THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY
OF SPACES.
WOJCIECH DORABIALA
Abstract.The goal of this paper is to show that if a smooth fiber bundle
has a compact Lie group as structure group, then the transfer map for the
algebraic K-theory of spaces satisfies analogs of the Mackey Double cose*
*t for-
mula and Feshbach's sum formula. We also prove a "cut and paste" formula
for parametrized Reidemeister torsion.
1.Introduction
Let p : E ! B be a fibration with fibers finitely dominated by finite CW
complexes. For these fibrations one can consider p* : A(B) ! A(E) the trans-
fer on Waldhausen's algebraic K-theory of spaces. The geometric transfer for h-
cobordisms and concordance are important tools in geometric topology. It is well
known that h-cobordisms over a manifold M are classified by a certain quotient
group of K1(Z[ss1(M)]). More generally the stabilized concordance space of a ma*
*n-
ifold M has an algebraic description in terms of Waldhausen's A(M), the algebra*
*ic
K-theory of spaces. The geometric transfers are induced by algebraic transfers,
p* : A(B) ! A(E). The transfer map for smooth pseudo-isotopies has been used
to study Diff(M) for compact manifolds, see [4]. Recall Waldhausen has proved
that there is a natural decomposition
A(X) = Q(X+ ) x W hd(X):
Dwyer, Weiss and Williams [8] have proved that p*|Q(X+) is the Becker-Gottlieb-
Dold transfer for a smooth fiber bundle p.
This naturally leads to the question: what is p*|Whd(B) when p is a smooth fi*
*ber
bundle with a compact Lie group G as structure group.
The recent paper of B.Williams [24] shows that the transfer for A-theory is
determined by an element O(p) in ss0A(E ! B). We call O(p) the parametrized
Euler characteristic for p. Here A(E ! B), roughly speaking, can be viewed as t*
*he
A-theory of B-parametrized families of finitely dominated retractive spaces over
the fibers of p. The key property of the classical Euler characteristic is tha*
*t if
X = X1 [X0 X2 where Xi is a finite sub-CW complex of X for i = 0; 1; 2 then
O(X) = O(X1) + O(X2) - O(X0). We have shown an analogous result for the
parametrized Euler characteristic.
Theorem. (Parametrized Euler Characteristic Additivity Theorem) Let
p : E ! B be a fibration such that p = p1 [p0p2 where pi: Ei! B are fibrations,
____________
Date: 1.16.00.
1991 Mathematics Subject Classification. Primary: 55, Secondary: 16.
Key words and phrases. A-theory, transfer.
1
2 WOJCIECH DORABIALA
E = E1 [E0 E2, and E0 ! Ei is a cofibration for i = 1; 2. Assume the fibers of *
*pi
for i = 0; 1; 2 are finitely dominated.
Then
O(p) = i1?O(p1) + i2?O(p2) - i0?O(p0)
where ij*: ss0A(Ej ! B) ! ss0A(E ! B) are the pushforward maps induce by the
obvious inclusions.
As a consequence of the above theorem we get a transfer additivity theorem.
Theorem (Transfer Additivity Theorem) Let p : E ! B be a fibration such
that p = p1 [p0p2 where pi: Ei! B are fibrations, E = E1 [E0 E2, and E0 ! E1
is a cofibration. Assume the fibers of pi for i = 0; 1; 2 are finitely dominate*
*d. Then
the A-theory transfer p* : A(B) ! A(E) can be written as a sum
p* = i1*p1*+ i2*p2*- i0*p0*
where ij*: A(Ej) ! A(E) are the maps induce by the obvious inclusions.
If X is path connected, and i = 0 or 1, then ssiA(X) = Ki(Z[ss1(X)]). Thus for
fibrations with fibers finitely dominated one has transfer maps
p! : Ki(Z[ss1(B)]) ! Ki(Z[ss1(E)]) for algebraic K-theory when i = 0; 1. It is
interesting that for i = 0; 1 the transfer map for A-theory agrees with the tra*
*nsfer for
algebraic K-theory studied by W.Luck [16], [17]. Results of the additivity theo*
*rem
are new even in dimensions 0 and 1.
