduce the family K-homology class of family signature operator along F , denoted by rD sgn E,B s, in the K-theory of the equivariant family localization algebra, and the family higher rho invariant ρpf ; Bq, associated to a fiberwise homotopy equivalence f : E 1 Ñ E, in the K-theory of the equivariant family obstruction algebra. The following theorem is a product formula for K-homology class of signature operator on fibered manifold, which implies the product formula for higher index of signature operator. Theorem 1.1. Let F Ñ E Ñ B be fibered manifold with base space B and fiber F . Denote the fundamental group of E by G, and the fundamental group of B by H. Let r E and r B be the universal covering spaces of E and B respectively. Write dim F " n and dim B " m. Let rD sgn E,B s be the family K-homology class of the family signature operator in K n pCLp r E, r Bq G q. We have the following product formula for family K-homology class of family signature operator k mn¨φ prD sgn B s b rD sgn E,B sq " rD sgn E s, where k mn " 1 when mn is even and k mn " 2 otherwise, and φ is the product We also obtain the following product formula for higher rho invariant of signature operator on fibered manifold.
where k mn " 1 when mn is even and k mn " 2 otherwise, and φ 0 is the product map As an application of Theorem 1.1, we give an alternative proof of the product formula of Chern, Hirzebruch and Serre (cf: [1]). Also, the product formula of higher rho invariant stated in Theorem 1.2 can be applied to study the behavior of higher rho map in [13] under fibration, and thus can be applied to study the topological nonrigidty of fibered manifold.
We mention that the product formula for higher index of signature operator has been obtained by Wahl in [11]. In this paper, we give a new proof of Wahl's product formula. On the other hand, product formula for higher rho invariant for positive scalar curvature metric on product manifolds has been proved by Siegel in his thesis [9], by Xie and Yu in [14], and by Zeidler in [17]. Their results and Theorem 6.8, 6.9 in [13] inspire us to study the product formula for higher rho invariant for signature operators.
The paper is organized as follows. In Section 2, we briefly recall some definitions of geometric C˚-algebras that we may use throughout the paper. In Section 3, we revisit the construction of several higher invariants associated to the signature operator. Next in Section 4, we prove the product formulas for higher index and higher rho invariant of signature operator on product manifolds. In Section 5, we generalize the product formulas to fibered manifold and prove Theorem 1.1 and 1.2. We shall define an auxiliary C˚-algebra consisting of operators that can be localized along the base manifold, and use the Mayer-Vietoris arguments.

Preliminary
The aim of this section is to briefly recall some basic definitions of geometric C˚-algebras used throughout the paper. For more details, we refer the readers to [13].
Let X be a proper metric space and G be a finitely represented discrete group. Suppose that G acts on X properly by isometries. For simplicity, we assume that the G-action is free. Let C 0 pXq be the C˚-algebra consisting of all complex-valued continuous functions on X that vanish at infinity. An X-module is a separable Hilbert space H X equipped with a˚-representation of C 0 pXq. It is called nondegenerate if the˚-representation is nondegenerate, and standard if no nonzero function in C 0 pXq acts as a compact operator. Additionally we assume that H X is equipped with a unitary representation of G which is compatible with C 0 pXq-representation, that is, @f P C 0 pXq, g P G, πpgqφpf q " φpg.f qπpgq where φ (resp. π) is the C 0 pXq(resp. G)-representation on H X and g.f pxq " f pg´1xq. Now let us recall the definitions of propagation of operator and locally compact operator.
Definition 2.1. Under the above assumptions, let T be a bounded linear operator acting on H X .
1. The propagation of T is defined by where SupppT q is the complement (in XˆX) of the set of points px, yq P XˆX such that there exists f 1 , f 2 P C 0 pXq such that f 1 T f 2 " 0 and f 1 pxqf 2 pyq ‰ 0; 2. T is said to be locally compact if both f T and T f are compact for all f P C 0 pXq.
In the following, we recall the definitions of the equivariant Roe algebra, localization algebra, and the obstruction algebra.

Definition 2.2.
