AXIOMS FOR HIGHER TORSION

KIYOSHI IGUSA

Abstract. This paper shows that nonequivariant higher torsion is characterizedby two axioms: additivity and transfer. Any characteristic class of smooth bun-dles satisfying these conditions must be a linear combination of the higher Franz-Reidemeister torsion and the higher Miller-Morita-Mumford classes.

Introduction

During September 1-7, 2003 there was a conference on higher torsion in Gottinghen.I gave a series of lectures on higher Franz Reidemeister (FR) torsion. These are lecturenotes from my last lecture given on September 7, 2003. Since this was the last lectureof the meeting, I had the advantage of being able to refer to all of the previouslectures.When I gave this lecture I did not have the complete statement. There was the

mystery fiber homotopy invariant ∆ = C that I conjectured was always zero. Fol-lowing suggestions of Bruce Williams and John Klein and using ideas given in thelectures of Williams and Michael Weiss, I was later able to prove this conjecture andcomplete the characterization of higher torsion as I had promised at the beginning ofthe meeting.These lecture notes have additional comments, the proof of the conjecture that

C = ∆ = 0 and an extension of the theorem without proof to a characterization ofhigher torsion for bundles with closed manifold fibers.I should thank everyone who encouraged me, in particular Sebastian Goette and

Xiaonan Ma who helped me to understand the properties of analytic torsion, WojciechDorabiala for helping me to understand the properties of topological torsion and Ialso owe a great debt to John Klein and Bruce Williams who helped me to completethe proof of the theorem. Those who remember John Klein’s comments during mylecture may recognize his contribution to the final step. Finally I would like tothank the organizers Thomas Schick, Ulrich Bunke and Sebastian Goette. They dida wounderful job making us feel at home.

1. Preliminaries

Consider compact smooth oriented bundle pairs

(F, ∂0) → (E, ∂0) → B

2000 Mathematics Subject Classification. Primary 55R40; Secondary 57R50, 19J10.Key words and phrases. higher Franz-Reidemeister torsion, analytic torsion, diffeomorphisms,

transfer, rational homotopy.Supported by NSF Grants DMS 02-04386, DMS 03-09480.

1

2 KIYOSHI IGUSA

where π1B acts trivially1 on H∗(F, ∂0F ;Q). By oriented we mean that the fibers areoriented and the action of π1B preserves the orientation. We are using the shorthand(X, ∂i) = (X, ∂iX).As an example, take F = Σg a closed surface of genus g. Then a smooth bundle

F → E → B as above is classified by a map B → BTg of B into the classifying spaceof the Torelli group Tg which is the group of all isotopy classes of diffeomorphisms ofΣg which induce the identity on homology.Recall that ∂vE = ∂0E∪∂1E is the vertical boundary of E. Let T vE be the vertical

tangent bundle of E. This is an n-dimensional vector bundle where n = dimF . Let

e(E, ∂0) ∈ Hn(E, ∂vE;Z)be the relative Euler class of (E, ∂0). This is the pull-back of the Thom class of T vEalong any vertical tangent vector field which points inward along ∂0E and outwardon ∂1E. Then we have the following easy observation.

Lemma 1.1. e(E, ∂1) = (−1)ne(E, ∂0).

Take the push-down in degree 4k + n:

p∗ : H4k+n(E, ∂vE;Z) → H4k(B;Z).

This comes from the spectral sequence of the fibration (F, ∂) → (E, ∂v) → B sincethere is a natural mapping

H4k(B;Hn(F, ∂;Z)) → H4k(B;Z)which is an isomorphism if F is connected. For real coefficients the push-down isgiven by integration of differential forms along the fibers.

Definition 1.2. We define the higher even Miller-Morita-Mumford (MMM) classesto be

M2k(E, ∂0) := p∗

((2k)!

2ch2k(T

vE ⊗ C) ∪ e(E, ∂0)

)∈ H4k(B;Z).

for k > 0.

This is an integral class for k ≥ 1 since

ch2k(TvE ⊗ C) = 2

∑ c1(λi)2k

(2k)!

if T vE is a sum of complex line bundles λi (plus a possible trivial real line bundle).For k = 0 the formula above gives:

M0(E, ∂0) = M0(F, ∂0) =n

2χ(F, ∂0) ∈ 1

2Z.

The original Miller-Morita-Mumford classes (from [Mum83], [Mor84], [Mil86]) aredefined for oriented surface bundles E → B by

κk(E) := p∗ (k! chk(TvE) ∪ e(E)) ∈ H2k(B;Z).

Note that, by Lemma 1.1, we have:

Proposition 1.3. M2k(E, ∂1) = (−1)nM2k(E, ∂0).

1We only need to assume the action is upper triangular in the sense that its matrix representationis upper triangular with 1’s on the diagonal.

AXIOMS FOR HIGHER TORSION 3

An immediate corollary which I forgot to mention in the lecture is:

Corollary 1.4. If the fiber F is an odd dimensional closed manifold then

M2k(E) = 0.

The higher MMM classes are related to the higher Franz-Reidemeister (FR)-torsionclasses τFR

2k (E, ∂0) ∈ H4k(B;R) by the following Poincare duality formula.

Theorem 1.5. [Igu03] For k ≥ 1 we have

τFR2k (E, ∂1) = (−1)n−1τFR

2k (E, ∂0) + (−1)n+k ζ(2k + 1)

(2k)!M2k(E, ∂0)

where ζ(2k + 1) =∑

m≥0 m−2k−1 is the Riemann zeta function.

Remark 1.6. Note that, by Poincare duality, π1B acts trivially on H∗(F, ∂0F ;Q) ifand only if it acts trivially on H∗(F, ∂1F ;Q). Thus τFR

2k (E, ∂1) is defined if and onlyif τFR

2k (E, ∂0) is defined.

Proof. The signs were wrong in the lecture. This proof is to show that the signsare now correct. Let f : (E, ∂1) → (I, 0) be a fiberwise framed function for (E, ∂1).Then −f is not in general framed but it is still a fiberwise generalized Morse function(GMF) with torsion related to the torsion of f by the involution formula

τFR2k (−f) = (−1)n−1τFR

2k (f) = (−1)n−1τFR2k (E, ∂1).

The framing principle say that the higher FR torsion of (E, ∂0) can be calculatedfrom the fiberwise GMF −f by

τFR2k (E, ∂0) = τFR

2k (−f) +M2k(E, ∂0)

where we use the shorthand notation

(1) M2k(E, ∂0) := (−1)kζ(2k + 1)

(2k)!M2k(E, ∂0)

The theorem follows from these three equations. !

Corollary 1.7. [Igu03] If F is an even dimensional closed manifold then

τFR2k (E) =

1

2M2k(E) =

1

2(−1)k

ζ(2k + 1)

(2k)!M2k(E)

for k ≥ 1.

For example, the higher FR-torsion of the universal bundle Σg → ETg → BTg isgiven by

τFR2k (ETg) =

1

2(−1)k

ζ(2k + 1)

(2k)!κ2k(Tg)

where κ2k(Tg) is the classical Miller-Morita-Mumford class of the Torelli group.