Suppose G is a compact Lie group, E ! B a bundle with structure group G and
fiber X a finite G-CW-complex. Let P ! B be the associated G-bundle. For any
G-map, x : G=H ! X let jx be the map P xG G=H ! P xG X = E.
The following theorem shows that the transfer p* : A(B) ! A(E) can be ex-
pressed as a sum of transfers p*G=H: A(B) ! A(P xG G=H).
Theorem (Sum Theorem)Let G be a compact Lie group. Let E ! B be a bundle
with fiber a finite G-CW complex X and structure group G. Then the transfer
p* : A(B) ! A(E)
can be written as the sum
X
p* = mxA(jx)p*G=H
where mx = integer defined on page 11 and we sum over isomorphism classes of
the component category (see page 10).
Notice this is analogous to Feshbach's sum formula (see [10]). We also get an
analogue to his Mackey Double coset formula. and Feshbach's sum formula. Let G
be a compact Lie group with H and K arbitrary closed subgroups of G. Then the
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 3
double coset formula expresses the composition
A(BK) J A(BH)99
JJp*(K;G)JJp*(H;G)tttt
JJ ttt
J%%J tt
A(BG)
as a finite sum of compositions of the form
A(B(Kg7\7H))O
p*(K\Hg;K)oooo OOp*(K\Hg;Hg)OO
oooo OOO
ooo OOO'' C(g-1)
A(BK) A(BHg) _____//A(BH)
Corollary. (Double Coset Formula) Let G be a compact Lie group and H, K
be closed subgroups of G. Then
X
p*(H;G)O p*(K;G)= nxC(g-1) O p*(K\Hg;Hg)O p*(K\Hg;K)
where nx is integer defined on page 11 and we sum over isomorphisms classes of
objects [x : K=(K \ Hg) ! G=H] in the component category (see page 10).
In section 6 we show how the A-theory double coset formula is different from Fe*
*s-
hbach's double coset formula for the homotopy transfer.
Let p : E ! B be a fibration with finitely dominated fibers Fb, and R any rin*
*g.
Let V be a bundle of f.g. projective R-modules on E. Suppose also that for each
b 2 B, the homology groups Hi(Fb; V |) vanish for all i. For such a fibration p*
* and
bundle V ; Dwyer,Weiss and Williams have constructed the homotopy parametrized
Reidemeister torsion (see section 7 for definition) as a map
ohV: B ! hB(E):
The next theorems give additivity for homotopy and smooth Reidemeister torsion.
Theorem (Homotopy Reidemeister Torsion Additivity Theorem) Let p :
E ! B be a fibration such that p = p1 [p0p2 where pi : Ei ! B are fibrations,
E = E1 [E0 E2, and E0 ! E1 is a cofibration. Let R be a ring and V a bundle of
f.g. projective left R-modules on E. Suppose that Hi(Ebi; V |Ei) = 0 for i = 0;*
* 1; 2
and all b 2 B. Then
ohV(p) = i1ohV(p1) + i2ohV(p2) - i0ohV(p0)
where ohV(?) is the homotopy Reidemeister torsion.
Theorem (Smooth Reidemeister Torsion Additivity Theorem) Let p : E !
B be a bundle of compact smooth manifolds. Let p be such that p = p1[p0p2 where
pi : Ei ! B are bundle of smooth manifolds, E = E1 [E0 E2, and E0 ! E1 is a
cofibration. Let R be a ring and V a bundle of f.g. projective left R-modules o*
*n E.
Suppose that Hi(Ebi; V |Ei) = 0 for i = 0; 1; 2 and all b 2 B. Then
4 WOJCIECH DORABIALA
odV(p) = i1odV(p1) + i2odV(p2) - i0odV(p0)
where odV(?) is the smooth Reidemeister torsion.
The author would like to thank Bruce Williams for calling our attention to th*
*is
problem and numerous helpful conversations.
2.Bivariant A-theory of spaces
In this section we present a quick review of the definition of bivariant A-th*
*eory
and how the transfer map for A theory is determined by an element in
ss0(A(E ! B)). Let <(X) be the the category of retractive spaces over the topo-
logical space X. Thus an object in <(X) is a diagram of topological spaces W AE*
* X
such that r O s = idX and s is a cofibration in the category of topological spa*
*ces. A
morphism in <(X) is a continuous maps over and relative to X. A morphism is a
cofibration in <(X) if the underlying map of spaces is a cofibration. A morphism
is a weak equivalence if the underlying map of spaces is a homotopy equivalence.