Let H X be a standard nondegenerate X-module and BpH X q the set of all bounded linear operators on H X .
1. The G-equivariant Roe algebra of X, denoted by C˚pXq G , is the C˚algebra generated by all G-equivariant locally compact operators with finite propagation in BpH X q.
3. The G-equivariant obstruction algebra CL ,0 pXq G is defined to be the kernel of the following evaluation map ev : In particular, CL ,0 pXq G is an ideal of CLpXq G .
Remark 2.3. Up to isomorphism, C˚pXq G does not depend on the choice of the standard nondegenerate X-module. The same holds for CLpXq G and CL ,0 pXq G .
When X is a Galois G-covering of a closed Riemannian manifold, L 2 pXq is a standard nondegenerate X-module. In this case, there is an equivalent definition of equivariant Roe algebra.
Definition 2.4. Let X be a Galois G-covering of a closed Riemannian manifold. Set CrXs G as a˚-algebra consisting of integral operators given by where k : XˆX Ñ C is uniformly continuous, bounded on XˆX, and has finite propagation, i.e.
Remark 2.5. If we remove the cocompactness of the G-action, Definition 2.4 actually gives us the uniform equivariant Roe algebra, which is different from the equivariant Roe algebra.
Suppose that T P CrXs G has corresponding Schwartz kernel kpx, yq. The support of T defined in Definition 2.2 is simply given by SupppT q " tpx, yq P XˆX : kpx, yq ‰ 0u. (2.2) Similarly we define the G-equivariant localization algebra CLpXq G to be the completion of all paths on t P r0,`8q with value in CrXs G , which are uniformly continuous and uniformly bounded with respect to the operator norm, and have propagation going to zero as t goes to infinity. Definition 2.4 can be easily generalized to the case where L 2 pXq is replaced by L 2 -section of an Hermitian vector bundle over X on which G acts by isometries. The above definitions coincide with Definition 2.2, as those C˚-algebras are independent of the choice of the standard nondegenerate X-module. Now we consider a product of two manifolds. Let M, N be two closed manifolds and Ă M , r N be their Galois G,H-covering spaces respectively. For any pair of integral operators in Cr Ă M s G and Cr r Ns H , their tensor product is well-defined as an operator in Cr Ă Mˆr N s GˆH . This induces the following product maps 3 Higher invariant associated to signature operator In this section, we recall a formula of the higher index of signature operator, which was obtained by Higson and Roe in [2] and [3]. After that we will give the construction of K-homology class of the signature operator, which was originally introduced by Weinberger, Xie and Yu in [13]. At last, we give the definition of the higher rho invariant of controlled homotopy equivalence of manifolds.
In this section, all manifolds mentioned are not assumed to be compact or connected unless otherwise noted.
Odd case. Suppose that M is odd dimensional. For any n P N`, let M n be the manifold M equipped with metric g n , which is n-times of the original metric g. Denote by š n M n the disjoint union of M n . The higher index of signature operator D sgn š Mn is represented by the invertible operator Denote by T n the restriction of T to M n . As shown in the previous subsection, T can be approximated by operators with finite propagation on š n M n .
Therefore with respect to the original metric g, tT n u can be approximated by a sequence of operators uniformly whose propagation goes to zero as n goes to infinity. To make the above sequence a continuous path on r1,`8q, we further consider š n M n`r for r P r0, 1s rather than š n M n , where the metric on M n`r is pn`rq-times of the original metric. This actually gives an invertible element in CLp Ă M q G .
Definition 3.1. We call the K-theory element in K 1 pCLp Ă M q G q represented by the invertible element defined above the K-homology class of signature operator, which will be denoted by rD sgn M s.
Even case. We sketch the construction of K-homology class of signature operator for the case of m being even but leave out the details. The higher index of signature operator D sgn š Mn is determined by the difference of projec- Mn q. Again, we consider the higher index of signature operator on š n M n`r for r P r0, 1s. This gives us a continuous path from r1,`8q to C˚p Ă M q G , which in turn defines a path of difference of projections lies in CLp Ă M q G , denoted by P M,`p tq´P M,´p tq, with respect to the original metric g.