4 KIYOSHI IGUSA

2. Axioms

Suppose that τ(E, ∂0) ∈ H4k(B;R) is a real characteristic class for smooth bundlepairs satisfying the conditions explained above. The words “characteristic class” referto a cohomology class in the base for allowable fibrations which is natural with respectto pull-backs, i.e.,

f ∗τ(E, ∂0) = τ(f ∗E, ∂0) ∈ H4k(B′;R)if f ∗E is the pull-back of E along a map f : B′ → B. We also assume that cornerscan be rounded off in some canonical way when needed.Consider the following two axioms (and variations) which may or may not be

satisfied.

(I) Additivity (gluing formula) Suppose that E = E1∪E2 where E1, E2 meet alonga portion of their boundaries: E1∩E2 = ∂0E2 ⊆ ∂1E1. (In other words, E2 is attachedto E1 by an attaching map ∂0E2 → ∂1E1.) Then

(2) τ(E, ∂0E1) = τ(E1, ∂0) + τ(E2, ∂0)

when all three terms are defined2.

Theorem 2.1 (Splitting Lemma). Higher FR torsion τFR2k and the higher MMM

classes M2k satisfy Axiom I.

One easy consequence of the additivity axiom is that τ = 0 on trivial bundle pairs.

Proposition 2.2. If τ satisfies Axiom I then for any smooth bundle E → B withvertical boundary ∂vE we have

τ(E × I, E × 0) = τ(E × I, E × 0 ∪ ∂vE × I) = 0.

Proof. Since (E × I, E × 0) is fiber diffeomorphic to (E × [0, 2], E × 0) its torsionτ satisfies 2τ = τ by additivity. Therefore τ = 0. When the corners are roundedoff, the bundle pairs (E × I, E × 0) and (E × I, E × 0 ∪ ∂vE × I) become fiberwisediffeomorphic. So they have the same torsion. !We also need the following variation of the additivity axiom.

(I’) Lateral additivity Suppose that E = E1 ∪ E2 with fiber dimension n and∂0E = ∂0E1 ∪ ∂0E2 with fiber dimension n − 1 where ∂0E1 ∩ ∂0E2 = ∂0E1 ∩ E2 =E1 ∩ ∂0E2 = X is a bundle of fiber dimension n− 2 and E1 ∩ E2 is diffeomorphic toX × I. Then

(3) τ(E, ∂0) = τ(E1, ∂0) + τ(E2, ∂0)

if all three terms are defined.

Proposition 2.3. Vertical additivity (I) implies lateral additivity (I’).

Proof. If we choose a collar neighborhood of X in ∂0E = ∂0E1 ∪ ∂0E2 we see that∂0E is fiber diffeomorphic to

Y := ∂0E1 ∪X×0 X × I ∪X×1 ∂0E2.

2For upper triangular actions, these terms are defined whenever two of them are defined (by thelong exact sequence of the triple (E,E1, ∂0E1)).

AXIOMS FOR HIGHER TORSION 5

Let E ′ be Y × I with E1, E2 attached by the mappings ∂0Ei → ∂0Ei × 1 ⊆ Y × 1.Then

τ(E ′, Y × 0) = τ(E1, ∂0) + τ(E2, ∂0) + τ(Y × I, Y × 0)

by vertical additivity. But (E ′, Y ×0) is fiber diffeomorphic to (E, ∂0) since attachingof the external collars ∂0Ei × I ⊆ Y × I does not change Ei and thickening theintersection E1 ∩ E2 by a factor of I also does not change their union (up to fiberdiffeomorphism). Also τ(Y × I, Y × 0) = 0 by Proposition 2.2. The statement (3)follows. !

(II) Transfer (composition formula) Consider the following diagram in which therows and columns are smooth fibration sequences.

F2=−−−→ F2&

&

(F3, ∂0) −−−→ (E2, ∂0) −−−→ B&

&q

&=

(F1, ∂0) −−−→ (E1, ∂0) −−−→p

B

Here the vertical sequences have horizontal instead of vertical boundaries in the sensethat ∂0E2 = q−1(∂0E1) and similarly for ∂0F3. Then the torsion of (E2, ∂0) consideredas a bundle pair over B should be related to the torsion of E2 as a bundle over E1 bythe following formula.

(4) τB(E2, ∂0) = χ(F2)τ(E1, ∂0) + p∗(τE1(E2) ∪ e(E1, ∂0))

where p∗ : H4k+n(E1, ∂0E1) → H4k(B) is the push-down.

Remark 2.4. Suppose that E2 = p∗E0 is the pull-back to E1 of a bundle E0 over B(with fiber F0 = F2). Equivalently, (E2, ∂0) is the fiber product

(E2, ∂0) = E0 ⊕ (E1, ∂0).

Then τE1(E2) = p∗(τ(E0)) by naturality and

p∗(p∗(τ(E0)) ∪ e(E1, ∂0)) = τ(E0) ∪ p∗(e(E1, ∂0)) = χ(F1, ∂0)τ(E0).

So, the transfer formula (4) becomes the product formula:

(5) τ(E0 ⊕ (E1, ∂0)) = χ(F0)τ(E1, ∂0) + χ(F1, ∂0)τ(E0).

Proposition 2.5. The higher MMM classes M2k satisfy Axiom II.

Proof. We use the fact that

q∗(q∗(x) ∪ e(E2)) = x ∪ q∗(e(E2)) = χ(F2)x

for any cohomology class x ∈ H∗(E1;R). The vertical tangent bundle of E2 consideredas a bundle over B is the direct sum

T vBE2

∼= q∗(T vE1)⊕ T vE1E2.

And the relative Euler class of (E2, ∂0) considered as a bundle pair over B is given by

eB(E2, ∂0) = q∗e(E1, ∂0) + eE1(E2).

6 KIYOSHI IGUSA

Since the Chern character is additive,

1

(2k)!M2k(E2, ∂0) = p∗q∗(ch2k(T

vBE2) ∪ eB(E2, ∂0))

is the sum ofp∗q∗(q

∗ch2k(TvE1) ∪ q∗e(E1, ∂0) ∪ eE1(E2))

= χ(F2)p∗(ch2k(TvE1) ∪ e(E1, ∂0)) =

χ(F2)

(2k)!M2k(E1, ∂0)

andp∗q∗(ch2k(T

vE1E2) ∪ q∗e(E1, ∂0) ∪ eE1(E2))

= p∗(q∗(ch2k(T

vE1E2) ∪ eE1(E2)) ∪ e(E1, ∂0)

)

= p∗

(1

(2k)!M2k(E2) ∪ e(E1, ∂0)

).

Multiplying by (2k)! we get the transfer formula for M2k. !There is, unfortunately, one problem with Axiom II. It has not been shown that

higher FR torsion (τFR2k ) satisfies this axiom. Even the product formula (5) is unknown

in general. Therefore, we take the following alternate axiom.

(II’)= the subset of Axiom II which holds for both τFR2k and M2k.

During the lecture, some people objected to this axiom as being ill-defined. There-fore I substituted another axiom which consists of those cases of Axiom II which areactually used in the proof.

(II”)= Axiom II in the following two cases.

(IIa) Axiom II in the case when E2 is a linear disk bundle over E1 with fiberF2 = Dn.

(IIb) Axiom II in the case when (E1, ∂0) is a linear disk bundle over B with fiberpair (F1, ∂0) = (Dn, Sn−1).

Lemma 2.6. Suppose that D(ξ) → B is the linear disk bundle of a real vector bundleξ over B. Then τFR

2k (D(ξ)) = 0 and

M2k(D(ξ)) =(2k)!

2ch2k(ξ ⊗ C).

