THE PRODUCT THEOREM FOR PARAMETRIZED
TOPOLOGICAL REIDEMEISTER TORSION
WOJTEK DORABIALA AND MARK W. JOHNSON
Abstract.The goal of this article is to prove the product formula for pa*
*rametrized
topological Reidemeister torsion. The theorem states that the product of*
* the
parametrized Euler characteristic of one fibration with the parametrized*
* Reide
meister torsion class of another fibration yields the parametrized Reide*
*meister
torsion class of the product fibration. In the process of establishing t*
*he the
orem, several new products must be defined involving (derivative theorie*
*s of)
parametrized Atheory and a detailed description of the coassembly map f*
*or
parametrized Atheory is included.
1.Introduction
Before moving into the statements of the main results, we would like to say a*
* few
words about our motivation for studying this question. The definition of topolo*
*gical
Reidemeister torsion used here is based on that of [5]. They conjecture that th*
*eir
definition should become equivalent to that of [7] once their class is pushed i*
*nto
cohomology using the Borel regulator technique. In fact, we would eventually li*
*ke to
establish that our definition, the definition from [7] and the definition from *
*[1] agree
in cohomology, which would generalize the work of [12] in the nonparametrized
case.
In [9], the authors give a completely algebraic analog of our product theorem.
They establish the fact that the Reidemeister torsion of the tensor product A *
* B
where B is contractible is equal to the Euler characteristic of A times the tor*
*sion
of B. Our goal is to give a topological lifting of this result to Whitehead spa*
*ces.
Moving into specifics, all fibrations considered here will be perfect fibrati*
*ons.
That is, the fibers of every fibration mentioned will be finitely dominated by *
*as
sumption. We will also assume the base space B comes equipped with an effi
cient triangulation, where each simplex contains only finitely many subcomplexe*
*s.
Clearly, this allows any differentiable compact manifold as a choice of B.
The first major result is an Atheory product formula for Euler characteristi*
*cs,
relying upon the external parametrized Atheory product
`E1 ' ` E2 ' ` E1xE2 '
~A : A # p1^ A # p2! A # p1x p2.
B1 B2 B1xB2
Theorem 1.1. Suppose p1 and p2 are perfect fibrations. Then
~A(ØA (p1), ØA (p2)) = ØA (p1 x p2).
____________
Date: March 3, 2003.
1991 Mathematics Subject Classification. Primary: 19D10; Secondary: 18F25, 1*
*9Exx, 55R70.
Key words and phrases. Reidemeister torsion, parametrized Atheory, parametr*
*ized Euler
characteristic, homotopy limit, Whitehead space, perfect fibration, retractive *
*space.
1
2 W. DORABIALA AND M. W. JOHNSON
In order to simplify notation, if a fibration p : E ! B is equipped with a ch*
*oice
of bundle of finitely generated, free Rmodules OE : V ! E, it will be referred*
* to
as a fibration with flat bundle . The phrase fibration with acyclic flat bundle*
* will
then imply the additional condition that H*(Eb; Vb) is zero for each b 2 B. For*
* the
purposes of our desired product formula for parametrized Reidemeister torsion, *
*the
following corollary is actually the key result.
Corollary 1.2. Suppose p1 : E1 ! B1 is a perfect fibration with flat bundle and
p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. Then
~Acy(ØA (p1), ØAcy(p2)) = ØAcy(p1 x p2).
The`next'result relies upon the existence of a restricted external multiplica*
*tion
E
for # p:
B
` E1 ' ` E2 ' ` E1xE2 '
_cy: # p1^ cy # p2 ! cy # p1x p2.
B1 B2 B1xB2
Theorem 1.3. Suppose p1 : E1 ! B1 is a perfect fibration with flat bundle and
p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. Then
_cy(Ø (p1), Øcy(p2)) ' Øcy(p1 x p2)
where ' means there exists a natural path connecting these points.
In section 6, we will establish the existence of a restricted product pairing
` E1 ' ` E2 ' ` E1xE2 '
: # p1^ Wh R2B2# p2 ! Wh R1BAR21xB2 # p1x p2
B1 B2 B1xB2
` E2 '
where Wh R2B2# p2 represents the Whitehead space associated to p2. Recall that
B2
the topological Reidemeister torsion øR2 (p2)may be viewed as a point in this
Whitehead space. The main result here is the product formula for torsion which
generalizes a classical result of Kwun and Szczarba in [9].
Theorem 1.4 (Product Formula). Suppose p1 : E1 ! B1 is a perfect fibration with
flat bundle and p2 : E2 ! B2 is a perfect fibration with acyclic flat bundle. T*
*hen
øR1 AR2 (p1 x p2)is defined and
(Ø (p1), øR2 (p2)) ' øR1 AR2 (p1 x p2).
Remark 1.5. Note that in the informal notation of [5], øR1 AR2 (p1 x p2)and
(Ø (p1), øR2 (p2)) both correspond to maps B1x B2 ! Wh R1BAR21xB2(E1 x(E2)this
is a different version of Whitehead space). The statement of the product formula
then becomes that these two maps are homotopy equivalent.
The Product Theorem is deduced as a consequence of Theorem 1.3 in section
2. The basic idea, again using the informal notation of [5], is that the follo*
*wing
diagram commutes up to homotopy.
AcyB;(E);
ØAcy(p)xxxx
xxx 
xxx fflffl
B øR(p)//_WhRB(E)
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 3
The authors would like to express their thanks to Bruce Williams for helpful
discussions concerning this material.
2. The Product Formula
This section is devoted to reducing the proofs of Theorems 1.1, 1.3, 1.4 and
Corollary 1.2 to the existence of certain products, which will be defined in se*
*ction
6, and several technical results which will be proven later.
The products we need will be the following.
` E1 ' ` E2 ' ` E1xE2 '
A # p1 ^ A # p2 __~A__//_A # p1x p2
B1 B2 B1xB2
` E1 ' ` E2 ' ` E1xE2 '
A # p1^ Acy # p2 ~Acy//_Acy # p1x p2
B1 B2 B1xB2
` E1 ' ` E2 ' ` E1xE2 '
# p1^ # p2 ______//_ # p1x p2
B1 B2 B1xB2
`E1 ' ` E2 ' _cy ` E1xE2 '
# p1 ^ cy # p2_____// cy # p1x p2
B1 B2 B1xB2
` E1 ' ` E2 ' ` E1xE2 '
# p1 ^ Wh R2B2# p2 _____//WhR1BAR21xB2# p1x p2
B1 B2 B1xB2
Also in Sections 5 and 6, we will construct natural öc assembly maps"
` E ' ` E '
A # p __ffi_//_ # p
B B
` E ' ` '
fficy cy E
Acy # p _____// # p
B B
and `
E ' ffl ` E '
cy # p _____//WhRB # p .
B B
We will define Ø (p)as the image under the coassembly map ffi of the parametr*
*ized
Atheory Euler characteristic ØA (p)and similarly for Øcy(p). At the end of Sec*
*tion
6, this will lead to a proof of the following.
Proposition 2.1. Suppose p2 : E2 ! B2 is a perfect fibration with acyclic flat
bundle. Then
øR2 (p2)= ffl2(Øcy(p2))
In Sections 6 and 7 and we will establish the following key technical result.
4 W. DORABIALA AND M. W. JOHNSON
Proposition 2.2. The following diagrams are each commutative up to a natural
homotopy:
` E1 ' ` E2 ' ` '
cy cyE1xE2
(1) A # p1^ Acy # p2 ~A__//_A # p1x p2
B1 B2 B1xB2
ffi1xfficy2 fficy1,2
` ' fflffl`' ` fflffl'
E1 E2 E1xE2
# p1^ cy # p2 __cy_// cy # p1x p2
B1 B2 _ B1xB2
and
` E1 ' ` E2 ' _cy ` E1xE2 '
(2) # p1 ^ cy # p2 _________//_ cy # p1x p2
B1 B2 B1xB2
1xffl2 ffl1,2
` ' fflffl` ' `fflffl '
E1 E2 R R E1xE2
# p1 ^ Wh R2B2# p2 _____//Wh1BA12xB2 # p1x p2.
B1 B2 B1xB2
In fact, the first statement of the propostion follows immediately from the f*
*ol
lowing theorem together with Lemma 2.5 below. Notice, we intend the product
rather than smash product versions of ~A and _ for technical reasons.
Theorem 2.3. The diagram
` E1 ' ` E2 ' ` E1xE2 '
A # p1 x A # p2__~A_//A # p1x p2
B1 B2 B1xB2
ffi1xffi2 ffi1,2
` ' fflffl`' ` fflffl'
E1 E2 E1xE2
# p1 x # p2 _____// # p1x p2.
B1 B2 _ B1xB2
commutes up to a natural homotopy.
We can now establish Theorem 1.1 and Corollary 1.2, which generalize classical
properties of the Euler characteristic. Recall the fold map
E t E ø E
`E '
defines an object of Retfd(p), hence a point in A # p which we denote by ØA (*
*p)
B
and think of as the sphere bundle of the trivial line bundle over E. One might
think of this as a parametrized version of the Euler characteristic of the fibe*
*r Eb
Since ~A is defined as the map induced by the biexact functor (by restriction
from [16]) external smash product of retractive spaces
^E1xE2 : Retfd(p1)x Retfd(p2)! Retfd(p1 x p2),
and the Euler characteristic is represented by
E t E ø E
in Retfd(p), Theorem 1.1 is a consequence of the following simple lemma, which *
*we
prove in the next section.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 5
Lemma 2.4. One has the identity
(E1 t E1)^E1xE2 (E2 t E2)= (E1 x E2)t (E1 x E2)
in Retfd(p1 x p2).
In order to prove Corollary 1.2, we need the following result on the relation*
*ship
between ~A and ~Acy.
Lemma 2.5. (1) The diagram
` E1 ' ` E2 ' ` E1xE2 '
A # p1^ Acy # p2 ~Acy//_Acy # p1x p2
B1 B2 B1xB2
 