Definition 3.2. We call the K-theory element determined by the formal difference rP M,`p tqs´rP M,´p tqs the K-homology class of signature operator D sgn M , which will be denoted by rD sgn M s P K 0 pCLp Ă M q G q.

Controlled homotopy equivalence and higher rho invariant
In this subsection, we recall the construction of higher rho invariant of signature operator associated to homotopy equivalence. Higher rho invariant associated to smooth homotopy equivalence was first introduced by Higson and Roe in [2][3][4]. Later in [8], Piazza and Schick gave an index theoretic definition of higher rho invariant of signature operator. In [18], Zenobi extended Higson and Roe, Piazza and Schick's work to define higher rho invariant associated to topological homotopy equivalence.
In [13], Weinberger, Xie and Yu constructed higher rho invariant of signature operator associated to homotopy equivalence by piecewise-linear approach. In this paper, we adapt their construction to give a differential geometric approach to the definition of higher rho invariant. It is not hard to see that our construction here is equivalent to the one given in [15]. t , t P r0, 1s (resp. h t , t P r0, 1s) the smooth homotopy between f g and id M 1 (resp. gf and id M ). We say that f is a controlled homotopy equivalence if there exists a positive constant C such that 1. the diameter of th 1 t paq|0 ď t ď 1u is bounded by C uniformly for all a P M 1 , 2. the diameter of th t pbq|0 ď t ď 1u is bounded by C uniformly for all b P M .
, which we will still denote by f . In general, the induced map is not a bounded operator. However, we may apply the Hilsum-Skandalis submersion (cf: [6,Page74], [12,Page 157], and [15, Page 34]) to construct a bounded operator T f out of f . Without loss of generality, we might as well assume that f is a bounded operator. Now let us recall the definition of higher rho invariant associated to controlled homotopy equivalence f according to the parity of dim M . We mention here that the construction is due to Higson and Roe (cf: [2], [4]), and Weinberger, Xie and Yu (cf: [13]).
Odd case We first assume that both M 1 and M are odd dimensional. Via conjugating by f on the first summand Λ˚p Ă M 1 q, we may identify D and S with their corresponding operators acting on Λ˚p Ă M q ' Λ˚p Ă M q. Under this identification, the invertible element defined by pD`SqpD´Sq´1ˇˇΛ The construction in Definition 3.1 gives rise to an invertible element pD t`St qpD t´St q´1ˇˇΛ In particular, we have D 1 " D and S 1 " S. Since f is a controlled homotopy equivalence, it gives rise to a canonical path that connects pD`SqpD´Sq´1 to the identity operator as shown by Higson and Roe in [2]. The path is constructed out of the following path S f ptq connecting S with´S, For any t P r0, 6s, S f ptq satisfying the following conditions: Definition 3.5. The higher rho invariant ρpf q is the K-theory class in Even case. The even dimensional case is parallel to the odd case above. The construction in Definition 3.2 gives rise to a path of difference of projections with P`pD 1`S1 q´P`pD 1´S1 q " P`pD`Sq´P`pD´Sq.
Let S f ptq be as above. Similarly, we have that: 1. D˘S f ptq are both invertible; 2. P`pD˘S f ptqq are of finite propagation, and Thus the formal difference The higher rho invariant associated to controlled homotopy equivalence f is defined as follows.
As Θ f,˘p tq are projections and their difference lies in M 2 pCL ,0 p Ă M q G q, the formal difference rΘ f,`s´r Θ f,´s defines a K-theory class ρpf q in K 0 pCL ,0 p Ă M q G q, called higher rho invariant.

Product formula
In this section, we will prove the product formula for higher rho invariant associated to the signature operator for homotopy equivalence. We only consider the case for product of manifolds for now. The general case for fibered manifolds will be discussed in the next section.
Proof. In the following, we omit the mention of ψ for simplicity. We avoid to use the fact that S 2 Ă M " 1 throughout the proof for the purpose of further generalization. Therefore, we have to consider four cases according to the parity of both dim M and dim N .
Even times odd. We first suppose that dim M is even and dim N is odd.