Proof. Both statements are obvious. For the higher FR-torsion, we take the functionf(x) = ∥x∥2. This is a fiberwise Morse function with one critical point of index 0 oneach fiber. This is framed since the negative eigenspace of D2f is zero. The cellularchain complex is constant giving zero as torsion. !Theorem 2.7. Higher FR-torsion satisfies IIa and IIb.

Proof. The first case (IIa) is more or less trivial. Let q : E2 → E1 be a lineardisk bundle over the total space E1 of a smooth bundle pair (E1, ∂0E1). Then anyfiberwise framed function f : (E1, ∂0) → (I, 0) gives another framed function (calledthe positive suspension) σ+f on (E2, ∂0) (considered as a bundle over B) by

σ+f(x) = f(q(x)) + ∥x∥2.

AXIOMS FOR HIGHER TORSION 7

This function behaves correctly on the vertical boundary in that its gradient pointsinward along ∂0E2 = q−1(∂0E1) and outward along ∂1E2. Also the cellular chaincomplex of σ+f is equal to the cellular chain complex of f so

τFR2k (E2, ∂0)B = τFR

2k (E1, ∂0).

This agrees with the transfer formula since χ(F2) = χ(Dn) = 1 and τFR2k (E2)E1 = 0

by the lemma.The second case (IIb) follows from the first case and Theorem 1.5. When F1 = Dn,

p : E1 → B is a homotopy equivalence. Consequently, the bundle E2 must be thepull-back along p : E1 → B of a bundle F0 = F2 → E0 → B and (E2, ∂0) must be afiber product:

(E2, ∂0) = (E1, ∂0)⊕ E0

The transfer formula becomes the product formula (5) which in this case is:

(6) τFR2k (E2, ∂0) = χ(F0)τ

FR2k (E1, ∂0) + (−1)nτFR

2k (E0).

By Proposition 2.5 we know that the analogous equation holds for M2k and thus forM2k:

(7) M2k(E2, ∂0) = χ(F0)M2k(E1, ∂0) + (−1)nM2k(E0).

By Case IIa proved above and Lemma 2.6 (τFR2k (E1) = 0) we know that

(8) τFR2k (E2, ∂1) = τFR

2k (E0, ∂) = (−1)mχ(F0)τFR2k (E1) + τFR

2k (E0, ∂)

where m = dimF0. But Theorem 1.5 implies that (6) is equal to (7) minus (−1)n+m

times (8). So, (7) and (8) imply (6) as required. !

3. Theorem

Theorem 3.1. Suppose that τ(E, ∂0) ∈ H4k(B;R) is a characteristic class of smoothbundle pairs (F, ∂0) → (E, ∂0) → B for which π1B acts trivially (or upper triangu-larly) on H∗(F, ∂0F ;Q). Suppose that τ satisfies Axioms I and II”. Then there existunique a, b ∈ R independent of (E, ∂0) so that

τ(E, ∂0) = aτFR2k (E, ∂0) + bM2k(E, ∂0).

Remark 3.2. (1) Note that the tangential data only affects the second componentM2k(E, ∂0) since τFR

2k (E, ∂0) does not change when E is replaced by the normal diskbundle of E.(2) During the lecture Sebastian Goette pointed out the following. By an analogous

argument, there are no nontrivial higher torsion theories in degrees 4k + 2 satisfyingAxioms I,II”. (See Definition 3.5 below.)

In the lecture I showed the following.

Lemma 3.3. Under the conditions of the above theorem there exist unique a, b ∈ Rso that

τ(E, ∂0) = aτFR2k (E, ∂0) + bM2k(E, ∂0) + C(E, ∂0)

where C(E, ∂0) is a fiber homotopy invariant of smooth bundle pairs.

In the last section I will prove the following which was stated as a conjecture inthe lecture.

8 KIYOSHI IGUSA

Lemma 3.4. C = 0 in the above lemma.

Theorem 3.1 has some interesting corollaries.

Definition 3.5. By a higher torsion theory we mean a real characteristic class

τ(E, ∂0) ∈ H4k(B;R)

for smooth bundle pairs (F, ∂0) → (E, ∂0) → B for which π1B acts trivially onH∗(F, ∂0F ;Q) satisfying Axioms I and II”.

Corollary 3.6. Suppose that τ is a higher torsion theory in the above sense. Thenτ extends unique to the upper triangular case. Furthermore this extension satisfiesAxiom II’.

Since the higher analytic torsion classes are know to satisfy the transfer axiom byXiaonan Ma and M2k satisfies this axiom by Proposition 2.5, we get the following.(See the last section.)

Corollary 3.7. If nonequivariant higher analytic torsion satisfies Axiom I then higherFR-torsion satisfies Axiom II.

Of course, we would also conclude that the higher analytic torsion is a linear com-bination of τFR

2k and M2k. Since this must be zero for closed even dimensional fibers,Corollary 1.7 tells us that the coefficients a, b should be related by:

b =a

2(2k)!(−1)k+1ζ(2k + 1).

For Bismut and Goette’s Chern normalization [BG00] the first coefficient should be

a =k!

(2π)k.

4. Parameters a, b

To prove the theorem we first need to find the parameters a and b.(a) Consider smooth oriented S1-bundles. These have structure group Diff0(S1)

consisting of orientation preserving diffeomorphisms of S1. It is an easy exercise toshow that SO(2) ∼= U(1) is a deformation retract of Diff0(S1). So all oriented circlebundles are equivalent to linear bundles and are classified by maps of the base B into

BDiff0(S1) ≃ BU(1) ≃ CP∞.

So, H4k(BDiff0(S1);R) ∼= R and the only characteristic classes for oriented circlebundles are scalar multiples of c2k1 = (2k)!ch2k. Thus there is an a0 ∈ R so that

τ = a0ch2k

for all oriented S1-bundles. In the first lecture I explained the following.

Lemma 4.1. For oriented linear S1-bundles we have

τFR2k = (−1)k+1ζ(2k + 1)ch2k.

AXIOMS FOR HIGHER TORSION 9

We also have M2k = 0 by Corollary 1.4. So, for oriented circle bundles we have

τ = (−1)k+1 a0ζ(2k + 1)

τFR2k + bM2k

for any value of b. Let a be the coefficient

(9) a := (−1)k+1 a0ζ(2k + 1)

of τFR2k in the above expression.

(b) Consider linear3 oriented D2-bundles. These are classfied by maps of B to

BSO(2) ≃ BU(1) ≃ CP∞.

So, again there is a b0 ∈ R so that τ = b0ch2k for all oriented linear D2-bundles. ByLemma 2.6 we have τFR

2k = 0 and

M2k = (2k)!ch2k

for these bundles. Thus τ = aτFR2k + bM2k where

(10) b :=b0

(2k)!.

The existence part of Lemma 3.3 can now be stated as follows.

Lemma 4.2.

∆(E, ∂0) := τ(E, ∂0)− aτFR2k (E, ∂0)− bM2k(E, ∂0)

is a fiber homotopy invariant.

The unique part of Lemma 3.3 is the following easy statement.

Lemma 4.3. No nontrivial linear combination of the higher FR torsion τFR2k and

higher MMM class M2k can be a fiber homotopy invariant.