 
` ' fflffl`' ` fflffl'
E1 E2 E1xE2
A # p1 ^ A # p2 __~___//_A # p1x p2
B1 B2 A B1xB2
commutes. ` E ' `E '
(2) The canonical map Acy # p ! A # p is an injection for every fibra
B B
tion p. ` ' ` '
E E
(3) The canonical map cy # p ! # p is an injection for every fibra
B B
tion p.
Proof of Corollary 1.2.The corollary follows from Theorem 1.1 and Lemma 2.5
along`with'the`fact'that ØAcy(p2)goes to ØA (p2)under the canonical inclusion
E2 E2
Acy # p2 ! A # p2.
B2 B2
This suffices to allow us to prove Theorems 1.3 and 1.4.
Proof of Theorem 1.3.The assertion is that
_cy(Ø (p1), Øcy(p2)) ' Øcy(p1 x p2).
which by our definition of Ø (p1)and Øcy(p2)is equivalent to
_cy(ffi1(ØA (p1)), fficy2(ØAcy(p2))) ' fficy1,2(ØAcy(p1 x p2)).
By diagram 1 in Proposition 2.2, it suffices to establish
~Acy(ØA (p1), ØAcy(p2)) = ØAcy(p1 x p2)
which is the statement of Corollary 1.2.
Proof of Theorem 1.4.The assertion is that
(Ø (p1), øR2 (p2)) ' øR1 AR2 (p1 x p2).
However, Proposition 2.1 implies this is equivalent to the statement
(Ø (p1), ffl2(Øcy(p2))) ' ffl1,2(Øcy(p1 x p2)).
Now Proposition 2.2 (2) implies this statement follows immediately from Theorem
1.3.
6 W. DORABIALA AND M. W. JOHNSON
A proof of Lemma 2.4 and some basic results on retractive spaces make up Sect*
*ion
3. Section 4 is devoted to defining all of the relevant theories, parametrized *
*Euler
characteristics and torsion. The technical details of Thomason homotopy limit
problem maps necessary for constructing the coassembly maps and understanding
their basic properties are handled in Section 5. The bulk of Section 6 is devot*
*ed
to defining the remaining products and establishing their properties, with proo*
*fs of
Proposition 2.1 and the second statement of Proposition 2.2 at the end. Finally*
*, in
Section 7 we give the proofs of Lemma 2.5 and of Theorem 2.3.
3.Functors on Retractive Spaces
In this section we would like to introduce the external smash product of retr*
*ac
tive spaces over a fibration along with two "change of base" functors related to
restriction to a subcomplex of the base space B. The external smash products
induce the external product in parametrized Atheory and their interaction with
the change of base functions is important to the proof of Proposition 2.2.
We begin by defining the category of retractive spaces over p, Retfd(p), as t*
*he
Waldhausen category of retractive spaces X over E where the composition X !
E ! B is a (Hurewicz) fibration whose fibers are finitely dominated. (See [17].*
*) The
Waldhausen category structure has weak equivalences the homotopy equivalences
which happen to be maps in this category and similarly cofibrations are the map*
*s in
this category which, as continuous maps, are closed embeddings with the homotopy
extension property. Recall that a retractive space over E is a pair X ø E where
the inclusion i : E ! X is a cofibration and ri = 1E . In this notation, Retfd(*
*E)
consists of finitely dominated retractive spaces over E, since it assumes the b*
*ase
B = *. In particular, notice Retfd(p)is a subWaldhausen category of Retfd(E).
Next, we introduce the external smash product.
Definition 3.1. Let X 2 Retfd(p1)and Y 2 Retfd(p2). Then their external smash
product is a space in Retfd(p1 x p2)which we define as the following pushout
X x E2 [ E1 x Y_______//_X x Y
 
 
fflffl fflffl
E1 x E2_________//X ^E1xE2 Y.
Proof of Lemma 2.4.Both terms can be identified with the pushout of the followi*
*ng
diagram.
E1 x (E2 t E2) [ (E1 t E1) x E2__//_E1 x E2
 
 
fflffl fflffl
(E1 t E1) x (E2 t E2)__________//P
Let Simp(B) denote the category of simplices of the chosen triangulation of t*
*he
base space B, where the morphisms are only the inclusions of subsimplices. If
ffi,oe 2 Simp (B), let iffioe: ffi ! oe and ioeB: oe ! B denote the natur*
*al inclusions.
Notice that iffioeis a homotopy equivalence, hence ~iffioe: p1(ffi) ! p1(oe) *
*will also be
a homotopy equivalence.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 7
fd fd
Definition 3.2. Given W 2 Retfdp1(ffi), Y 2 Ret p1(oe) and X 2 Ret (E)
we define pullbacks and pushforwards as below.
(1) The pushforward of W along iffioeis the following pushout
~iffioe
p1(ffi)____//p1(oe)
 
 
fflffl fflffl
W ______//_iffioe*(W )
where ~iffioe: p1(ffi) ! p1(oe) is the induced inclusion (which is a h*
*omotopy
equivalence).
(2) Similarly, we can define ioeB*(Y ) as the pushout of Y along the induced
inclusion ~ioeB.
(3) The pullback of Y along iffioeis the following pullback
*
iffioe(Y_)____//Y
 
 
fflffl~iffifflffloe
p1(ffi)____//p1(oe)
where ~iffioe: p1(ffi) ! p1(oe) is the induced inclusion (which is a h*
*omotopy
equivalence).
(4) Similarly, we can define ioeB*(X) as the pullback of X along the induced
inclusion ~ioeB.
To see that the first two operations land in the correct categories, see [17].
For the pullback operations, it is important to note the fact that iffioeis a*
* closed em
bedding implies the fibers of iffioe*(X) are simply certain fibers of X itself*
* (thereby
preserving the finitely dominated condition). A result of Lück [10] implies the*
* in
clusion being a cofibration is preserved by the pullback operation as well. Fin*
*ally,
the fibration condition is preserved under pullbacks.
The following lemma is largely a consequence of the universal properties of
pushouts and pullbacks.
Lemma 3.3. Given ffi oe, the pullback and pushforward over iffioeyield an adj*
*oint
pair of functors
i * j
iffioe*: Retfdp1(ffi)! Retfdp1(oe), iffioe: Retfdp1(oe) ! Retfdp1(ffi)
*
Hence, there is a natural homotopy equivalence iffioe*iffioe(Y ) ! Y called th*
*e unit
of adjunction and similarly for Z ! iffioe*iffioe*(Z). The functor iffioe*is *
*an exact
functor, while the maps of spaces iffioe*(Y ) ! Y and Z ! iffioe*(Z) are homo*
*topy
equivalences. Finally, the functors iffioe*and ioeB*preserve homotopy equival*
*ences.
Notice, the fact that X ! oe or X ! B is assumed to be a fibration implies
the pullback preserves homotopy equivalences. We now have a result about the
interaction between change of base and external smash products.
fd
Lemma 3.4. Suppose Wn 2 Retfdp1n(ffin)and Xn 2 Ret p1n(oen)with ffin
oen for n = 1,2. Then there are natural homotopy equivalences
8 W. DORABIALA AND M. W. JOHNSON
(1)
* *
iffi1oe1(X1) ^p11(ffi1)xp12(ffi2)iffi2oe2(X2)