M for short. On the product manifold Ă Mˆr N , the differential operator d Ă Mˆr N is given by where E Ă M is the even-odd grading operator for Λp Ă M q. Therefore As dim N is odd, the Hodge˚-operator as well as the Poincaré duality operator S r N reverses the parity of Λp r N q. Note that S 2 r N : Λ p p r N q Ñ Λ p p r N q is a multiple of identity. Therefore we identify Under this identification, the higher index of signature operator is represented by the following invertible operator Since BẰ M is invertible, we define a path of bounded operators For the invertible operator BĂ . We see that Since D r N anti-commutes with S r N , we have r N ą 0. Thus W`, s is a path of invertible operator for every s in r0, 1s.
Similarly we define a path of invertible operator W´, s . Thus via the homotopy W`, s pW´, s q´1, the higher index of signature operator on MˆN is also represented by the invertible operator We rewrite the expression above using 1 " P`pBĂ The last two equalities follow from the definition of product of K-groups and the formula of the higher index of signature operator in Section 3.1.
Odd times even. Suppose that M is odd dimensional and N is even dimensional. Straightforward computation shows that Let Λ˘p r N q be the˘1 eigenspace of S r N . We make the following identifications 1. Under the decomposition 2. Under the decomposition Λ even p Ă MˆN q " With these identifications, we have re positive invertible operators. It follows that d Ă

Mˆr NdĂ
Mˆr N`S Ă Mˆr N is homotopic tô where the matrix form is written with respect to the decomposition Λp r N q " Λ`p r N q ' Λ´p r N q.
Since D r N is off-diagonal, d Ă
1`x 2 . In the meantime, d Ă It follows that where S 1 " S r N , and S 2 " gpD r N q`S r N f pD r N q. Note that S 2 S 1 S 2 is a symmetry, i.e. S 2 S 1 S 2 can be approximated by finite propagation operators, and pS 2 S 1 S 2 q 2´1 belongs to C˚p r N q H . We define Now one can see that the higher index of D sgn MˆN is actually represented by as computed in [3].
Even times even. Suppose that both M and N are even dimensional.
In [13], it is shown that Note that IndpD sgn R q is the generator of

Now we have
It follows that IndpD sgn M q b IndpD sgn N q " IndpD sgn MˆN q.
Odd times odd. Let M and N be both odd dimensional manifolds. In this case, as shown in [13], we have 2IndpD sgn N q b IndpD sgn R q " IndpD sgn NˆR q, IndpD sgn MˆN q b IndpD sgn R q " IndpD sgn MˆNˆR q.

Now we have
2IndpD sgn M qq b IndpD sgn N q b IndpD sgn R q "IndpD sgn M q b IndpD sgn NˆR q "IndpD sgn MˆNˆR q "IndpD sgn MˆN q b IndpD sgn R q.
It follows that Note that in the proof of Proposition of 4.1, we actually do not demand that S 2 Ă M " 1. It follows that given the definition of K-homology class of signature and higher rho invariant, the argument above can be easily generalized to show the following Proposition and Theorem:

Proposition 4.2. With the same notations, under the product map
there is a product formula of K-homology class of signature operator on M and N goes as follows, 1, mn is even, 2, mn is odd.
there is a product formula goes as follows, k mn¨ψL,0 pρpf q b rD sgn N sq " ρpfˆI N q, where I N : N Ñ N is the identity map, and 1, mn is even, 2, mn is odd.

Product formula for fibered manifolds
In this section, we generalize the product formula given in the previous section to fibered manifolds. We will first introduce a series of family geometric C˚algebras with respect to the fibration. Next we define the family version of K-homology class and higher rho invariant of fiberwise signature operator in K-theory of these C˚-algebras. At last, We will prove Theorem 1.1 and 1.2.