Proof. Suppose that C = xτFR2k + yM2k is a fiber homotopy invariant. Since disk

bundles are always fiber homotopy equivalent to trivial bundles and τFR2k = 0 for

linear disk bundles we must have y = 0. But Hatcher’s example is a disk bundlewith τ2k = 0 which is tangentially fiber homotopy equivalent to a trivial bundle. SoC = M2k = 0 which implies x = 0. !

5. Proof of Lemma

The proof of Lemma 4.2 is based on the following two facts.

Lemma 5.1. Scalar multiples of the Chern character

1

2ch2k(ξ ⊗ C)

are the only additive real characteristic classes of oriented real vector bundles ξ. Inother words, if c ∈ H4k(BSO;R) so that c(ξ ⊕ η) = c(ξ) + c(η) then c is a scalarmultiple of the Chern character.

3In the lecture I used the fact that all smooth D2 bundles are fiber diffeomorphic to linear bundles.But we can just assume that we start with a linear D2-bundle.

10 KIYOSHI IGUSA

Proof. This follows easily from the splitting principle for oriented real vector bundles.!

Theorem 5.2 (Borel,Waldhausen,I). The classifying space BDiff0(Dn) has the ra-tionally homotopy type of BSO ×BSO in a stable range (n/3 plus a constant).

This theorem combines Borel’s computation of the rational K-theory of Z [Bor74],Waldhausen’s theorem relating A(∗) to KZ [Wal87] and my stability theorem [Igu88].Farrell and Hsiang [FH78] computed the rational homotopy type ofDiff0(Dn rel ∂Dn)which requires additional work but their result is not being used here.

Remark 5.3. This theorem implies that, for n >> 4k, the cohomology groupH4k(BDiff0(Dn);R) has only two linearly independent additive classes.

We will use the following version of the above theorem which is due to Bokstedt.

Theorem 5.4. [Bok84] The mapping from G/O to the stable space of h-cobordismsof a point given by Hatcher’s construction is a rational homotopy equivalence.

Remark 5.5. Bruce Williams defined the stable space of h-cobordisms of a point inhis lectures:

H(∗) = lim→

BDiff0(Dn)/O(n).

By the h-cobordism theorem this space has only one component. By the abovetheorem it has the rational homotopy type of BO.

Proof. As Bokstedt pointed out in his paper, we have a mapping G/O → H(∗)between two spaces which we already know are rationally homotopy equivalent byTheorem 5.2. The rational/real homotopy groups are also know:

π4k(G/O)⊗ R ∼= π4kH(∗)⊗ R ∼= R.(And other homotopy groups are rationally trivial.) Thus it suffices to show that themapping is rationally nontrivial on π4k. But this is true since it is detected by thehigher FR-torsion of the disk bundle over S4k given by Hatcher. !These facts imply that ∆(E) = 0 for smooth oriented disk bundles Dn → E → B.

We prove this first for linear bundles.

Corollary 5.6. Let D(ξ), S(ξ) be disk and sphere bundles associated to an orientedlinear n-plane bundle ξ. Then ∆(D(ξ)) = ∆(S(ξ)) = ∆(D(ξ), S(ξ)) = 0.

Proof. If n = 2 then ∆ = 0 by definition of a and b. In general we may assume bythe splitting principle that ξ is the direct sum of oriented 2-plane bundles ξ = ⊕λi

(plus a trivial line bundle if n is odd). Since the product formula for disk bundlesholds for τ, τFR

2k ,M2k it also holds for ∆. So,

∆(D(ξ)) = ∆(⊕D(λi)) =∑

∆(D(λi)) = 0.

Now consider the sphere bundle Sn−1 → S(ξ) → B. If n = 1, then S(ξ) is trivialand ∆ = 0. If n = 2 then ∆ = 0 by definition of the parameters a, b. For n ≥ 3 wemay assume that ξ is a direct sum of two oriented subbundles ξ = ξ1⊕ ξ2. Then S(ξ)can be decomposed as a union:

S(ξ1 ⊕ ξ2) = S(ξ1)⊕D(ξ2) ∪D(ξ1)⊕ S(ξ2)

AXIOMS FOR HIGHER TORSION 11

with intersection S(ξ1)⊕ S(ξ2). By Axioms I, IIa, IIb we have

∆(S(ξ1 ⊕ ξ2)) = ∆(S(ξ1)⊕D(ξ2)) +∆((D(ξ1), S(ξ1)⊕ S(ξ2))

= ∆(S(ξ1)) +∆(S(ξ2))

= 0 + 0

by induction on n. !

Corollary 5.7. ∆(E) = 0 for smooth oriented disk bundles Dn → E → B.

Proof. If D is any oriented linear disk bundle over B then we just proved that ∆(D) =0. So, by Axiom IIa we have

∆(E ⊕D) = ∆(E) +∆(D) = ∆(E)

and it suffices to show that ∆(E ⊕D) = 0 for some D. Choose D so that the fiberproduce E ⊕ D is tangentially trivial. This can be done by first making the fiberdimension very large, choosing a section and taking a complementary vector bundleto the vertical tangent bundle along the section.Replacing E with E⊕D we can now assume that n >> dimB and E is tangentially

fiber homotopically trivial. This implies that there is a smooth section B → E whoseimage we can take to lie in the vertical boundary of E so that the bundle E is trivialin a neighborhood of the section. Thus the vertical boundary of E contains a trivialn− 1 disk bundle

∂0E = B ×Dn−1.

Since E ∼= E ∪ ∂0E × I, additivity (Axiom I) implies that

∆(E) = ∆(E, ∂0) +∆(∂0E × I) = ∆(E, ∂0).

These bundles are classified by mappings of B into the space of h-cobordisms of Dn−1

which is BDiff(Dn rel ∂0Dn). However, this space has the rational homotopy typeof G/O in low degrees. Therefore, ∆ (on these bundles) is given by an element ofH4k(G/O;R).We look again at Hatcher’s construction. Generators for H4k(G/O;R) are repre-

sented by mappings B → G/O = SG/SO where B is a closed smooth 4k-manifold.Such a mapping is given by an oriented real vector bundle ξ (say of dimension n) overB together with a homotopy trivialization of the associated sphere bundle. This canbe realized as a fiberwise smooth embedding

g : S(ξ) → B × Sn−1 ×Dm+ ⊆ B × Sn−1 × Sm

which by Hatcher’s construction gives an n+m dimensional disk bundle

E = B × Sn−1 ×Dm+1 ∪D(ξ)⊕D(η)

with a trivial subbundle ∂0E ⊆ B × Sn−1 ×Dm− where η is the m-dimensional com-

plementary bundle for ξ and Dm+ , D

m− are the northern and southern hemispheres of

Sm. The bundle pair (E, ∂0) represents the composite mapping

B → G/O → H(∗)

12 KIYOSHI IGUSA

and therefore gives a generator of H4k(H(∗);R). But the value of ∆ on this generatoris zero (Axiom I for line 1, Axiom IIb for line 2 and Corollary 5.6 for line 3):

∆(E, ∂0) = ∆(B × Sn−1 ×Dm+1, ∂0E) +∆((D(ξ), S(ξ))⊕D(η))

= 0 +∆(D(ξ), S(ξ)) + (−1)n∆(D(η))

= 0.

Therefore ∆ = 0 for all disk bundles. !Lemma 5.8. If D is a linear disk bundle over E and ∂0D is the restriction to ∂0Ethen ∆(D, ∂0) = ∆(E, ∂0).