Let H (x)=W[x]H); where XH (x) is the component of XH
which contains the point x(1H), where W[x]H is the subgroup of W H acting on
XH (x), and where X>H (x) = XH (x) \ X>H .
Thus
X
OGuniv([x]) = (-1)pdimQ (Hp(XH (x)=W[x]H; X>H (x)=W[x]H; Q)):
p>0
This can also be expressed by counting cells. If #p([x]) is the number of equiv*
*ariant
p-dimensional cells of orbit type G=H which meet the component XH (x), then
X
(6) OGuniv(X)([x]) = (-1)p#p([x])
p>0
Notice
X
(7) OGuniv(X) = mxxOG (G=H)
where
(8) mx = OGuniv(X)[x]:
12 WOJCIECH DORABIALA
Corollary 1.For a G-space X of the homotopy type of a finite G CW-complex
there is a unique natural map
: UG (X) ! AG (X)
which takes the equivariant Euler characteristic OGunivto the equivariant A-the*
*ory
Euler characteristic OGA.
Proof.It suffices to show that (AG ; OGA) is a functorial additive invariant. *
*Since
homotopy invariance and normalization conditions are clearly satisfy by (AG ; O*
*GA)
we will only comment on additivity.
Let
g
X0B____//_X2
BB |
i|| BBk0BBk2|
fflffl|!!Bfflffl|k1
X1 _____//X
be a push-out in the category of G-spaces of the homotopy type of a finite G CW-
complexes where i is a cofibration. The additivity for OGAwill follow from Theo*
*rem
3, but first we need to replace g by a G-cofibration. Let
X0 __i2_//EM(g)
EE
i|| Ek0EEEEr||
fflffl|E""fflffl|l1
X1 ______//X0
be a G push-out and M(g) the mapping cylinder of g. There is G-homotopy
equivalence f : X0 ! X. Let f? : AG (X0) ! AG (X) be a map induce by f.
Then f?OGA(X0) = OGA(X).
By Theorem 3
f?OGA(X0) = f?(l1?OGA(X1) + r?OGA(M(g)) - (l1i)?OGA(X0))
We obtain
o f?l1?OGA(X1) = k1?OGA(X1)
o f?r?OGA(M(g)) = k2?OGA(X2)
o f?(l1i)?OGA(X0) = k0?OGA(X0)
from which the desired equality follows.
OGA(X) = k1?OGA(X1) + k2?OGA(X2) - k0?OGA(X0)
|___|
Note that when we replace the category C by the category D of G-spaces of the
homotopy type of finitely dominated G-CW-complexes then (W aG UG ); (wG ; OG ))
is the universal functorial additive invariant for D (see Luck [15]). Here W aG*
* stands
for equivariant Wall's finiteness obstruction group.
Corollary 2.For a G-space X of the homotopy type of a finitely dominated G
CW-complex there is natural map
fd : W a(X) UG (X) ! ss0AG (X)
which takes the sum of the equivariant Euler characteristic OGunivand the equiv*
*ariant
Wall's finiteness obstruction wG to the equivariant A-theory Euler characterist*
*ic
OGA.
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 13
5.Additivity for transfer for Algebraic K-theory of spaces.
In this section we show additivity for the A-theory transfer. Let p : E !
B be a fibration with fibers homotopy dominated by finite CW complexes. As
we mentioned in the introduction one has the transfer map p* : A(B) ! A(E).
B.Williams have showed that the transfer map is determined by cupping with the
parametrized Euler characteristic O(p) = [E t E AE E] which is consider as an
element in ss0(A(E ! B))
Theorem 4. Let p : E ! B be a fibration such that p = p1[p0p2 where pi: Ei! B
are fibrations, E = E1 [E0 E2, and E0 ! Ei is a cofibration for i = 1; 2. Assume
the fibers of pi for i = 0; 1; 2 are finitely dominated. Then the A-theory tra*
*nsfer
p* : A(B) ! A(E) can be written as a sum
p* = i1*p1*+ i2*p2*- i0*p0*
where ij*: A(Ej) ! A(E) are the maps induce by the obvious inclusions.