fflffl
(iffi1oe1x iffi2oe2)*(X1 ^p11(oe1)xp12(oe2)X2)
(2)
(iffi1oe1x iffi2oe2)*(W1 ^p11(ffi1)xp12(ffi2)W2)


fflffl
iffi1oe1*(W1) ^p11(oe1)xp12(oe2)iffi2oe2*(W2)
Proof.We will display the second map in detail, while the first is dual. To beg*
*in,
note there is a commutative diagram
W1 xOW2O______________________//iffi1oe1*(W1)OxO iffi2oe2*(W2)
 
 
 
W1 x p12(ffi2) [ p11(ffi1)_x/W2/_iffi1oe1*(W1) x p12(oe2) [ p11(oe1) x iff*
*i2oe2*(W2)
 
 
fflffl fflffl
p11(ffi1) x p12(ffi2)______________//_p11(oe1) x p12(oe2)
where the horizontal maps are all homotopy equivalences and both maps to the top
row are cofibrations. Thus, there is an induced homotopy equivalence on pushouts
ffi ffi
W1 ^p11(ffi1)xp12(ffi2)W2 ! ioe11*(W1) ^p11(oe1)xp12(oe2)ioe22*(W2)
which factors through an induced map
ffi *
* ffi
(iffi1oe1x iffi2oe2)*(W1 ^p11(ffi1)xp12(ffi2)W2) ! ioe11*(W1) ^p11(oe1)xp*
*12(oe2)ioe22*(W2)
by construction. Since
W1 ^p11(ffi1)xp12(ffi2)W2 ! (iffi1oe1x iffi2oe2)*(W1 ^p11(ffi1)xp12*
*(ffi2)W2)
is also a homotopy equivalence, the statement follows.
4.Parametrized Theories, Euler Characteristics and Torsion
Classes
` E '
Definition 4.1. Let A # p = wS. Retfd(p) where Retfd(p)is the Wald
B
hausen category of retractive spaces over E with the fibers of the composition *
*pr
finitely dominated, where r : X ! E is the retraction.
Definition 4.2. Suppose p is a perfect fibration with flat bundle V . Then
` E '
Acy # p = wS. Retfd,cy(p)
B
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 9
where Retfd,cy(p)is the full subWaldhausen category of Retfd(p)consisting of t*
*hose
objects X ø E where r*(V ) ! X makes each fiber Xb ! * a perfect fibration with
acyclic flat bundle.
Definition 4.3. Suppose p is a perfect fibration with flat bundle V . Let
` E '
AR # p = wS. RetfdR(p)
B
where RetfdR(p)is the category Retfd(p)with a different Waldhausen category str*
*uc
ture where the weak equivalences are the local homology chain equivalences with
local coefficients in r*V and cofibrations are as usual. (See [4] for details.)
`E '
Recall Acy # p is homotopy equivalent to the homotopy fiber of the map
B
induced by the identity, (which is an exact functor in this direction)
`E ' ` E '
A # p ! AR # p
B B
as in [4].
` E '
Definition 4.4. We set # p = holim wS. Retfdp1(oe).
B oe2Simp(B)
Let
f~
W _____//V
 
 
fflfflfflffl
X __f__//Y
be a pullback diagram where X is equipped with the flat bundle W and Y is
equipped with the flat bundle V . Then we will say (f; ~f) is a bundle morphism
from (X; W ) to (Y ; V ). Let D denote the category whose objects are pairs (X;*
* W )
where X has flat bundle W and morphism set D ((X; W )(Y ; Vt))he set of bundle
maps (f, ~f) as above.
We would like to think of Acy as a functor from D to T op. Thus, we need
to understand the map Acy(X; W )! Acy(Y ; V )induced by a bundle morphism
(f, ~f).
We define a functor f* from Retfd(X)to Retfd(Y )by taking Z ø X to Z[X Y ø
Y . We would like to know that, restricting the source to the full subcategory
Retfd,cy(X)(W), the target lies in the full subcategory Retfd,cy(Y()V.)This req*
*uires
a statement from homological algebra; specifically, H*(Z, X; W ) = 0 must imply
H*(f*(Z), Y ; V ) = 0. However, this follows from the definition of homology wi*
*th
local coefficients. (See [3].)
Now we would like to know the functor f* is exact. See Lemma 3.3 for the fact
that f* is exact in our case when considered as a functor
f* : Retfd(X)! Retfd(Y ).
However, since both cyclic categories are subWaldhausen categories, the exactn*
*ess
in our case follows by restriction. Thus, f* induces a map Acy(X) ! Acy(Y ).
Now we define a functor P : Simp(B) ! D associated to any fibration p with fl*
*at
bundle V (so (E; V ) 2 D) by sending oe 7! (p1(oe); V p1(oe)). Any morphism *
*ffi ! oe
10 W. DORABIALA AND M. W. JOHNSON
in Simp (B)is the inclusion of a subcomplex. Thus, the horizontal rectangles and
right squares in the following diagram are pullbacks, which implies the left sq*
*uares
are pullbacks as well.
V p1(ffi)_//V p1(oe)_//V
  