Proof. By Axiom IIb we have

∆(D, ∂0) = ∆(E, ∂0) + p∗(∆E(D) ∪ e(E, ∂0)).

But ∆E(D) = 0 by Corollary 5.6 since D is a linear disk bundle over E. !Lemma 5.9. If (F, ∂0) is an h-cobordism (∂0F ≃ F ), then ∆(E, ∂0) = 0.

Proof. Choose a smooth fibered embedding of ∂0E into B × DN for some large N .Let ∂0D be the normal disk bundle of ∂0E in B × DN . Since E is fiber homotopyequivalent to ∂0E, this linear disk bundle extends to a disk bundle D over all of E ina unique way up to homotopy. By Lemma 5.8 we have ∆(E, ∂0) = ∆(D, ∂0). So, itsuffices to show that the latter is zero.Let Z be the closure of the complement of ∂0D in B ×DN . Then

B ×DN = ∂0D ∪ Z

and ∂0D ∩ Z = X is the vertical boundary of ∂0D. We assume that ∂0D has beenrelieved of its corners but that D has a corner set along X. Choose an external(vertical) collar for X in the boundary of D. This is diffeomorphic to X × I. We canuse this external collar to paste together D and Z × I:

W = D ∪X×I Z × I, ∂0W = ∂0D ∪ Z = B ×DN .

It is easy to see that ∂0W is a fiberwise deformation retract of W . However, the fiberof ∂0W over B is DN . So W is also a disk bundle over B by the h-cobordism theorem.So,

∆(W, ∂0) = 0

by Corollary 5.7. But lateral additivity (Axiom I’) which follows from Axiom I impliesthat

∆(W, ∂0) = ∆(D, ∂0) +∆(Z × I, Z × 0)

and ∆(Z × I, Z × 0) = 0 since it is trivial. So, ∆(D, ∂0) = ∆(E, ∂0) = 0. !We are now ready to prove the main lemma.

Proof of Lemma 4.2. Suppose that (E1, ∂0) and (E2, ∂0) are smooth bundle pairs overB for which our higher torsion invariant τ is defined. Suppose they are fiber homotopyequivalent. Then we want to show that

∆(E1, ∂0) = ∆(E2, ∂0).

AXIOMS FOR HIGHER TORSION 13

If we replace (E2, ∂0) by a large dimensional disk bundle (D2, ∂0) we can approximatethe fiber homotopy equivalence by a fiberwise smooth embedding

g : (E1, ∂0) → (D2, ∂0)

which is also a fiber homotopy equivalence. Let (D1, ∂0) be the normal disk bundleof g(E1, ∂0) in (D2, ∂0). Then ∆(Di, ∂0) = ∆(Ei, ∂0) by Lemma 5.8. So, it suffices toshow that ∆(D1, ∂0) = ∆(D2, ∂0).However, (D2, ∂0) is the union of (D1, ∂0) with a fiberwise h-cobordism. To see

this let Z be the closure of ∂0D2 − ∂0D1 so that ∂0D2 = Z ∪ ∂0D1. Let D3 be theclosure of D2 −D1 with ∂0D3 = Z ∪ ∂1D1. Let (Fi, ∂0) be the fiber pair for (Di, ∂0).Then (F3, ∂0) has the same homology as (F2, F1) which is a deformation retract and(F1, ∂1) is highly connected. Therefore (F3, ∂0) is an h-cobordism and

∆(D2, ∂0) = ∆(D1, ∂0) +∆(D3, ∂0)

by a variation of Axiom I and ∆(D3, ∂0) = 0 by Lemma 5.9.We need to justify this variation of the axiom. Attach a copy of ∂0D2 × I to ∂0D2

along ∂0D2 × 1. Then, by Axiom I and since ∆(∂0D2 × I, ∂0D2 × 0) = 0, we have:

∆(D1, ∂0) = ∆(D1 ∪ ∂0D2 × I, ∂0D2 × 0)

∆(D2, ∂0) = ∆(D2 ∪ ∂0D2 × I, ∂0D2 × 0)

= ∆(D1 ∪ ∂0D2 × I, ∂0D2 × 0) +∆(D3, ∂0)

= ∆(D1, ∂0).

This concludes the proof of the lemma. !

6. Proof of Theorem

We have show that ∆ is a fiber homotopy invariant of smooth bundles. We willnow show that it is always zero.Since ∆ is a fiber homotopy invariant, it is well defined on any fibration pair

(X,A) → (Z,C) → B

so that

(1) π1B acts trivially (or upper triangularly4) on H∗(X,A;Q).(2) (Z,C) is fiber homotopy equivalent to a smooth bundle pair (E, ∂0) with

compact manifold fiber (F, ∂0).

We will call fibrations smoothable if they satisfy both conditions and we call (E, ∂0)a smoothing of (Z,C).

Lemma 6.1. Given a smoothable fibration pair as above, let Z/BC be the fibrationover B given by collapsing C to B, i.e., collapsing A to a point in each fiber. Then(X/A, ∗) → (Z/BC,B) → B is also smoothable with the same value of ∆:

∆(Z/BC,B) = ∆(Z,C).

Remark 6.2. When C = B we will refer to (Z,B) as a pointed fibration. This lemmasays that it suffices to show that ∆ = 0 for pointed smoothable fibrations.

4We are proving two different versions of the theorem at the same time.

14 KIYOSHI IGUSA

Proof. Take the corresponding smooth bundle pair (E, ∂0) → B, embed ∂0E fiberwiseinto B ×DN and thicken it up just as we did in the proof of Lemma 5.9. !Lemma 6.3. For any smoothable fibration pair (Z,C) the fiberwise suspension

(ΣB(Z/BC), B) → B

(with fiber (Σ(X/A), ∗)) is smoothable with

∆(ΣB(Z/BC), B) = −∆(Z,C).

Proof. Choose a smoothing (E, ∂0) for (Z,C). If we attach a fiberwise external collar∂0E × I along ∂0E × 1 to ∂0E we get a fiberwise diffeomorphic bundle pair

(E ′, ∂0) = (E ∪ ∂0E × I, ∂0E × 0)

Another smoothing for (Z,C) is given by

(D, ∂0) = (E ′, ∂0)× [0, 3] = ((E ∪ ∂0E × I)× [0, 3], (∂0E × 0)× [0, 3])

which can be expressed as a union of two subbundles (D, ∂0) = (D1, ∂0) ∪ (D2, ∂0)with ∂0D2 ⊆ ∂1D1 where

(D1, ∂0) = (E ′, ∂0)× [0, 1] ∪ (∂0E × I, ∂0E × 0)× [1, 2] ∪ (E ′, ∂0)× [2, 3]

(D2, ∂0) = (E × [1, 2], E × 1 ∪ ∂0E × [1, 2] ∪ E × 2).

By additivity this gives

∆(Z,C) = ∆(D, ∂0) = ∆(D1, ∂0) +∆(D2, ∂0)

= 2∆(Z,C) +∆(D2, ∂0).

So, ∆(D2, ∂0) = −∆(Z,C). But D2/B∂0D2 is the fiberwise suspension of E ′/B∂0E ′ ≃Z/BC. By the previous lemma this is smoothable with

∆(ΣB(Z/BC), B) = ∆(D2/B∂0D2, B) = ∆(D2, ∂0) = −∆(Z,C).

(The first equality follows from the fiber homotopy invariance of ∆.) !The following lemma and its proof goes along the line of the discussion I had with

John Klein on the evening before my lecture. It uses ideas from Bruce Williams’lectures).