Proof.Since p*(x) = O(p) o x : ss*(A(B)) ! ss*(A(E)) where O(p) is an element in
ss0A(E ! B). From the corollary 1 we get on the ss0 level
p*(x)= O(p) o x
= (i1O(p1) + i2O(p2) - i0O(p0)) o x
= i1*p1*(x) + i2*p2*(x) - i0*p0*(x)
|___|
6. Sum decomposition
In this section we prove a sum decomposition analog of the sum decomposition
for Becker-Gottlieb transfer proven by Feshbach [10] and G.Lewis [13]
P
Theorem 5. Let X be a finite G-CW-complex and let mx = (-1)p#p[x] (see
section 5 for definition of #p[x]).
Then X
OGA(X) = mxxOGA(G=H)
x2Is0(G;X)
where x : AG (G=H) ! AG (X) is the map induced by x : G=H ! X.
Proof.Given a G-space X an element j 2 UG (X) is represented by a function
Is0(G; X) ! Z. From (7)
X
OG (X) = mxxOG (G=H):
x2Is0(G;X)
where [x : G=H ! X] 2 Is0(G; X), and OG (G=H) 2 UG (G=H) is the function
such that OG (G=H)[x] = 1 when x = id : G=H ! G=H and 0 otherwise. From
Corollary 1
X
OGA(X) = OG (X) = mxxOG (G=H)
X
(9) = mxxOG (G=H)
X
= mxxOGA(G=H)
|___|
14 WOJCIECH DORABIALA
Suppose G is a compact Lie group, E ! B a bundle with structure group G
and fiber X a finite G-CW-complex. Let P ! B be the associated G-bundle. For
any G-map, G=H ! X let jx be the map P xG G=H ! P xG X = E, and let
pG=H : P xG G=H ! B.
Corollary 3.Let G be a compact Lie group. Let E ! B be a bundle with fiber a
finite G-CW complex X and structure group G. Then the transfer
p* : A(B) ! A(E)
can be written as the sum X
p* = mxA(jx)p*G=H
x2Is0(G;X)
Proof.This formula follows immediately from theorem 5. |___|
As a corollary to the sum decomposition one can get an analogue to Feshbach`s
double coset formula. Let G be a compact Lie group and H and K arbitrary closed
subgroups of G. Then double coset formula expresses the composition
A(BK) J A(BH)99
JJp*(K;G)JJp*(H;G)tttt
JJ ttt
J%%J tt
A(BG)
as a finite sum of compositions of the form
A(B(K7\7Hg))O
p*(K\Hg;K)oooo OOp*(K\Hg;Hg)OO
oooo OOO
ooo OOO'' C(g-1)
A(BK) A(BHg) _____//A(BH)
Corollary 4.Let G be a compact Lie group and H, K be closed subgroups of G.
Let nx = OKuniv(G=H)[x] for [x : K=K [ Hg ! G=H] (see (6)). Then
X
p*(H;G)O p*(K;G)= nxC(g-1) O p*(K\Hg;Hg)O p*(K\Hg;K)
where C(g-1) : A(BHg) ! A(BH) is induce by conjugation with g-1 and we sum
over isomorphisms classes of objects [x : K=(K \ Hg) ! G=H] in the component
category.
Proof.This formula follows from the observation that the diagram
EK xK G=H --j--!BH
? ?
q?y ?yp(H;G)
p(K;G)
BK ----! BG
is a pullback and the sum decomposition.
Note that bundle q : E = EK xK G=H ! BK is a bundle with fiber G=H and
structure group K. Thus from the sum theorem
X
q* = nxA(jx)p*(L;K)
where nx = OKuniv(G=H)[x] for x : K=L ! G=H and we sum over isomorphism
classes of component category 0(K; G=H), and jx : EKxK K=L ! EKxK G=H =
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 15
E. Since the isotropy group of gH with respect to the K action is K \ Hg for
arbitrary gH 2 G=H it follows that mx = 0 if [x : K=L ! G=H] is such that
L 6= K \ Hg. Therefore
X
(10) q* = nxA(jx)p*(L;K)
where we sum over isomorphism classes of objects [x : K=K \ Hg ! G=H] in the
component category. By naturality of transfer and (10)
X
(11) p*(H;G)O p*(K;G)= A(j) O q* = nxA(j)A(jx)p*(K\Hg;K):
There is a commutative diagram
p(K\Hg;Hg)
B(K \ Hg) --------! B(Hg)
? ?