Family algebras
In this subsection, we introduce family geometric C˚-algebras associated to a fibered manifold. Let π : E Ñ B be a fibration with fiber F and base space B. Assume that E, F and B are closed connected oriented Riemannian manifolds. The fibration induces a long exact sequence of homotopy groups¨¨/ / π 2 pBq B / / π 1 pF q / / π 1 pEq π˚/ / π 1 pBq / / 0 Denote by r E and r B the universal covering of E and B. From the exactness of the above sequence, we see that Bpπ 2 pBqq is a normal subgroup of π 1 pF q. Write Γ " π 1 pF q{Bpπ 2 pBqq. The above exact sequence shows that r E is also a fibration on r B with fiber projection r π : r E Ñ r B and fiber r F , the Galois Γ-covering of F .
From now on, we will write G " π 1 pEq and H " π 1 pBq for short. Recall that the equivariant Roe algebras C˚p r Eq G is defined to be the completion of Gequivariant, locally compact operators with finite propagation as in Definition 2.2. Now let us define the equivariant family Roe algebra.
First we construct an equivariant Roe algebra bundle over B. View the fiber bundle E over B as gluing many pieces of local trivialization by a series of diffeomorphisms of F . More precisely, there exists an open cover tV α u of B and continuous maps ϕ αβ : V α X V β Ñ DiffpF q such that the fiber bundle E is equivalent to the tuple pV αˆF , ϕ αβ q, that is, every continuous section s of E is equivalent to a series of continuous maps s α : V α Ñ F satisfying the cocycle condition, ϕ αβ pxqs α pxq " s β pxq for any x P V α X V β .
By our previous arguments, when turning to the universal covering, r E is also an r F -bundle over r B.
Assume that every open set in tV α u is small enough so that it lifts to a π 1 pBq-equivariant open cover tU j u of r B, each open set of which is homeomorphic to the Euclidean space and trivialize r E. Also, the transition map lifts to tψ ij u with ψ ij : U i X U j Ñ Diffp r F q. Recall the subspace Cr r F s Γ of the equivariant Roe algebra C˚p r F q Γ as in Definition 2.4. For any x P U i X U j , ψ ij pxq induces an isomorphism ψ ij,˚p xq of C˚p r F q Γ by conjugation, which maps Cr r F s Γ to itself. This induces the following fiber bundle.
Definition 5.1 (Equivariant family Roe algebra). Recall that G " π 1 pEq and Γ " π 1 pF q{Bpπ 2 pBqq. A continuous section of the fiber bundle given by ptU i u, tψ ij,˚u q is defined by a series of norm-continuous maps s i : U i Ñ C˚p r F q Γ satisfying the cocycle condition, ψ ij,˚p xqs i pxq " s j pxq for any x P U i X U j . Let Cr r E, r Bs G be the collection of uniformly norm-bounded and uniformly normcontinuous sections that are invariant under π 1 pBq-action, and have uniformly finite propagation on r B. The norm of such a section ts j u is defined to be sup j sup xPU j }s j pxq}. Denote the completion of Cr r E, r Bs G by C˚p r E, r Bq G .
We mentioned that Definition 5.1 is related to the "Groupoid Roe algebra" given by Tang, Willett and Yao ( [10, Definition 3.6]).
It is easy to verify that the above definition is independent of the local trivialization. Similarly, we define the corresponding equivariant family localization algebra.
Definition 5.2 (Equivariant family localization and obstruction algebras). The equivariant family localization algebra CLp r E, r Bq G is the completion of uniformly norm-bounded and uniformly norm-continuous paths s : r0,`8q Ñ C˚p r E, r Bq G such that the propagation of sptq goes to zero as t goes to infinity uniformly on r B, where the norm of sptq is defined to be sup tPr0,`8q ||sptq||. The equivariant family obstruction algebra CL ,0 p r E, r Bq G is then defined to be the kernel of the family assembly map:

Product map of K-theory
In this subsection, we construct the productive map on the family C˚-algebras.
Theorem 5.3. Recall that G " π 1 pEq and H " π 1 pBq. There are product maps Proof. Without loss of generality, we assume that both m and n are even. We will only give in details the construction of Suppose that f t P pCLp r Bq H q`represents a K 0 -class, which has finite propagation that goes to zero as t goes to infinity, and is a 1{10-projection, that is, ft " f t and }f 2 t´f t } ă 1{10. Similarly, we suppose that g t P pCL ,0 p r E, r Bq G qì s a 1{10-projection, has finite propagation that goes to zero uniformly as t goes to infinity, and satisfies that g 1 " 1. Furthermore, we assume that f t´1 and g t´1 are given by kernel operators acting on L 2 -sections as in Definition 2.4.