  
fflffl fflffl fflffl
p1(ffi)____//p1(oe)___//_E
  
  
fflffl fflffl fflffl
ffi_________//_oe_____//B
Hence, we have defined a morphism
(p1(ffi); V p1(ffi)) ! (p1(oe); V p1(oe))
associated to each morphism ffi ! oe, thereby making our assignment P a functor.
We can now define the composite functor
cy
Simp (B) _P__//_D_A__//T op
and this is the functor whose holimappears in the following definition.
` E '
Definition 4.5. We set cy # p = holim wS. Retfd,cyp1(oe).
B oe2Simp(B)
Replacing the functor Acy : D ! T op with the functor AR : D ! T op we may
form a similar holim.
` E '
Definition 4.6. We set R # p = holim wS. RetfdRp1(oe).
B oe2Simp(B)
`E '
Definition 4.7. Let ØA (p)2 A # p denote the point corresponding to
B
E t E ø E
as an object of Retfd(p), with the fold map as retraction.
Definition`4.8.'Suppose p is a fibration with acyclic flat bundle V . Let ØAcy(*
*p)2
E
Acy # p denote the point corresponding to EtE ø E as an object of Retfd,cy(p).
B
Notice that H*((E t E)b, Eb; Vb) is naturally isomorphic to H*(Eb, Vb) = 0, so
the assumption of acyclic flat bundle is simply a restatement of the fact that *
*the
sphere bundle of the trivial`line'bundle, E t E ø E, is an object of Retfd,cy(p*
*).
E
Now define Ø (p)2 # p as the image of ØA (p)under the coassembly map
B
ffi we will describe in detail in section 5. (This agrees with the definition o*
*f Ø (p)
given in [5].) Similarly, set Øcy(p)= fficy(ØAcy(p)).
Next, we define the parametrized Whitehead space associated to a fibration p
with flat bundle V . Notice the inclusions p1(oe) ! E will yield a map
` E '
# p = holim A p1(oe)! holim A (E).
B Simp(B) Simp(B)
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 11
Follow this by the map
holimA (E) ! holim K (R)
Simp(B) Simp(B)
induced by the linearization map associated to V . Denote the composition
` E '
V : # p ! holim K (R).
B Simp(B)
`E '
Definition 4.9. Let Wh RB # p denote the homotopy fiber of V.
B
For the sake of precision, we should mention our model for K(R) is wS.Ch*(R*
*) 
where Ch*(R) is the category of bounded complexes of projective modules given
a Waldhausen structure where injections are cofibrations and quasiisomorphisms
(homology equivalences) are the weak equivalences. Then the linearization map
is induced by the functor Retfd(E) ! Ch*(R) sending X to the relative chain
complex with local coefficients and compact supports C*(X, E; r*(V )).
It will be shown at the end of the next section that there is a canonical pat*
*h fi
in holim K (R) from V(Ø (p)) to the basepoint provided the flat bundle over p *
*is
Simp(B)
assumed to be acyclic.
Definition 4.10. Suppose p is a`fibration'with acyclic flat bundle V . Then the
E
Euler characteristic Ø (p)2 # p together with the path fi define a point in
` ' B
E
Wh RB # p by definition of homotopy fiber. We will refer to this point as the
B
torsion class øR (p)or the topological Reidemeister torsion.
5. Thomason Homotopy Limit Problems
We would like to understand the definition of the coassembly map
`E ' ` E '
ffi : A # p ! # p.
B B
This comes from a formal trick which travels under the name of a Thomason ho
motopy limit problem which we will describe in an abstract context. In order to*
* be
careful with the properties of the realization of a category, in this section o*
*nly, we
will make explicit the composition of the nerve functor and the geometric reali*
*zation
of a simplicial set.
Suppose C is a small category and G : C ! Cat is a functor. There is always
another important functor C ! Cat, defined by C 7! C=C sending each object in C
to the category of objects over it in C. Clearly this is a covariant functor, d*
*efined
on morphisms by postcomposition. Given two functors with the same source and
target, one can define the category of natural transformations between them, wh*
*ich
we'll denote by Nat (C=?, G).
Recall that one normally views a natural transformation as a point in a large
product of mapping sets satisfying certain additional conditions. More precise*
*ly,
natural transformations lie in the equalizer of the two maps induced by precomp*
*o
sition and postcomposition respectively. (See section IX.5 of [11] for the tran*
*slation
as an "end".)
12 W. DORABIALA AND M. W. JOHNSON
Thus, it is natural to define
Y
Nat (C=?, G) F un(C=C, G(C))
C2C
as the equalizer in Cat of two functors. The first functor
Y Y
F un(C=C, G(C)) ! F un(C=C, G(D))
C2C ':C!D2C
is induced by the collection of functors
G(')* : F un(C=C, G(C)) ! F un(C=C, G(D))'.
The second functor is defined similarly using the precomposition,
(C=')* : F un(C=D, G(D)) ! F un(C=C, G(D))'.
In other words, a natural transformation is a collection of functors
(C) : C=C ! G(C)
where the following diagram commutes for each morphism ' : C ! D in C:
(C)
C=C _____//_G(C)
C=' G(')
fflffl fflffl
C=D _(D)//_G(D).
Similarly, a morphism in the category Nat (C=?, G)consists of a natural transfo*
*r
mation in each factor which must be compatible in the sense that a certain cubi*
*cal
extension of this diagram commutes.
The nerve functor is a right adjoint, hence it preserves equalizers. Thus, ap*
*plying
nerve to Nat (C=?, G)yields an equalizer in the category of simplicial sets. Ta*
*king
the geometric realization does not preserve arbitrary products, but does preser*
*ve
equalizers. Hence, there is a natural isomorphism
N(Nat (C=?, G)) eq(N( ), N( )).
However, one must keep in mind that N( ) and N( ) no longer decompose as
the products of N(?) applied to the expected factors. We let ~ and ~ denote
the same constructions where the product is taken after the realization and the
components are of the form
N(F un(C=C, G(C))) and N(F un(C=C, G(D))).
Notice there is a natural map N( ) ! ~ given by the universal property of the
product in the definition of ~ and similarly for .
Now suppose there is another small category D together with a functor
F : D ! Nat (C=?, G).
Then taking nerves yields a natural map
N(F) : N(D) ! N(Nat (C=?, G))
and taking geometric realization gives a natural map
N(D) ! N(Nat (C=?, G))
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 13
One could avoid the finiteness assumption in the following result by working
throughout in a convenient category of topological spaces in the sense of Steen*
*rod
[15].
Lemma 5.1. Suppose C has the property that each overcategory C=C is finite. Then
there is a natural map
N(Nat (C=?, G)) ! holimCN(G).
Proof.To begin, recall that holimCN(G) is itself an equalizer of two maps
Y Y
Map (N(C=C), N(G(C))) ! Map(N(C=C), N(G(D))).
C2C ':C!D2C
These maps are constructed as with and , using the maps N(G('))* and
N(C=C)*.
Now, we will describe a collection of maps
N(F un(C=C, G(D))) ! Map (N(C=C), N(G(D)))
which are natural in the sense that they send the map ~ described above to the
map induced by the maps N(C=C)* and similarly for .
There is an evaluation functor
e : F un(C=C, G(D)) x C=C ! G(D)
whose nerve defines a morphism
N(e) : N(F un(C=C, G(D))) x N(C=C) ! N(G(D))
since the nerve functor commutes with products. Now, apply geometric realization
which commutes with finite products to yield a natural map
N(e) : N(F un(C=C, G(D))) x N(C=C) ! N(G(D))
which is adjoint to the required map. Notice the adjoint map exists because the
finiteness assumption implies N(C=C) is a compact Hausdorff space.
Since geometric realization does not commute with arbitrary products, we must
be careful that the maps N(e) described above induce a map from the equalizer
of the pair ~, ~ rather than N( ) or N( ). This gives
eq(~ , ~) ! holimCN(G).
Fortunately, there is still a natural map
eq(N( ), N( )) ! eq(~ , ~)
given by the universal property of the products involved in defining the target*
*. The
composition is the required natural map.
By Lemma 5.1, the composite displayed above yields a map
N(D) ! holimCN(G).
This is the map usually called a Thomason homotopy limit problem map.
The case of interest for us is when C = Simp (B)and G : Simp (B)! Cat is
defined by oe 7! w Retfdp1(oe). In particular, notice our assumption on B from
the introduction implies C satisfies the assumption in the statement of Lemma 5*
*.1.
Notice the natural group completion map N(G(oe)) ! A p1(oe) yields a natural
14 W. DORABIALA AND M. W. JOHNSON
` E '
map holimCN(G) ! # p as well. Precomposing with our Thomason homo
B ` '
E
topy limit problem map, this gives a map N(D) ! # p. For the category
B
D = w Retfd(p)and the functor w Retfd(p)! Nat (C=?,`G)discussed'in detail be
E
low, this will yield a map N(w Retfd(p)) ! # p whose target is an infinite
B
loop space by construction.`Thus,'we can extend over the natural group completi*
*on
E
map N(D) ! A # p to build the desired coassembly map
B
`E ' ` E '
ffi : A # p ! # p.
B B
In order to complete the definition of the coassembly map ffi it suffices to de*
*scribe the
relevant functor F : w Retfd(p)! Nat (C=?, G). We will describe this relationsh*
*ip
by saying ffi is the group completion of the Thomason homotopy limit problem map
associated to F.
We require two other general results in this context.
Lemma 5.2. (1) Suppose ' is a natural transformation from G to G0. Then
the induced maps make the following diagram commute.
N(Nat (C=?, G))___//N(Nat (C=?, G0))
 
 
fflffl fflffl
holimCN(G)________//_holimCN(G0)
(2) There is a pseudoproduct in Nat (?, !)which makes the following diagram
commute.
N(Nat (C1=?, G1)) x N(Nat (C2=?,_G2))//_N(Nat (C1=? x C2=?, G1 x G2))
 