j*?y ?yC(g-1)
EK xK G=H - -j--! BH
Thus the double coset formula follows from above diagram and (11). __
|__|
In what follows we explain a key difference between the A-theory double coset
formula and Feshbach's double coset formula. For a compact Lie group G and a
compact G-CW-complex X we defined the equivariant homotopy Euler character-
istic OG;!(X) 2 !G0(X+ ) to be the composite
SV !j X+ ^ X*+!flX*+^ X+ id^-!X+ ^ X+ ^ X*+j^id-!SV ^ X+
where j is the equivariant Spanier-Whitehead duality map and j is dual to j.
First we recall a description of the Becker-Gottlieb transfer map in terms of t*
*he
equivariant homotopy Euler characteristic OG;!(X). Then we get the Feshbach
sum formula using a natural transformation UG (X) ! !G0(X+ ) which send OGuniv
to OG;!.
Homotopy_transfer_for_fiber_bundle_with_structure_group G
Let p : E ! B be a fiber bundle with structure group G and fiber X, where X
is a finite G-CW-complex. Consider the following two pull back diagrams
E ____//_EG xG X QB (E) ____//_EG xG Q(X+ )
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
B _______//_BG B ____________//BG
The homotopy transfer is given by a composition
etr
(12) tr : B----! QB (E) ----! Q(E+ )
Furthermore, etris the pull back of the universal transfer
gtrU2 EG xG Q(X+ ) ----! BG = Q(X+ )hG
For any G-space Z, ZhG denotes the homotopy fixed point set MapG (EG; Z).
16 WOJCIECH DORABIALA
Let
QG (X+ ) = lim-!VMap*(SV ; SV ^ X+ )
Since Q(X+ ) = lim-!nMap*(Sn; Sn ^ X+ ), the natural map Q(X+ ) ! QG (X+ ) is
an equivariant map that is a (nonequivariant) homotopy equivalence. Such a map
become an equivalence when the homotopy fixed point functor is applied.
Consider the composition
(13) QG (X+ )G----! QG (X+ )hG---w-! Q(X+ )hG
If we apply ss0, we get a homomorphism.
!G0(X) ----! ss0(EG xG Q(X+ ) ----! BG)
which sends OG;!(X) to gtrU
Thus the above shows that homotopy transfer for bundle with structure group G
and fiber finite G-CW-complex is determined by OG;! 2 !G0(X+ ). It is interesti*
*ng
to compare the above construction of the transfer with those given in [1] and (*
* [14]
see p.175)
Construction_of_natural_transformation_UG_(X)_!_!G0(X+_)
Lemma 3. For a G-space X of the type of a finite G CW-complex there is a unique
natural map
(14) : UG (X)----! !G0(X+ )
which takes the equivariant Euler characteristic OGunivto the equivariant homot*
*opy
Euler characteristic OG;!.
Proof.It is sufficient to show that (!G0; OG;!) is a functorial additive invari*
*ant.
Homotopy invariance, normalization are clearly satisfy and additivity is proved_
in [14] Theorem 2.9 where OG;! is called the pretransfer. |_*
*_|
Notice that OG;!(G=H) is a image of 1 2 !G0under the transfer map for G=H ! *.
If W H is not finite, Feshback (see [10]) has shown this transfer is trivial. *
*This
yields the following result.
________
Definition 1.Let Conj(G) be the set of conjugacy classes of subgroups H such
that W H is discrete.
Proposition 1. The following diagram commutes
P projectionP
Z[Isoss0(G; X)] = Z[ss0(XH )=W H]-------! Z[ss0(XH )=W H]
?Conj(G) ____Conj?
=?y ?ys
UG (X) ----! !G0(X+ )
where s is the tom-Dieck splitting isomorphism.
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 17
Proof.See tom-Dieck [22] II 7.6 and II 8.13.7. |___|
From the description of the homotopy transfer and Proposition 1 we obtain a sum
formula for the homotopy transfer.