Choose r ą 0 small enough such that for any x P B, the restriction of the fiber bundle E to the r-ball near x is trivial. Since f t and f t`M are homotopic for any M ą 0, we may assume that the propagation of f t is smaller than r. By the local triviality, we define φprf t s b rg t sq " rpf t´1 q b pg t´1 q`1s, where pf t´1 q b pg t´1 q`1 P pCL ,0 p r Eq G q`is given bý`p pf t´1 qpx, x 1 q b pg t´1 q x 1 py, y 1 qhpx 1 , y 1 qdy 1 dx 1 , with h P L 2 p r Eq. The above expression makes sense as the propagation of f t is small enough. It is easy to verify that pf t´1 q b pg t´1 q`1 is at most a 3{10-projection, which gives rise to a K 0 -class. Now passing to the matrix algebra and the Grothendieck group, we obtain the product map.

Family higher invariants
In this subsection, we introduce the family version of higher invariants of the signature operator on fibered manifold, and prove Theorem 1.1 and 1.2.
On the fibered manifold E, the vertical differentials and Poincaré duality are well-defined as they are compatible with the transition maps. Thus the family K-homology class of the vertical signature operator rD sgn E,B s P K dim F pCLp r E, r Bq G q is defined similarly as in Definition 3.1 and 3.2.
Theorem 5.4. We have the following product formula for family K-homology class of family signature operator along F holds: where k B,F " 1 when dim B¨dim F is even, and k B,F " 2 otherwise, and φ is the product map We shall also define the family higher rho invariant of a fiberwise homotopy equivalence. Suppose we have two fibrations over the same base Let f : E 1 Ñ E be a fiberwise homotopy equivalence, that is, the following diagram commutes as well as replacing f with its homotopy inverse and the corresponding homotopy. Using the vertical differential and the Poincaré duality operator, we define a family higher rho invariant ρpf ; Bq P K dim F pCL ,0 p r E, r Bq G q as in Definition 3.5.
Theorem 5.5. With the same notation as above, we have the following product formula for family higher rho invariant associated to fiberwise homotopy equivalence holds: k B,F φprD sgn B s b ρpf ; Bqq " ρpf q, where k B,F " 1 when dim B¨dim F is even, and k B,F " 2 otherwise, and φ 0 is the product map In the following, we only prove Theorem 5.5 in details. The proof for Theorem 5.4 is similar.
We need some definitions to prepare for the proof of Theorem 5.5.
Definition 5.6. For any element T P C˚p r Eq G , we define the propagation of T along the base space r B by prop r B pT q " suptdpr πpxq, r πpyqq : px, yq P SupppT qu, where r π is the lift of the fiber projection π : E Ñ B.
We need the following C˚-algebra generated by elements in CL ,0 p r Eq G that can be localized horizontally. This is a generalization of the equivariant localization algebra defined in Definition 2.2. p r Eq G to be the C˚-algebra generated by paths f : r1,`8q Ñ CL ,0 p r Eq G such that f psq is uniformly norm-continuous and uniformly norm-bounded, and its propagation along r B is finite and goes to zero uniformly as s goes to 8. The norm of f P Cr B,L,0 p r Eq G is given by the supreme of its norm in CL ,0 p r Eq G , that is, }f } " sup sě1 }f psq}.
There is an evaluation map ev : Cr B,L,0 p r Eq G Ñ CL ,0 p r Eq G , which induces a K-theoretical map denoted by ev˚. If X is a closed Riemannian manifold, the equivariant localization algebra CLp r Xq π 1 X admits a Mayer-Vietoris sequence for a partition of X. More precisely, if U 1 , U 2 are two open sets on X and Ă U 1 , Ă U 2 are their lifts to r X, then we have the following six-term exact sequence (cf: [16,Proposition 3.11]).