 
fflffl fflffl
holimCN(G1) x holimN(G2)___________//holimN(G1) x N(G2)
1 C2 C1xC2
Lemma 5.3. Suppose F : D ! Nat (C=?, G)and F0 : D ! Nat (C=?, G)are
functors together with a natural transformation
_ : F ! F0.
Then the two Thomason homotopy limit problem maps
N(D) ! holimCN(G)
are naturally homotopic, as are their group completions.
Proof.The point here is that N(_) induces the necessary homotopy between the
two maps N(F) and N(F0) : N(D) ! N(Nat (C=?, G)). The remaining exten
sions by the composite
N(Nat (C=?, G)) ! eq(~ , ~) ! holimCN(G)
will then be homotopic as well, so the group completions will also be homotopic.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 15
Since Nat (C=?, G)is defined as an equalizer of and inQCat, defining a fu*
*nctor
into Nat (C=?, G)is equivalent to defining a functor into C2CF un(C=C, G(C)) *
*so
that the two composite functors
Y
D ! F un(C=C, G(D))'
':C!D2C
agree. As each of these target categories is a product, the universal propertie*
*s imply
it suffices to check on each projection. In other words, we should define a ser*
*ies of
functors
FC : D ! F un(C=C, G(C))
such that each diagram
FC
D _______________________//_F un(C=C, G(C))
FD  G(')*
fflffl fflffl
F un(C=D, G(D))______(C=')*____//_F un(C=C, G(D))'
commutes.
In our specific situation, this means we need to define functors
1
Foe: w Retfd(p)! F un(Simp (B)=oe, w Retfdp (oe)).
The functors we have in mind send X ø E to the functor
ffi ffi* 1
ffi 7! ioe*(iB ) (X) ø p (oe).
Functoriality of this assignment requires a natural weak equivalence
ffi ffi* ø ø *
ioe*(iB ) (X) ! ioe*(iB ) (X)
whenever ffi ø. However, uniqueness of inclusions implies
ffi* ø * i ffij*
iø iB (X) = iB (X)
and
ø ffi ffi
ioe* iø *= ioe*
so
ffi ffi* ø ffi ffi*ø *
ioe*(iB ) (X) = ioe*iø * iø iB (X).
Hence, iøoe*applied to the unit of adjunction gives the required natural weak
equivalence. Clearly, this is natural with respect to further inclusions in ei*
*ther
direction.
Notice that C=C in this case simply consists of the subcomplexes of oe, and t*
*he
functors C=' for ' : ø ! oe simply become the inclusion of the subcomplexes of ø
as subcomplexes of some larger simplex oe. Hence the associated functor
1 fd 1
G(') : w Retfdp (ø) ! w Ret p (oe)
should be the pushforward functor iøoe*.
16 W. DORABIALA AND M. W. JOHNSON
Now the required commutative diagrams
Fø fd 1
w Retfd(p)________________//_F un(Simp (B)=ø, w Ret p (ø) )

Foe 
fflffl 
F un(Simp (B)=oe, w Retfdp1(oe)) ((iøoe)*)*
XXXXX ø* 
X(Simp(B)=ioe)XXXXXX 
XXXXXXX 
XX++X fflffl
F un(Simp (B)=ø, w Retfdp1(oe))iøoe
are reduced to the statement that
ø i ffii ffij* j ffii ffij*
ioe* iø* iB (X) = ioe* iB (X).
Thus, our assignment gives the required functor
F : w Retfd(p)! Nat (C=?, G).
If we choose a different initial functor G, sending oe`to w Retfd,cyp1(oe), *
*the
E ' ` E '
same process yields the cyclic coassembly map fficy: Acy # p ! cy # p as the
B B
group completion of the Thomason homotopy limit problem map of a modification
of the functor F above. Similarly, one can alter G to send oe to w RetfdRp1(oe)
and notice the functor F described above also yields a functor
F : w RetfdR(p)! Nat (C=?, G)
in this case. Thus, we have another coassembly map
` E ' ` E '
ffiR : AR # p ! R # p .
B B
In order to define the Reidemeister torsion, we needed the existence of a can*
*onical
path fi from V(Ø (p)) to the basepoint in holimK (R). Using the machinery of t*
*his
section, it is straightforward to describe this path.
To begin, notice any object X of Retfd,cy(p)sitting in RetfdR(p)has the prope*
*rty
that the retraction is a weak equivalence by the definition of Retfd,cy(p). Hen*
*ce,
there is a natural map from X to E in`w RetfdR(p), which leads to a natural path
E '
from the point associated to X in AR # p to the basepoint.
B i j
Now suppose X represents an object in Nat C=?, w RetfdRp1(?)which comes
i j
from an object in Nat C=?, w Retfd,cyp1(?)under the "inclusion" functor.
i j
There is another object of Nat C=?, w RetfdRp1(?)which consists of the con
stant functors to p1(?) and serves as the äb sepoint". Once again, the combina*
*tion
of all of the relevant retractions will provide a natural morphism`from'X to p*
*1(?).
E
Since we will choose the image of p1(?) as a point of R # p as the basepoin*
*t,
` ' B
E
this gives a natural path fi0 in R # p from the point associated to X to the
B
basepoint.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 17
Our choice for X will`be the'image`of'E t E under the functor which builds
E E
the coassembly map Acy # p ! cy # p , which is only possible if the bundle
B B ` '
E
involved is an acyclic flat bundle . Then fi0gives a natural path in R # p f*
*rom
B
the image of Øcy(p) to the basepoint, since Øcy(p) was defined as the image of
ØAcy(p)(which itself corresponds to the retractive space E t E) under the cyclic
coassembly map. We define our natural path fi in holimK (R) from V(Ø (p)) to
the basepoint as V applied to the the path fi0.
Alternatively, we could define fi by a construction quite similar to that of *
*fi0,
using Nat (C=?, wCh*(R) )which corresponds to the holim of the constant functor
K (R). Since the postcomposition by an exact functor preserves the zero map, the
natural map from X to the äb sepoint" will be sent to the (unique) natural map
from the image of X to the constant functor on the zero chain complex, which se*
*rves
as the basepoint in Ch*(R) . Taking group completions then says this definition
of fi agrees with that given above. The interested reader may also see the path
given by [5], Observation 6.4, along with Propositions 6.6 and 6.7 is yet anoth*
*er
description of fi.
6. Products and Natural Maps
The purpose of this section is to define the multiplications referred to prev*
*iously
in full detail.
First, recall the definition of ~A as induced by the external smash product of
retractive spaces defined in Section 3. In fact, this defines a biexact functo*
*r, hence
a natural map
`E1 ' ` E2 ' ` E1xE2 '
~A : A # p1x A # p2! A # p1x p2
B1 B2 B1xB2
which clearly descends to the smash product by construction to give
`E1 ' ` E2 ' ` E1xE2 '
~A : A # p1^ A # p2! A # p1x p2.
B1 B2 B1xB2
We would like to use the external smash product again to define
`E1 ' ` E2 ' ` E1xE2 '
~Acy: A # p1^ Acy # p2 ! Acy # p1x p2 .
B1 B2 B1xB2
However, it is not yet clear that X^E1xE2Y lies in the subcategory Retfd,cy(p1 *
*x p2).
In other words, the construction yields a map
` E1 ' ` E2 ' ` E1xE2 '
A # p1 ^ Acy # p2 ! A # p1x p2
B1 B2 B1xB2
which currently has the wrong target to be our ~Acy. One should also keep in mi*
*nd
that we need flat bundles OE1 : V1 ! E1 and OE2 : V2 ! E2 in order to even defi*
*ne
the spaces involved in ~Acy. To deal with this problem we first need a technic*
*al
result.
When describing relative chain complexes with local coefficients, the symbol
C*(A, B; V )will indicate that V is a flat bundle over B, A is retractive over*
* B
and the bundle in question over A is the pullback of V over the retraction, whi*
*ch
remains a flat bundle. Also, given OE1 : V1 ! E1 and OE2 : V2 ! E2, the symbol
18 W. DORABIALA AND M. W. JOHNSON
V1^ V2 will denote the tensor product of the two bundles over E1 x E2 given by
pulling back each Vi over the relevant projection map.
Lemma 6.1. Suppose X 2 RetfdR1(p1). Then the functor
X^E1xE2? : RetfdR2(p2)! RetfdR1 AR2(p1 x p2)
preserves weak equivalences.
Proof.Suppose f : Y ! Z is a weak equivalence in RetfdR2(p2). This means f
induces a quasiisomorphism
C*(Y, E2, V2)! C*(Z, E2, V2)
with local coefficients. Thus, tensoring with C*(X, E1, V1)yields another quas*
*i
isomorphism
C*(X, E1, V1) C*(Y, E2, V2)! C*(X, E1, V1) C*(Z, E2, V2)
since V1 is a flat bundle. However, the relative EilenbergZilber Theorem with *
*local
coefficients (exercise 8 on page 282 of [14] or [6]) then implies the existence*
* of a
quasiisomorphism
* *
C* X x Y, X x E2 [ E1 x Y, r1(V1^ V2)! C* X x Z, X x E2 [ E1 x Z, r2(V1^ V2)
with r1 : X x E2 [ E1 x Y ! E1 x E2 and similarly for r2. Using the relative
MayerVietoris sequence with local coefficients and compact supports (see page
412, exercise 8 of [3]) together with the fact that
X x E2 [ E1 x Y ! X x Y
is a cofibration, one concludes that there is a natural quasiisomorphism
*
C* X x Y, X x E2 [ E1 x Y, r1(V1^ V2) C* X ^E1xE2 Y, E1 x E2, V1^ V2
and similarly for Z. Thus, transitivity implies the map f induces a quasiisomo*
*rphism
C* X ^E1xE2 Y, E1 x E2, V1^ V2! C* X ^E1xE2 Z, E1 x E2, V1^ V2
as well.
Lemma 6.2. Suppose X 2 Retfd(p1)and Y 2 Retfd,cy(p2). Then
X ^E1xE2 Y 2 Retfd,cy(p1 x p2).
Proof.The statement that Y 2 Retfd,cy(p2)is equivalent to saying Y 2 RetfdR2(p2)
with Y ! E2 a weak equivalence in this structure. Then Lemma 6.1 implies
X ^E1xE2 Y ! X ^E1xE2 E2 E1 x E2
is a weak equivalence in RetfdR1,2(p1 x,p2)or equivalently that
X ^E1xE2 Y 2 Retfd,cy(p1 x p2).
Lemma 6.3. (1) There is a natural external multiplication
`E1 ' ` E2 ' ` E1xE2 '
~AR : AR1 # p1^ AR2 # p2! AR1 AR2 # p1x p2
B1 B2 B1xB2
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 19
(2) The following diagram commutes.
` E1 ' ` E2 ' ` E1xE2 '
A # p1^ A # p2 __~A___//_A # p1x p2
B1 B2 B1xB2
 