Suppose G is a compact Lie group, E ! B a bundle with structure group G and
fiber X a finite G-CW-complex. Let P ! B be the associated G-bundle. For
any G-map, G=H ! X let jx be the map P xG G=H ! P xG X = E, and let
pG=H : P xG G=H ! B.
Lemma 4. Let G be a compact Lie group. Let E ! B be a bundle with fiber a
finite G-CW complex X and structure group G. Then the transfer
p* : Q(B+ ) ! Q(E+ )
can be written as the sum
X
(15) p* = mxQ(jx)p*G=H
where mx = integer defined on page 11 and we sum over isomorphism classes
[G=H ! X] of objects in the component category such that W H is not discrete.
From lemma 4 we get Feshbach's double coset formula
Corollary 5.Let G be a compact Lie group and H, K be closed subgroups of G.
Let nx = OKuniv(G=H)[x] for [x : K=K [ Hg ! G=H] (see (6)). Then
X
p*(H;G)O p*(K;G)= nxC(g-1) O p*(K\Hg;K)O p*(K\Hg;Hg)
where C(g-1) : Q(BHg) ! Q(BH) is induce by conjugation with g-1 and we sum
over isomorphisms classes of objects [x : K=(K \ Hg) ! G=H] in the component
category such that NK (K \ Hg)=(K \ Hg) is discrete.
The vanishing of the OG;!(G=H) when W H is not discrete shows that the Becker-
Gottlieb transfer for a fiber bundle p : E ! B with compact Lie group G as
structure group and fiber G=H is trivial. This can be easily be reduced to the
special case G = S1 and H is the trivial subgroup. Thus it is natural to ask
whether the analogues result is true for the A-theory transfer. However, result*
* of
Oliver [20] imply that even in this case the A-transfer can be nontrivial on ss*
*1.
Thus the A-theory Euler characteristic for OS1A(S1) is nontrivial. This shows w*
*hy
in the A-theory formula we need to take sum over all isomorphisms classes of the
component category. This also shows that in A-theory the double coset formula
summation is over bigger set than in Fesbach's double coset formula.
7. Additivity for Reidemeister Torsion
In this section, we recall the notion of parametrized Reidemeister torsion fr*
*om [8]
and present our proof of additivity.
The Reidemeister torsion is a classical invariant of non-simply connected com-
plexes. Let X be a CW -complex with fundamental group ss. Let F be a field.
For a representation ae : ss ! GL(n; F ) such that the cohomology H?(Z; Eae) of*
* Z
18 WOJCIECH DORABIALA
with the local system Eaeinduced from ae vanishes, one can defined the classical
Reidemeister torsion o(Z; ae) (see [18]) as an element of
W hae1(F ) = Cokernel x _ss__[ss;!ss]K1(F:)
Later Wagoner suggested there should be a notion of higher Reidemeister torsi*
*on.
Suppose that one has a fiber bundle Z ! M ! Sk with compact connected fiber
Z. Let ss0 be the fundamental group of M and suppose that one has a representa-
tion ae0: ss0 ! U(n; F ) such that the cohomology of the fibers with ae0-coeffi*
*cients
vanish. To such a bundle and representation, Wagoner [25] proposed to assign a
higher Reidemeister torsion o(M; ae0), an element of a certain quotient K0k+1(F*
* ) of
Kk+1(F ). He then suggested this could be used to detect elements of homotopy
groups of the diffeomorphism group of Z which fix a base-point and induce the
identity on ss0=Ker(ae0):
The first parametrized Reidemeister torsion was introduced by Igusa and Klein.
Let p : E ! B be a bundle of smooth compact manifolds. Let ae : V ! E be a
bundle of f.g. projective left R-modules such that for each b 2 B the homology
group Hi(Fb; V |) vanish. Then assuming E has a fiberwise Morse function, Igusa
and Klein defined Reidemeister torsion of p : E ! B. Their Reidemeister torsion
is a map from B to the homotopy fiber of j : Q(E+ ) ! K(R) (see [11], [12] for *
*the
definition of j).
An independent method of constructing parametrized Redemeister torsion is
given in [8]. The advantage of the definition in [8] is simplicity. It is not*
* clear
at this stage whether the parametrized Reidemeister torsion produced in [11], [*
*12]
and [8] are in agreement. let p : E ! B be a fibration with finitely dominated
fibers Fb, and R any ring. Let V be a bundle of f.g. projective R-modules on E.