As the C˚-algebra Cr B,L,0 p r Eq G is generated by elements that can be localized along B, it also admits a Mayer-Vietoris sequence as above for two open sets on the base space.
Proof. We sketch the proof of Proposition 5.8 as follows, which is essentially the same as the proof of Proposition 3.11 in [16]. For any open subset Y of B, we define pCr B,L,0 p r Eq G q Y to be the C˚-subalgebra of Cr B,L,0 p r Eq G generated by all paths f : r1, 8q Ñ CL ,0 p r Eq G such that for any s, t P r1, 8q, proppf ps, tqq ă 8 as an operator in Cr r Es G and for any ε ą 0, there exists S ą 0 for any s ą S and t P r1, 8q, Supppf ps, tqq lies in the ε-neighborhood of r Proof. Without loss of generality, we assume that both m and n are zero.
With the same notations as in the proof of Theorem 5.3, we define φ L,0 by φprf t s b rg t sq " rpf t`s´1´1 q b pg t´1 q`1s, where pf t`s´1´1 q b pg t´1 q`1 P pCL ,0 p r Eq G q`is given bý`p x, x 1 q b`g t´1˘x1 py, y 1 qhpx 1 , y 1 qdy 1 dx 1 , with h P L 2 p r Eq. Here t P r1,`8q is the parameter in CL ,0 p r Eq G and s P r1,`8q is the extra parameter induced in Cr B,L,0 p r Eq G . The expression makes sense as we may assume that the propagation of f t is small enough. After passing to the matrix algebra and the Grothendieck group, we obtain the map φ L,0 . The commuting diagram follows directly from the definition.
Lemma 5.10. With the same notations, for the fiberwise homotopy equivalence f , there exists a K-theory class ρ L pf q P K dim E pCr B,L,0 p r Eq G q such that Proof. Denote by g B , g E 1 and g E the metric on B, E 1 and E respectively. For any r P r0, 1s and n P N`, let š n B n`r be the disjoint union of countably many B's, where B n`r is equipped with the metric pn`rqg B . Similarly, we define š n E 1 n`r and š n E n`r , where E 1 n`r and E n`r are equipped with the metric g E 1`p n`rqπ 1˚gB and g E`p n`rqπ˚g B . Now we have the following fibrations Hence for some ε ą 0 small enough, ρp š n f n`r q admits a K 0 (resp. K 1 )representative which is a ε-almost projection (resp. unitary) with finite propagation uniformly in n P N`. Along r B the propagation of such representative restricted to r E n`r goes to zero as n goes to infinity. When r varies in r0, 1s, the above construction gives rise to a path ρ L pf qpsq for s P r1,`8q, along which the propagation along r B goes to zero as s goes to infinity. Therefore the path defines a class in K dim E pCr B,L,0 p r Eq G q, which we will also denote by ρ L pf qpsq. Furthermore, ρ L pf qp1q represents the same K-theory class as ρpf q in K dim E pCL ,0 p r Eq G q by definition. This finishes the proof. Now we are ready to prove Theorem 5.5. We will go through the proof in details only for the case where the dimension of B and F are both even. The other cases are totally similar.