 
` ' fflffl` ' ` fflffl'
E1 E2 E1xE2
AR1 # p1^ AR2 # p2 _~__//_AR # p1x p2
B1 B2 AR B1xB2
Proof.The multiplication ~AR is induced by the same external smash product
^E1xE2 : RetfdR1(p1)x RetfdR2(p2)! RetfdR(p1 x p2).
With the homology chain equivalences as weak equivalences, we must show this
remains a biexact functor. Since we have not changed any other portion of the
Waldhausen structure, we only need to check that fixing an element X 2 RetfdR1(*
*p1),
X^E1xE2? preserves the new class of weak equivalences (and similarly for ?^E1xE2
Y ). However, this follows immediately from Lemma 6.1.
Once we know the same functor induces the multiplication, the commutativity of
the diagram follows from the fact that the vertical maps are induced by the ide*
*ntity
functor at the level of retractive spaces.
The product
` E1 ' ` E2 ' ` E1xE2 '
_ : # p1^ # p2 ! # p1x p2
B1 B2 B1xB2
is defined as the following composition:
1 1
holim A p1 (oe1)^ holim A p2 (oe2)
oe12Simp(B1) oe22Simp(B2)


fflffl
holim A p11(oe1)^ A p12(oe2)
oe1xoe22Simp(B1xB2)
(~A)*
fflffl
holim A (p1 x p2)1(oe1 x oe2).
oe1xoe22Simp(B1xB2)
where the first map comes from the interaction of holim and products.
Suppose p1 is a perfect fibration with flat bundle V1 and p2 is a perfect fib*
*ration
with acyclic flat bundle V2. We can now describe
` E1 ' ` E2 ' ` E1xE2 '
_cy: # p1^ cy # p2 ! cy # p1x p2
B1 B2 B1xB2
20 W. DORABIALA AND M. W. JOHNSON
similarly as the following composition.
1 1
holim A p1 (oe1)^ holim Acy p2 (oe2)
oe12Simp(B1) oe22Simp(B2)


fflffl
holim A p11(oe1)^ Acy p12(oe2)
oe1xoe22Simp(B1xB2)
(~Acy)*
fflffl
holim Acy (p1 x p2)1(oe1 x oe2)
oe1xoe22Simp(B1xB2)
` E ' `E '
Next is the definition of ffl : cy # p ! Wh RB # p. We begin with a lemma
B B
from the nonparametrized case.
Lemma 6.4. The linearization map factors through A R(E) and its generalized
linearization map. That is, the following diagram commutes.
A (E) ____//_AR(E)
II
III ~V
~V III$$Ifflffl
K (R)
See [4] for details.
Now, consider the following diagram which commutes as a result of Lemma 6.4.
=
holimA p1(oe)_____//_holimAp1(oe)
 
  V
fflffl fflffl
holimAR (p1(oe))_V___//_holimK(R)
As mentioned previously, Acy p1(oe) is homotopy equivalent to the homotopy
fiber of
1 1
A p (oe)! AR p (oe).
` E*
* '
Since homotopy fibers commute with homotopy inverse limits, this implies cy #*
* p
B
is homotopy equivalent to the homotopy fiber`of the'left vertical map in the pr*
*e
E
vious diagram. However, by definition Wh RB # p is the homotopy fiber of the
B
right vertical in the previous`diagram.'Thus,`commutativity'of the diagram above
E E
induces a map ffl : cy # p ! Wh RB # p.
B B
We now need a technical lemma.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 21
Lemma 6.5. (1) Given a diagram in T op
f
T ^ W _____//Y
1^p q
fflffl fflffl
T ^ X _g___//Z
together with a pointed homotopy H from qf to g(1 ^ p) there is a natural
map 'H making the following extension commute:
T ^ hofiberp'H__//hofiberq
 
 
fflffl fflffl
T ^ W __________//Y.
(2) Suppose one has a diagram in T op
T ^ M _e___//N
1^i j
fflfflf fflffl
T ^ W _____//Y
1^p q
fflffl fflffl
T ^ X _g___//Z
where je = f(1 ^ i) and H is a pointed homotopy as above. Then one has
a homotopy commutative diagram
T ^ hofiberi______//_hofiberj
 
 
fflffl fflffl
T ^ hofiber(pi)'H//_hofiber(qj).
Proof.Recall, a point in hofiberp consists of a pair (ff, w) where w 2 W and
ff : I ! X is a path from p(w) to the basepoint. Define 'H (t, ff, w) as the p*
*air
(g(t, ff)H(t, p(w), ?), f(t, w)), where the first component means the concatena*
*tion
of these two paths in Z.
For the second statement, define the map
T ^ hofiberi ! hofiberj
by a simpler variation of 'H , namely (t, ff, m) is mapped to (f(t, ff), e(t, m*
*)). (No
tice this is homotopic to the variation of 'H using a constant homotopy.) In or*
*der
to establish the homotopy commutativity of the diagram, it suffices to show tha*
*t the
paths qf(t, ff) and g(t, p(ff))H(t, i(m), ?) are homotopic relative to the endp*
*oints.
If we use s to denote the time variables for ff and H, and r as a time variable*
* for
our homotopy, the relevant formula is:
( 1
_;
(3) K(s, r) = H(t, ff(s  rs), 2rs),if s 2 1
H(t, ff(s  r + rs), r),if s _2.
22 W. DORABIALA AND M. W. JOHNSON
The multiplication
` E1 ' ` E2 ' ` E1xE2 '
: # p1^ Wh R2B2# p2 ! Wh R1BAR21xB2 # p1x p2
B1 B2 B1xB2
is defined by applying Lemma 6.5 to the homotopy commutative diagram given by
the following proposition.
Proposition 6.6. (1)There is a natural commutative diagram as below:
` E1 ' ` E2 ' ` E1xE2 '
# p1 ^ # p2 _________//_ # p1x p2
B1 B2 B1xB2
 