Lemma 6.4. ∆(Z,B) = 0 for pointed smoothable fibrations with rationally trivialfibers.

Proof. By the previous lemma we may assume that the fiber X has been suspendedmany more times than the dimension of B. Let (E, ∂0) be a smoothing of (Z,B) withfiber (M, ∂0) where ∂0M = Dn. Then M is highly connected (compared to dimB)and we may assume that dimM = n + 1 is much bigger than the dimension of thespine of M . This implies that both ∂M and ∂1M are highly connected.Consider the following homotopy fibration sequence (from Bruce Williams’ lecture).

(11) C(∂1M) → Diff(M rel ∂0M)r−→ Emb+(M,M − int(∂1M) rel ∂0M)

where r is the restriction map (r(g) = g f0 for some fixed f0 ∈ Emb+), rel ∂0Mmeans keeping ∂0M fixed pointwise and

C(X) = Diff((X × I)∩ rel X × 0)

AXIOMS FOR HIGHER TORSION 15

where (X × I)∩ is X × I with the corners at ∂X × 1 rounded. The actual fiber of thefibration (11) over f0 ∈ Emb+ is the relative diffeomorphism group

r−1(f0) = Diff(M rel f0(M)) ≃ C(∂1M).

Here Emb+ is the space of embeddings which are homotopy equivalences. In lowdegrees this embedding space has the homotopy type of the space F0(M) of stablepointed self homotopy equivalences of M . Since M is rationally trivial the spaceF0(M) is rationally trivial and has a finite number of components. Since ∂1Mis simply connected, C(∂1M) is also connected according to Cerf. Consequently,Diff(M rel ∂0M) has only finitely many components.Let Diff0(M rel ∂0M) be the identity component of Diff(M rel ∂0M). Then the

classifying space BDiff0(M rel ∂0M) is a finite covering of BDiff(M rel ∂0M). Thisimplies that the induced map in real cohomology is a monomorphism:

H4k(BDiff(M rel ∂0M);R) " H4k(BDiff0(M rel ∂0M);R).

The smooth bundle (E, ∂0) is classified by a mapping B → BDiff(M rel ∂0M) and,by naturality of ∆, ∆(E, ∂0) is the pull-back along this mapping of a universal class∆ ∈ H4k(BDiff(M rel ∂0M);R). To show that this universal class is zero it sufficesto show that its image in H4k(BDiff0(M rel ∂0M);R) is zero.However, the space BDiff0(M rel ∂0M) is rationally homotopy equivalent (in low

degrees) to the space of h-cobordisms of ∂1M . Thus ∆ = 0 on this space by Lemma5.9. The vanishing of this universal class implies that ∆(E, ∂0) = 0 proving thelemma.We will clarify the last paragraph. The fibration (11) restricted to the path com-

ponent Emb0 = Emb0(M,M − int(∂1M) rel ∂0M) of the fixed element f0 deloops tothe following fibration sequence.

(12) Emb0 → BC(∂1M)π−→ BDiff0(M rel ∂0M).

A specific model for this fibration sequence can be given as follows. Let

E := Emb(M,R∞ rel ∂0M)

be the space of all embeddings of M into R∞ which are equal to some fixed em-bedding on ∂0M . This space is contractible and the diffeomorphism group Diff0 =Diff0(M rel ∂0M) acts freely on the right on this space with local section. (SeeAppendix of Cerf’s [Cer68].) Consequently, E/Diff0 is a model for the classifyingspace BDiff0. The elements of E/Diff0 are submanifolds W of R∞ containing afixed copy of ∂0M which are diffeomorphic to M together with an isotopy class ofdiffeomorphisms with M rel ∂0M . Similarly, a model for BC(∂1M) is given by:

BC(∂1M) = E/Diff(M rel f0(M)).

An element of this space is submanifoldW of R∞ with ∂0W = ∂0M fixed together withan embedding g : f0(M) → W − int ∂1W (rel ∂0M) which is a homotopy equivalence.Since C(∂1M) is connected, such an embedding g extends uniquely up to isotopy toa diffeomorphism M ∼= W rel ∂0M . Therefore, the mapping π in (12) is defined.The fiber of π is the connected embedding space

Diff0(M rel ∂0M)/Diff(M rel f0(M)) ∼= Emb0

16 KIYOSHI IGUSA

which is rationally trivial. Therefore π is a rational homotopy equivalence and ∆ ∈H4k(BDiff0(M rel ∂0M);R) is zero if and only if its pull-back to BC(∂1M) is zero.Therefore, we may assume that our smooth bundle (E, ∂0) is classified by a mapping

B → E/Diff(M rel f0(M)).

This means that (E, ∂1) is the union of the constant bundle B × (f0(M), ∂0) with afiberwise h-cobordism. By additivity and Lemma 5.9 we conclude again (but withmore confidence) that ∆(E, ∂0) = 0. !The rest of the proof of the theorem follows suggestions of Bruce Williams who

explained rational homotopy theory to me during lunch on the day before my lecture.We use the well-know fact that stable homotopy groups are rationally isomorphic

to homology. By Lemma 6.1 the main theorem will follow from the following lemma.

Lemma 6.5. ∆(Z,B) = 0 on all smoothable pointed fibration.

Proof. By Lemma 6.4 it suffices to reduce the rank of the rational homology of thefiber X of a pointed smoothable fibration. Let n be maximal so that the reducedrational homology Hn(X;Q) is nonzero. Choose one generator which is fixed by theaction of π1B. Assuming that X has been suspended a large number of times, amultiple of this generator is represented by some α ∈ πn(X). Multiplying by theorder of the torsion subgroup of πn(X) we may assume that α represents a rationallynontrivial element of πn(X) which is fixed by the action of π1B.The homotopy groups πi(X) will be torsion and thus finite for all n < i < n+dimB.

Let m be the product of the orders of all of these groups. Then we claim that mα isrepresented by a fiber preserving continuous mapping

f : (B × Sn, B × ∗) → (Z,B).

This is an exercise in obstruction theory. Choose a triangulation of B and over eachvertex v let

fv : Sn → Xv

be a pointed map representing the homotopy class α. Then, by construction of α, fvextends to a fiber preseving map

f1 : (B1 × Sn, B1) → (Z1, B1)

where Bi is the i-skeleton of B and Zi = p−1(Bi). The obstruction to extending f1to the 2-skeleton of B is a 2-cocycle in B with coefficients in the finite abelian groupπn+1(X). Therefore m1f1 extends to a fiber preserving map

f2 : (B2 × Sn, B2) → (Z2, B2)

where mi = |πn+i(X)|. Continuing in this way we see that∏

mif1 extends to amapping f : B × Sn → Z.Passing to the smoothing (E, ∂0), we can represent f by a smooth fiberwise em-

bedding

f : B × Sn → ∂1E

which we can use to attach a handle B × Sn ×DN to the top of E. (We can assumethat E has a trivial vertical tangent bundle by replacing it with the normal diskbundle.) By additivity the new bundle will be smooth with the same value of ∆

AXIOMS FOR HIGHER TORSION 17

but with fibers having rational homology of rank less than the original bundle. Thisproves the lemma and the main theorem. !