Suppose also that for each b 2 B the homology groups Hi(Fb; V |) vanish for all*
* i.
To such a fibration p and bundle V , Dwyer, Weiss and Williams have constructed
the homotopy parametrized Reidemeister torsion in the following way. Let J be a
continuous homotopy invariant functor. Suppose that p : E ! B is classified by
ae : B ! BG(F ). Then we get a new fibration JB (E) ! B which is classified by
ae := BJ O ae :aBe_//BG(F )_BJ//_BG(J(F )). Roughly speaking the fibration
JB (E) ! B is create by applying J to the fibers of p. In particular for the fu*
*nctor
A(?) we get the fibration
AB (E)_____//B
where the fiber over b 2 B is A(p-1(b).
Consider
oh(p) V
B ____//_AB (E)___//K(R):
Here oh(p) : B ! AB (E) is the section of the fibration AB (E)_____//Bgiven by
the fiberwise Euler characteristic, and V is the composition
0V
V : AB (E) ____//_A(E)____//K(R)
where the first map AB (E) ! A(E) is such that for any b 2 B, the composition
A(Fb) ! AB (E) ! A(E) is induced by the canonical maps Fb ! E. The second
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 19
map 0V : A(E) ! K(R) is induced by a "linearization "functor. For a homotopy
finitely dominated retractive space
X AE E
let 0V(X AE E) be the relative singular chain complex of the pair (X; E) with
coefficients in the bundle of modules r?(V ). The functor 0V induces a map of
K-theory spaces A(E) ! K(R).(see p.40 [8] for details)
Using the linearization functor one gets a new subcategory of weak equivalenc*
*es
in Ret(E) by specifying (Y AE E) ! (Z AE E) to be a weak equivalence if and only
if 0V((Y AE E)) ! 0V((Z AE E)) is a homology equivalence. We let wRet(E)
denote the category of weak equivalences defined by 0V and AR (E) the K-theory
of the new category. Note the linearization map factors thru AR (E)
A(E) P
| PPPP
| PPPP
| PPPP
| ((
0V| AR (E)
| nn
| nnnn
| nnn
fflffl|vvnnn
K(R)
Since Hi(Fb; V ) = 0 for all b 2 B. It follows that the composition
oh(p) 0V
B ____//_AB (E)___//K(R)
is canonically nullhomotopic. In other wards we have lifted oh(p) to ohV(p) : B*
* !
hB(E) (where hB(E) = hofiber(AB (E) ! K(R)). Following [8] we call ohV(p) the
homotopy Reidemeister torsion.
Recall a regular manifold Mn is a topological manifold together with an n-disc
bundle q : L ! M, and an open embedding j : U ! L over M,where U is an open
neighborhood of the diagonal in M x M.
If p happens to be a bundle of compact regular manifolds then the composition
Oh(p) V
B ____//_QB (E)___//K(R)
(where the first map is Poincare dual to the fiberwise Euler class) is canonica*
*lly
nullhomotopic (see [8]) and the lift of od(p) to odV(p) : B ! dB(E) (where dB(E*
*) =
hofiber(QB (E) ! B)) we call odV(p) the regular Reidemeister torsion of p.
Note that when F is a field with involution and B = ?, then the set of path
components of dB(E) is canonically isomorphic to the target group for the class*
*ical
Franz-Reidemeister torsion.
Let p : AB (E) ! B be the space of sections of the fibration p : AB (E) ! B.
B.Williams introduce generalized coassembly map [24]
ff? : A p : E ! B ! p : AB (E) ! B
which is an example of one of Thomason`s limit problems map [22].
Remark 1. Note that if p : E ! B is a fibration with fibers finitely dominated*
*, then
ff? sends O(p) 2 A E ! B to the parametrized Euler characteristic constructed
in [8]. Recall the parametrized Euler characteristic oh(p) in sense of [8] is a*
* section
of the fibration AB (E) ! B.
20 WOJCIECH DORABIALA
Next we state additivity for parametrized Euler characteristic oh(p) : B !
AB (E).