 
` ' fflffl` ' ` fflffl '
E1 E2 E1xE2
# p1 ^ R2 # p2_____//R1 AR2 # p1x p2.
B1 B2 B1xB2
(2) There is a natural choice of homotopy between the two compositions in the
following diagram:
` E1 ' ` E2 ' ` E1xE2 '
# p1 ^ R2 # p2_____//R1 AR2 # p1x p2
B1 B2 B1xB2
1^ V  V
` ' fflffl 
E1 fflffl
# p1 ^ holimK (R2)______//_holimK(R1 A R2).
B1
Proof.We begin with the diagram
1 1
A p1 (oe1)^ A p2 (oe2)_________//A(p1 x p2)1(oe1 x oe2)
 
 
fflffl fflffl
AR1(p11(oe1)) ^ AR2(p12(oe2))//_AR1 AR2((p1 x p2)1(oe1 x oe2))
which commutes by Lemma 6.3. Taking holimover the category Simp (B1 x B2)
then implies the following diagram commutes
1 1
holim A p1 (oe1)^ A p2 (oe2)
WW
 WWWWWWW
 WWWWWWW
 WWWWW++
 holimA (p1 x p2)1(oe1 x oe2)

 
 
fflffl 
holim AR1(p11(oe1)) ^ AR2(p12(oe2)) 
WWWWW 
WWWWWWW 
WWWWWW 
WWW++ fflffl
holimAR1 AR2 ((p1 x p2)1(oe1 x oe2)).
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 23
Of course, one also has a commutative diagram associated to holim of products
1
holim A p1 (oe1)
Simp(B1) 1 1
^ ______//_holimSimp(B1xB2)Ap1^(oe1)Ap2 (oe2)
holim A p12(oe2)
Simp(B2) 
 
 
fflffl 1 
Sholimimp(B1)AR1(p1 (oe1)) fflffl
1 1
^ _____//Sholimimp(B1xB2)AR1(p1 (oe1)) ^ AR2(p2 (oe2))
holim AR2(p12(oe2))
Simp(B2)
which taken together yields a commutative diagram which we reinterpret as the
following
` E1 ' ` E2 ' ` E1xE2 '
# p1^ # p2 ____________________________// # p1x p2
B1 B2 B1xB2
 
 
` ' fflffl`' ` fflffl '
E1 E2 E1xE2
# p1^ R2 # p2 R1 AR2 # p1x p2.
B1 B2P 77 B1xB2
PPPP nnnnn
PPPP nnnn
PP''P`' ` nnn'
E1 E2
R 1 # p1 ^ R2 # p2
B1 B2
For the second diagram, begin with the following diagram (coming from the
nonparametrized case) which commutes up to a natural homotopy
1 1 ~A
A p1 (oe1)^ AR2(p2 (oe2))___//_AR1 AR2((p1 x p2)1(oe1 x oe2))
~V1^~R2V2 ~R1VAR21V2
fflffl fflffl
K(R1) ^ K(R2) _______jK________//_K(R1 A R2)
where the bottom map is Loday's pairing on Ktheory [13]. The natural homotopy
in this case comes from the EilenbergZilber map on chain complexes. Now take
holimover the category Simp (B1 x B2)and proceed as above, keeping in mind
that the upper left corner in each diagram is not symmetric. The statement about
naturality of the homotopy then follows from the naturality of the EilenbergZi*
*lber
map.
We can now prove the second statement of Proposition 2.2.
24 W. DORABIALA AND M. W. JOHNSON
Proposition 6.7. The diagram
` E1 ' ` E2 ' _cy ` E1xE2 '
# p1 x cy # p2 _________//_ cy # p1x p2
B1 B2 B1xB2
1xffl2 ffl1,2
` ' fflffl` ' ` fflffl '
E1 E2 R R E1xE2
# p1 x Wh R2B2# p2 _____//Wh1BA12xB2 # p1x p2.
B1 B2 B1xB2
commutes up to a natural homotopy.
Proof.The homotopy commutative square comes from applying Lemma 6.5 to the
diagrams provided by Proposition 6.6.
We can now give a proof of Proposition 2.1.
Proof of Proposition 2.1.Consider the diagram
1 = 1
holimA p2 (oe2)______//holimAp2 (oe2)
 
 V
fflffl fflffl
holimAR2(p12(oe2))_V___//holimK(R2)
` *
*E2 '
involved in defining ffl. By definition, øR2 (p2)is a lift of Ø (p2)to Wh R2B2*
* # p2
*
*B2
associated to a specific choice of path fi from V(Ø (p2)) to the basepoint, di*
*scussed
at the end of section 5. It is important to keep in mind this path arises as th*
*e image
of a similar path to the basepoint fi0 in each A R2(p12(oe2)), given by effect*
*ively
the same construction.` Since'the`pair'(Øcy(p), fi0) is the image of Øcy(p)in t*
*he
E E
homotopy fiber F of # p ! R # p , the composite
B B
` E ' ` E '
# p ! F ! R # p
B B
will send Øcy(p)to the point in the Whitehead space corresponding to the pair
(Ø (p), fi). However, this point was our definition of the Reidemeister torsion*
* class
øR2 (p2).
7.The coassembly map is Multiplicative
The purpose of this section is to prove that the diagram
` E1 ' ` E2 ' ` '
cy cy E1xE2
(4) A # p1^ Acy # p2 _~A__//A # p1x p2
B1 B2 B1xB2
ffi1xfficy2 fficy1,2
` ' fflffl`' ` fflffl'
E1 E2 E1xE2
# p1^ cy # p2 __cy_// cy # p1x p2.
B1 B2 _ B1xB2
is commutative up to a natural homotopy. We begin with the proof of Lemma 2.5.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 25
Proof of Lemma 2.5.Part 1 follows from the fact that the multiplications ~A and
~Acyare both defined by the same biexact functor at the level of retractive sp*
*aces.
To see part 2, recall that Retfd,cy(p)! Retfd(p)is the inclusion of a sub
Waldhausen category. Hence,
wS. Retfd,cy(p)! wS. Retfd(p)
is the inclusion of a subsimplicial category. This implies
` E ' `E '
Acy # p = wS.Retfd,cy(p) ! wS.Retfd(p) = A # p
B B
is an injection.
Finally, part 3 follows from the nonparametrized analogue`of'part`2 and'the
E E
standard model for the homotopy inverse limit. Since Acy # p ! A # p is an
B B
injection, it should be clear that for each oe 2 Simp(B) one has
1 1
Map ( Simp(B) =oe, Acy p (oe)) ! Map ( Simp(B) =oe, A p (oe))
an injection as a postcomposition by an injection. This implies the product of
such maps is also an injection. However, the standard model for the homotopy
inverse limit is the appropriate subspace of such a product and the restriction*
* of
an injection remains an injection.
As a consequence of Lemma 2.5, it should be clear that the first portion of
Proposition 2.2 is a corollary of Theorem 2.3, where we dealt with the product
rather than smash product versions of ~A and _ for technical reasons.
We begin working toward this proof with another technical lemma.
Lemma 7.1. The product map
`E1 ' ` E2 '
w Retfd(p1) x w Retfd(p2) ! A # p1x A # p2
B1 B2
retains the group completion property.
Proof.The definition of the S. construction implies that for Waldhausen categor*
*ies
C and D one has a natural isomorphism
S.(C x D) S.(C) x S.(D).
Of course, this assumes the Waldhausen structure on C x D is that coming from
the product.
Clearly, the inclusion
i j i j
w Retfd(p1)x Retfd(p2) ! wS. Retfd(p1)x Retfd(p2)
has the group completion property by [16]. However, the previous paragraph leads
us to conclude the target is isomorphic to
`E1 ' ` E2 '
wS.(Retfd(p1)) x wS.(Retfd(p2)) A # p1x A # p2.
B1 B2
On the other hand, we also have an isomorphism
i j
w Retfd(p1) x w Retfd(p2) w Retfd(p1)x Retfd(p2).
26 W. DORABIALA AND M. W. JOHNSON
Thus, it suffices to notice that the composite of these maps
`E1 ' ` E2 '
w Retfd(p1) x w Retfd(p2) ! A # p1x A # p2
B1 B2
is the product of the maps
i j `Ei '
w Retfd(pi) ! wS. Retfd(pi) = A # pi.
Bi
There are now two functors
i j
w Retfd(p1)x w Retfd(p2)! Nat Simp(B1 x B2)=?, w Retfd((p1 x p2)1(?).
The functor F1 comes from following the external smash product (which induces
the external product in parametrized Atheory)
^E1xE2 : w Retfd(p1)x w Retfd(p2)! w Retfd(p1 x p2)
with the functor
i j
w Retfd(p1 x p2)! Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)1(?)
described after Lemma 5.3.
The second functor F2 comes from first taking the product of the functors
i j
w Retfd(pi)! Nat Simp(Bi)=?, w Retfdp1i(?)
as above, followed by a functor
i j
Nat Simp(B1)=?, w Retfdp11(?)
i x j
Nat Simp(B2)=?, w Retfdp12(?)