7. Restriction to closed fibers

In this section we express the axioms for higher torsion strictly in terms of smoothbundles with closed oriented manifold fibers. The purpose of this is to make it easierto determine if the higher nonequivariant analytic torsion satisfies these conditions.For example, we know from X. Ma that the transfer axiom is satisfied if both fibersare closed. Thus it will suffice to verify the closed fiber version of Axiom I.Higher torsion is determined by its value on smooth bundles with closed fibers

because of the following simple formulas.

Proposition 7.1. For any higher torsion theory τ satisfying Axioms I and II” wehave

τ(E, ∂0) = τ(E)− τ(∂0E)

τ(E) =1

2(τ(DE) + τ(∂vE))

where DE is the double of E given by pasting two copies of E together along theirvertical boundaries provided that the terms are defined. (See below.)

Proof. Since E ∼= (I × ∂0E) ∪ (E, ∂0), we have by additivity that

τ(E) = τ(I × ∂0E) + τ(E, ∂0).

However, τ(I × ∂0E) = τ(∂0E) by Axiom IIa. The first formula follows.For the other equation we note that DE ∼= E ∪ (E, ∂v). So,

τ(DE) = τ(E) + τ(E, ∂v) = 2τ(E)− τ(∂vE).

Solving for τ(E) we get the second equation. !In this section we need to assume the upper triangular rational homology condition.

We consider H∗(F ;Q) as a module over the group ring Qπ1B. We say this moduleis upper triangular if it has a composition series whose quotients are modules withtrivial action of π1B. By the upper triangular rational homology condition on E wemean this condition on the homology groups H∗(F ;Q) and ,H∗(∂0F ;Q). Since uppertriangular modules form a Serre category, the long exact homology sequence of a pairimplies that H∗(F, ∂F ;Q), H∗(DF ;Q) and H∗(∂F ;Q) will also be upper triangular.Now consider a characteristic class τ(E) ∈ H4k(B;R) for smooth bundles

F → E → B

where F is a closed oriented manifold and the upper triangular rational homologycondition is satisfied.

Axiom Ic: Suppose that F1, i = 1, 2, 3, 4 are compact smooth manifolds with thesame boundary ∂Fi = M . Let Ei, i = 1, 2, 3, 4 be smooth bundles over B withfiber Fi and with the same vertical boundary ∂vEi = E0 so that E0, · · · , E4 satisfythe upper triangular rational homology condition. Then the higher torsion of thebundles Ei ∪ Ej with closed fibers Fi ∪ Fj should be related by:

(13) τ(E1 ∪ E2) + τ(E3 ∪ E4) = τ(E1 ∪ E3) + τ(E2 ∪ E4).

18 KIYOSHI IGUSA

Axiom II for closed fibers: Suppose that F1 → E1p−→ B and F2 → E2

q−→ E1

are bundles with closed fibers Fi satisfying the upper triangular rational homologycondition. Then the higher torsion τB(E2) of E2 as a bundle over B is given by

(14) τB(E2) = χ(F2)τ(E1) + p∗(τE1(E2) ∪ e(E1))

Theorem 7.2. [Ma97] Nonequivariant higher analytic torsion satisfies Axiom II forclosed fibers.

As before, we do not know whether higher FR-torsion satisfies this condition so wealso consider the weaker statement:

Axiom IIc: Axiom II holds in the case when either p : E1 → B or q : E2 → E1 is anoriented linear sphere bundle.

Proposition 7.3. Higher FR-torsion τFR2k and the higher MMM classes M2k satisfy

Axioms Ic, IIc.

Theorem 7.4. Suppose that τ is a higher torsion theory satisfying Axioms Ic, IIc.Then there exist unique real numbers a, b so that

τ = aτFR2k + bM2k.

Remark 7.5. The numbers a, b are determined by the value of τ on oriented linearS1 and S2 bundles since:

(1) τFR2k = (−1)k+1ζ(2k + 1)ch2k and M2k = 0 for oriented S1 bundles.

(2) τFR2k = (−1)kζ(2k + 1)ch2k, M2k = 2(2k)!ch2k for oriented linear S2 bundles.

The proof of Theorem 8.4 and Proposition 8.3 will be given in another paper. Itis not hard but I need a different version of the main theorem so that it will implyTheorem 8.4. For this I need to rewrite the first part of this paper.

AXIOMS FOR HIGHER TORSION 19

8. Restriction to closed fibers

In this section we express the axioms for higher torsion strictly in terms of smoothbundles with closed oriented manifold fibers. The purpose of this is to make it easierto determine if the higher nonequivariant analytic torsion satisfies these conditions.For example, we know from X. Ma that the transfer axiom is satisfied if both fibersare closed. Thus it will suffice to verify the closed fiber version of Axiom I.Higher torsion is determined by its value on smooth bundles with closed fibers

because of the following simple formulas.

Proposition 8.1. For any higher torsion theory τ satisfying Axioms I and II” wehave

τ(E, ∂0) = τ(E)− τ(∂0E)

τ(E) =1

2(τ(DE) + τ(∂vE))

where DE is the double of E given by pasting two copies of E together along theirvertical boundaries.

Proof. Since E ∼= (I × ∂0E) ∪ (E, ∂0), we have by additivity that

τ(E) = τ(I × ∂0E) + τ(E, ∂0).

However, τ(I × ∂0E) = τ(∂0E) by Axiom IIa. The first formula follows.For the other equation we note that DE ∼= E ∪ (E, ∂v). So,

τ(DE) = τ(E) + τ(E, ∂v) = 2τ(E)− τ(∂vE).

Solving for τ(E) we get the second equation. !In this section we will be using the equations in the above proposition as definitions

of the relative torsion τ(E, ∂0) given the value of τ on bundles with closed manifoldfibers satisfying the upper triangular rational homology condition detailed below.We considerH∗(F ;Q) as a module over the group ring Qπ1B. We say this module is

upper triangular if it has a composition series whose quotients are modules with trivialaction of π1B. By the upper triangular condition on (E, ∂0E) we mean this conditionon the homology groups H∗(F ;Q), H∗(∂0F ;Q), H∗(∂1F ;Q) and H∗(∂0F ∩ ∂1F ;Q).Since upper triangular modules form a Serre category, the long exact homology se-quence of a pair implies that H∗(F, ∂0F ;Q), H∗(DF ;Q) and H∗(∂F ;Q) will also beupper triangular.Now consider a characteristic class τ(E) ∈ H4k(B;R) for smooth bundles

F → E → B

where F is a closed oriented manifold and the upper triangular rational homologycondition is satisfied.

Axiom Ic: Suppose that F1, i = 1, 2, 3, 4 are compact smooth manifolds with thesame boundary ∂Fi = M . Let Ei, i = 1, 2, 3, 4 be smooth bundles over B withfiber Fi and with the same vertical boundary ∂vEi = E0 so that E0, · · · , E4 satisfythe upper triangular rational homology condition. Then the higher torsion of thebundles Ei ∪ Ej with closed fibers Fi ∪ Fj should be related by:

(15) τ(E1 ∪ E2) + τ(E3 ∪ E4) = τ(E1 ∪ E3) + τ(E2 ∪ E4).

20 KIYOSHI IGUSA

Axiom II for closed fibers: Suppose that F1 → E1p−→ B and F2 → E2

q−→ E1

are bundles with closed fibers Fi satisfying the upper triangular rational homologycondition. Then the higher torsion τB(E2) of E2 as a bundle over B is given by

(16) τB(E2) = χ(F2)τ(E1) + p∗(τE1(E2) ∪ e(E1))

Theorem 8.2 (X.Ma). Nonequivariant higher analytic torsion satisfies Axiom II forclosed fibers.