Lemma 5. Let p : E ! B be such that p = p1 [p0p2 where pi : Ei ! B are
fibrations, E = E1 [E0 E2, and E0 ! E1 is a cofibration. Then
oh(p) = i1oh(p1) + i2oh(p2) - i0oh(p0)
where ii : AB (Ei) ! AB (E) are maps induced by inclusions ii : Ei ! E for
i = 0; 1; 2.
Proof.From theorem 3 there is a path in A E ! B from O(p) to i1O(p1) +
i2O(p2) - i0O(p0). Applying the coassembly
ff? : A p : E ! B ! p : AB (E) ! B
map we get a path from ff?O(p) = oh(p) to ff?i1O(p1) + ff?i2O(p2) - ff?i0O(p0) *
*by
naturality of coassembly map, we get
ff?i1O(p1) + ff?i2O(p2) - ff?i0O(p0) = i1oh(p1) + i2oh(p2) - i0oh(p0)
|___|
For the purpose of the proof of additivity for homotopy Reidemeister torsion *
*we
use the following two lemmas.
Lemma 6. There is a homotopy fibration
Acy(E)_____//A(E)____//_AR (E)
where Acy(E) is the K-theory of the category whose objects are retractive space*
*s Y
over E s.t C(Y; E; V ) is contractible.
Proof.It follows from Waldhausen fibration theorem. |___|
Let AcyB(E) ! B be the fibration gotten by applying the functor Acy(?) to the
fibration p : E ! B.
Lemma 7. There is a homotopy fibration
AcyB(E)____//_AB (E)___//AR (E)
Proof.It follows from lemma 6 and from Puppe's theorem (see theorem 1). |__*
*_|
We now return to the investigation of additivity for homotopy Reidemeister
torsion.
Theorem 6. Let p : E ! B be a fibration such that p = p1[p0p2 where pi: Ei! B
are fibrations, E = E1 [E0 E2, and E0 ! E1 is a cofibration. Let R be a ring and
V a bundle of f.g. projective left R-modules on E. Suppose that Hi(Ebi; V |Ei) *
*= 0
for i = 0; 1; 2 and all b 2 B. Then
ohV(p) = i1ohV(p1) + i2ohV(p2) - i0ohV(p0)
where ohV(?) is the homotopy Reidemeister torsion.
THE DOUBLE COSET FORMULA FOR ALGEBRAIC K-THEORY OF SPACES. 21
Proof.Since the linearization map factors thru AR (E) it follows that ohV(p) fa*
*ctors
thru the homotopy fiber of the map AB (E) ! AR (E). By lemma 7 the fiber of
AB (E) ! AR (E) is AcyB(E). (see diagram below)
AcyB(E)____//hB(E)
DD;; ss99
;; sss |
h h s;;ss |
ocy(p) oV(p)ss ;; |
sss ;; |
sss ;; |
sss ;; |
sss oh(p) AEAE;fflffl|
B ________________//_AB (E)I
| III
|| III
| II$$
V| AR (E)
| u
| uuu
| uuu
fflffl|zzuu
K(R)
|___|
Therefore to prove additivity for ohV(p) it suffices to prove additivity for oh*
*cy(p).
Note ohcy(p) is a lift of oh(p) to AcyB(E) ( such a lift exists because the com*
*position
B ! AB (E) ! AR (E) is nullhomotopic).
The proof of the additivity for ohcy(p) is similar to the proof of additivity*
* for oh(p)
in lemma 5. Namely we construct a space Acy(E ! B) together with a coassembly
map ff?1: Acy(E ! B) ! (AcyB(E) ! B). Then the proof of theorem follows from
the following lemma (we postpone the proof of lemma for a while).
Lemma 8. Let p : E ! B be such that p = p1 [p0p2 where pi : Ei ! B are
fibrations, E = E1 [E0 E2, and E0 ! E1 is a cofibration. Let R be a ring and V
a bundle of f.g. projective left R-modules on E. Suppose that Hi(Ebi; V |Ei) *
*= 0
for i = 0; 1; 2 and all b 2 B. Then there is a path in Acy(E ! B) between Ocy(p)
and i1Ocy(p1) + i2Ocy(p2) - i0Ocy(p0). where ij*: Acy(Ej) ! Acy(E) are the maps
induce by the obvious inclusions.
We define a space Acy(E ! B) as the K-theory of Waldhausen category