i fflffl j
Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)1(?)
given by the pseudoproduct in Nat (?, !)followed by the functor induced by con
sidering the external smash product as a natural transformation.
Lemma 7.2. (1) The composite ffi1,2(~A) from Theorem 2.3 is the group com
pletion of the Thomason homotopy limit problem map associated to F1.
(2) The composite _ (ffi1x ffi2) from Theorem 2.3 is the group completion of*
* the
Thomason homotopy limit problem map associated to F2.
Proof.For the first claim, recall that ffi1,2is the group completion of the sec*
*ond
functor in the definition of F1 and ~A is the group completion of the external
smash product.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 27
To verify the second claim, it suffices by the uniqueness of group completion*
*s to
establish commutativity of the following diagram
` E1 ' ` E2 '
w Retfd(p1) x w Retfd(p2)__________//A # p1 x A # p2
 B1 B2
 
 ffi1xffi2
i fflffl j 
Nat Simp(B1)=?, w Retfdp11?  ` ' fflffl`'
E1 E2
i x j___________//_ # p1 x # p2
Nat Simp(B2)=?, w Retfdp12?  B1 B2
 
 _
 
i fflffl j ` E1xEfflffl2'
Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)1?_______// # p1x p2.
B1xB2
The commutativity of the top square follows from the fact that group completions
commute with products by naturality. Thus, it remains only to show the bottom
square commutes. However, by the construction of F2, this bottom square itself
factors as the group completion of a pair of squares,
i j
Nat Simp(B1)=?, w Retfdp11?  ` E1 ' ` E2 '
i x j___________// # p1 x # p2
Nat Simp(B2)=?, w Retfdp12?  B1  B2
 
 
i fflffl j 
fd 1 fd 1 
Nat Simp(B1 x B2)=?, w Ret p1 ? x w Ret p2 ?  
UUUU 
UUUUU 
UUUUU 
UUUUUU** fflffl
holim A p11(oe1)
Simp(B1xB2)
x
A p12(oe2)
and
i j
Nat Simp(B1 x B2)=?, w Retfdp11?x w Retfdp12? 
 U
 UUUUU
 UUUUU
 UUUUU
 UUUU**holim A p1(oe1)
 Simp(B1xB2) 1
 x
 1
 A p2 (oe2)

 
 
i fflffl j ` E1xfflfflE2'
Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)1?_______// # p1x p2.
B1xB2
28 W. DORABIALA AND M. W. JOHNSON
each of which commutes by Lemma 5.2. The first comes from the pseudoproduct
in Nat (?, !)and the second from the map induced by the external smash product
considered as a natural transformation.
Proposition 7.3. There is a functor
i j
F0 : w Retfd(p1)xw Retfd(p2)! Nat Simp(B1 x B2)=?, w Retfd(p1 x p2)1(?)
together with natural transformations (_n) : F0 ! Fn for n = 1,2.
Proof.We return to the notation of section 5 in order to describe F1 and F2 alo*
*ng
with their effect on an object (X1iø E1,jX2*ø E2) of w Retfd(p1)xiwjRetfd(p2).
To simplify notation, let (in)* = ioenBnand let (jn)* = iffinoen. First, in b*
*uilding
*
F1 one sends this pair to the functor
(ffi1, ffi2) 7! (j1 x j2)*(i1 x i2)*(X1 ^E1xE2 X2).
Next, in building F2 one sends the pair to the functor
(ffi1, ffi2) 7! (j1)*(i1)*(X1) ^p11(oe1)xp12(oe2)(j2)*(i2)*(X2).
The new functor F0 is defined by
i j
(ffi1, ffi2) 7! (j1 x j2)* (i1)*(X1) ^p11(ffi1)xp12(ffi2)(i2)*(X2).
The natural transformation _1 is built from the natural weak equivalences which
arise by applying (j1xj2)* to the natural weak equivalence of type (1) in Lemma*
* 3.4.
Similarly, the natural transformation _2 is built from the natural weak equival*
*ences
of type (2) in Lemma 3.4.
This suffices to allow us to prove Theorem 2.3.
Proof of Theorem 2.3.Lemma 5.3 together with Proposition 7.3 imply the Thoma
son homotopy limit problem maps associated to F1 and F2 are naturally homo
topic (via the Thomason homotopy limit problem map associated to F0). However,
Lemma 7.2 then implies
` E1 ' ` E2 ' ` E1xE2 '
A # p1 x A # p2__~A_//A # p1x p2
B1 B2 B1xB2
ffi1xffi2 ffi1,2
` ' fflffl`' ` fflffl'
E1 E2 E1xE2
# p1 x # p2 _____// # p1x p2.
B1 B2 _ B1xB2
commutes up to a natural homotopy.
8.Open Questions
There are two main directions (aside from the general goal mentioned in the
introduction) in which we would like to proceed in the future. The first would *
*be
to give a definition of smooth parametrized torsion after [5]. This would lead *
*to an
appropriate definition of higher Reidemeister torsion as in [8]. Specifically, *
*Theorem
6.6.1 of [8] and possibly conjecture 6.6.7 would follow from the generalization*
* of
Theorem 1.4 to higher torsion.
PRODUCT THEOREM FOR PARAMETRIZED REIDEMEISTER TORSION 29
To see Theorem 6.6.1 of [8], consider the product of a perfect fibration p1 w*
*ith
the constant map p2 : N ! * for a manifold N. Then Theorem 1.4 gives
(Ø (p2), øR1 (p1)) ' øR1 AR2 (p1 x p2).
However, from [5] we know Ø (p2)= Ø (N), the standard Euler characteristic of t*
*he
manifold N.
The second logical direction is to pursue a definition of Reidemeister torsion
which does not require the acyclicity assumption. A careful analysis of the res*
*ults
in this paper suggest the effective role of the acyclicity assumption is to est*
*ablish
E t E as a multiplicatively natural element of Retfd,cy(p). Without the acycli*
*c
ity assumption, one might still hope to construct such a multiplicatively natur*
*al
element X(p) in Retfd,cy(p). In that case, the element X(p) could play the role
of X in the discussion at the end of Section 5, giving an analog of the path fi*
*(p)
associated to X(p). The pair (X(p), fi(p)) would then be a natural choice of a *
*point
in the Whitehead space, hence a possible notion of Reidemeister torsion. With t*
*his
definition, the proofs of this article should remain effective and yield an app*
*ropriate
product formula without the acyclicity assumption.
Since the identity functor Retfd(p)! RetfdR(p)induces a localization functor,
one way to try to make a choice for X(p) which would be natural comes from an
attempt to define a functor Retfd(p)! Retfd,cy(p)analogous to the construction
of the kernel of the localization map in [2]. Unfortunately, it seems optimisti*
*c to
suggest such an element would be multiplicative as well, a requirement for any
product theorem.
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Institute of Mathematics, Szczecin University, ul. Wielkopolska 15, 70451 Sz*
*czecin,
Poland
Current address: Department of Mathematics, Penn State Altoona, Altoona, PA 1*
*66013760
Email address: wud2@psu.edu
Department of Mathematics, Penn State Altoona, Altoona, PA 166013760
Email address: mwj3@psu.edu