As before, we do not know whether higher FR-torsion satisfies this condition so wealso consider the weaker statement:

Axiom IIc: Axiom II holds in the case when either p : E1 → B or q : E2 → E1 is anoriented linear sphere bundle.

Proposition 8.3. Higher FR-torsion τFR2k and the higher MMM classes M2k satisfy

Axioms Ic, IIc.

Theorem 8.4. Suppose that τ is a higher torsion theory satisfying Axioms Ic, IIc.Then there exist unique real numbers a, b so that

τ = aτFR2k + bM2k.

Remark 8.5. The numbers a, b are determined by the value of τ on oriented linearS1 and S2 bundles since:

(1) τFR2k = (−1)k+1ζ(2k + 1)ch2k and M2k = 0 for oriented S1 bundles.

(2) τFR2k = (−1)kζ(2k + 1)ch2k, M2k = 2(2k)!ch2k for oriented linear S2 bundles.

The rest of this paper is devoted to the proof of Theorem 8.4 (and Proposition 8.3).

Lemma 8.6. Suppose that E → B is a smooth bundle so that E and ∂vE satisfy theupper triangular condition. Then Axioms Ic and IIc imply that

τ(∂vE) = τ(∂v(E ×D2)).

Also, Axioms I,IIa,IIb imply this same equation.

Proof. First we note that E×D2 has corners. It is the union of two manifold bundlesE × S1 and ∂vE ×D2 which meet along their common vertical boundary ∂vE × S1.By Axiom Ic, twice the torsion of E ×D2 is the sum of the torsions of the doubles ofeach part:

2τ(∂(E ×D2)) = τ(∂vE × S2) + τ(DE × S1).

However, the product formula for sphere bundles (which follows from IIc) tells us that

τ(SN ×X) = χ(SN)τ(X).

Therefore, τ(∂vE × S2) = 2τ(∂vE) and τ(DE × S1) = 0. This proves the lemma inthe first case.In the second case, we prove the following equivalent form of the equation:

τ(E, ∂v) = τ(E ×D2, ∂v).

AXIOMS FOR HIGHER TORSION 21

(The equivalence follows from Proposition 8.1 and from Axiom IIa which impliesτ(E) = τ(E ×D2).) By the main theorem we may assume that τ is a linear combi-nation of τFR

2k and M2k. For M2k the statement is easy. For τFR2k we use Theorem 1.5

which implies:τFR2k (E, ∂v) = (−1)n−1τFR

2k (E) + (−1)nM2k(E)

τFR2k (E ×D2, ∂v) = (−1)n+1τFR

2k (E ×D2) + (−1)n+2M2k(E ×D2).

The right hand terms are equal. So the left hand terms are also equal. !Proof of Theorem 8.4 and Proposition 8.3. We first extend τ to manifold bundles E →B so that E and ∂vE have upper triangular rational homology bundles. In this casethe fiberwise double DE also satisfies the upper triangular condition and we candefine τ(E) by

(17) τ(E) :=1

2(τ(DE) + τ(∂vE)).

Next, suppose that (E, ∂0) = (E, ∂0E) → B is a smooth bundle pair satisfying theupper triangular condition. Then τ(E, ∂0) can be defined by

(18) τ(E, ∂0) := τ(E)− τ(∂0E).

To prove the theorem it suffices to show that τ(E, ∂0) satisfies Axioms I, IIa, IIb asdescribed earlier. To prove the proposition it suffices to show the converse, namely,if τ(E, ∂0) satisfies I, IIa, IIb then τ satisfies Axioms Ic and IIc.We begin with the additivity axiom:

(19) τ(E1 ∪ E2, ∂0E1) = τ(E1, ∂0) + τ(E2, ∂0) ?

Expressed in terms of absolute torsion using (18) this is:

(20) τ(E1 ∪ E2) = τ(E1) + τ(E2)− τ(E1 ∩ E2) ?

Expand each term using the defining equation (17):

τ(E1) :=1

2(τ(DE1) + τ(∂vE1))(21)

τ(E2) :=1

2(τ(DE2) + τ(∂vE2))(22)

τ(E1 ∪ E2) :=1

2(τ(D(E1 ∪ E2)) + τ(∂v(E1 ∪ E2)))(23)

τ(E1 ∩ E2) :=1

2(τ(∂v(E1 ∩ E2)) + τ(D(E1 ∩ E2))).(24)

The order of the terms in the last equation is reversed so that it matches the followingtwo examples of Axiom I for closed fibers:

τ(∂vE1) + τ(∂vE1) = τ(∂v(E1 ∪ E2)) + τ(D(E1 ∩ E2))

τ(DE1) + τ(DE2) = τ(D(E1 ∪ E2)) + τ(D(I × (E1 ∩ E2)))

However, the last term is equal to

τ(D(I × (E1 ∩ E2))) = τ(∂v(D2 × (E1 ∩ E2))) = τ(∂v(E1 ∩ E2))

by Lemma 8.6. Therefore, the sum of the RHS of (21) and (22) is equal to the sumof the RHS of (23) and (24). So, the sum of the LHS of (21) and (22) is equal to thesum of the LHS of (23) and (24) as required. Thus τ(E, ∂0) satisfies Axiom I.

22 KIYOSHI IGUSA

Conversely, given τ satisfying Axioms I and IIa, IIb, we know by Proposition 8.1that (20) holds. Axiom Ic follows immediately.Next we examine the transfer axioms. To prove the theorem we need to show

that Axioms Ic,IIc imply axioms IIa and IIb. To prove the proposition we need toprove that I,IIa,IIb imply both cases of IIc. In both directions we may assume bothadditivity statements (I and Ic) since we just proved that they both hold in bothcases.Recall that we have the following diagram.

F2=−−−→ F2&

&

(F3, ∂0) −−−→ (E2, ∂0) −−−→ B&

&q

&=

(F1, ∂0) −−−→ (E1, ∂0) −−−→p

B

The transfer equation (Axiom II) is:

(25) τB(E2, ∂0) = χ(F2)τ(E1, ∂0) + p∗(τE1(E2) ∪ e(E1, ∂0))

The transfer formulas for the double DE2 and vertical boundary ∂vE2 of E2 as abundle over E1 are

(26) τB(DE2, ∂0) = χ(DF2)τ(E1, ∂0) + p∗(τE1(DE2) ∪ e(E1, ∂0))

(27) τB(∂vE2, ∂0) = χ(∂F2)τ(E1, ∂0) + p∗(τE1(∂

vE2) ∪ e(E1, ∂0))

By (17) and the corresponding formula for the Euler characteristic we see that (26)and (27) imply (25) and we conclude the following.If the transfer formula (25) holds for closed F2 then it holds for all F2 with (E1, ∂0)

held fixed.For the case when q : E2 → E1 is a linear disk bundles, by induction on the

dimension of the sphere, if the transfer formula holds when q : E2 → E1 is a lineardisk bundle then it holds when q is a linear sphere bundle. This gives the following.The transfer formula holds for all oriented linear disk bundles E2 → E1 if and only

if it holds for all oriented linear sphere bundles E2 → E1 with (E1, ∂0) held fixed.Now we look at how the transfer formula changes as we vary E1. We have !

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