Proof of Theorem 5.5. Let ρ L pf q be as constructed in the proof of Lemma 5.10. We shall show that by the Mayer-Vietoris arguments. And the theorem follows from Lemma 5.9 and 5.10. We first assume a special case where E " FˆB, a trivial fiber bundle over B. In this case, the family algebra CL ,0 p r E, r Bq G is isomorphic to The product map φ L,0 and the localized higher rho invariant ρ L pf q are constructed in Lemma 5.9 and 5.10 respectively. Using the same construction as in Section 4, we will obtain line (5.3) for this trivial case. Now we turn to the general situation. For simplicity, we assume that the base space B admits a triangulation that makes it a simplicial complex. Assume that the diameter of every simplex on B is small enough so that the restriction of E on every simplex is trivial. Let B pkq be a small open neighborhood of the k-skeleton of B, which contain the k-skeleton of B as a deformation retraction. In particular, B pkq " B when k is dim B. Denote the lift of B pkq to r B by r B pkq and the restriction of r E to r B pkq by r E r B pkq . For any K-theory element in K˚pCr B,L,0 p r Eq G q, its restriction to r E r B pkq is well defined by multiplying the element by the characteristic function of r E r B pkq on both side. Similarly for K˚pCLp r Bq H q and K˚pCLp r E, r Bq G q. We will prove that line (5.3) holds when restricted to r E r B pkq by induction on k. When k is zero, B p0q is a disjoint union of small balls in B, to which the restriction of E is trivial. Therefore line (5.3) holds on r E r B p0q . Now we assume that line (5.3) holds on r E r B pkq . Let ∆ be the disjoint union of the interior of every k`1-simplex in B pk`1q . Denote the lift of ∆ to r B by r ∆ and the restriction of r E to r ∆ by r E r ∆ . Note that B pk`1q " ∆ Y B pkq . By Proposition 5.8, we have the following six-term exact sequence: From the assumption that the diameter of each simplex of B is small, the restriction of E to ∆ or ∆ X B pkq is a disjoint union of trivial bundles. Direct computations show that p r E r B pkq q G q ' K 0 pCr B,L,0 p r E r ∆ q G q Ñ K 0 pCr B,L,0 p r E r B pk`1q q G q.
Then along with the inductive hypothesis, the K-theory classes represented by φ L,0 prD sgn B s b ρpf ; Bqq´ρ L pf q restricted to r E r ∆ , r E r B pkq vanishes, which shows that φ L,0 prD sgn B s b ρpf ; Bqq´ρ L pf q is the image of trivial class. Now line (5.3) follows when k " dim B.

For special fiber bundle
In this section, we show that Theorem 5.4 implies the product formula of numerical signature on fibered manifold given by Chern, Hirzebruch and Serre in [1]. Consider the fiber bundle π : E Ñ B with fiber F , with all those spaces are 4k-dimensional oriented closed Riemannian manifolds. Assume that π 1 pBq acts trivially on Hd R pEq, the de Rham cohomology of E. We would like to use our product formula to prove the original formula introduced by Chern, Hirzebruch and Serre in [1], namely sgnpBqˆsgnpF q " sgnpEq.
Consider the K-theoretic index map ind E : K 0 pCLpẼq π 1 pEq q Ñ K 0 pCLpptqq -Z induced by the map that crashes the whole space to a point and forgets the group action. Under this the localized index of signature operator will be mapped to its graded Fredholm index, i.e. sgnpEq. Besides we replace E by the base space B and obtain ind B : K 0 pCLpBq π 1 pBq q Ñ K 0 pCLpptqq -Z.
Recall the equivariant family localization algebra CLpẼ,Bq π 1 pEq is the collection of some sections of a C*-bundle over B. Any element sptq P CLpẼ,Bq π 1 pEq is viewed as a family of operators sptq x P CLpF q Γ for x PB. Thus we define a family index map ind E,B : K 0 pCLpẼ,Bq π 1 pEq q Ñ K 0 pCpBqq π 1 pBq q -K 0 pCpBqq by taking indices along the fiber.
From the construction above, we see that the following diagram commutes.
Since F is even-dimensional, the family index ind E,B prD E,B sq living in K 0 pCpBqq can be viewed as a virtual vector bundle over B. The local picture of such vector bundle is rkerpD F qs´rcokpD F qs. As we have assumed that π 1 pBq acts on Hd R pEq trivially, the virtual bundle is indeed a trivial bundle, i.e. it comes from the inclusion Z -K 0 pCpptqq Ñ K 0 pCpBqq. Moreover, the preimage of ind E,B prD E,B sq under the inclusion is actually dim kerpD F q´dim cokpD F q " sgnpF q.
Thus the pairing map x¨,¨y is simplified as followed xrD B s, ind E,B prD E,B sqy " xrD B s, sgnpF qy " ind B prD B sqˆsgnpF q " sgnpBqˆsgnpF q.
From this we obtain the classical product formula of signature of Chern, Hirzebruch and Serre.