Geometry & Topology 11 (2007) 215–249 215
Absolute Whitehead torsion
ANDREW KORZENIEWSKI
We refine the Whitehead torsion of a chain equivalence of finite chain complexes in an iso additive category ށ from an element of Ke1 .ށ/ to an element of the absolute group iso K1 .ށ/. We apply this invariant to symmetric Poincaré complexes and identify it in terms of more traditional invariants. In the companion paper (joint with Ian Hambleton and Andrew Ranicki) this new invariant is applied to obtain the multiplicativity of the signature of fibre bundles mod 4.
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Introduction
The Whitehead torsion of a homotopy equivalence f X Y of finite CW complexes W ! is an element of the Whitehead group of 1.X / 1.Y / D D .f / .fe C.Xe/ C.Ye// Wh./ K1.ޚŒ/= ; D W ! 2 D f˙ g with fe the induced chain equivalence of based f.g. free cellular ޚŒ–module chain complexes. The Whitehead torsion of a finite n–dimensional Poincare´ complex X is n .X / .ŒX C.Xe/ C.Xe// Wh./: D \ W ! 2 In this paper we extend the methods of Ranicki [8] to consider absolute Whitehead torsion invariants for homotopy equivalences of certain finite CW complexes and finite Poincare´ complexes, which take values in K1.ޚŒ/ rather than Wh./. We shall also be extending the round L–theory of Hambleton–Ranicki–Taylor [2], which is the algebraic L–theory with absolute Whitehead torsion decorations. In the paper Hambleton–Korzeniewski–Ranicki [1] absolute Whitehead torsion in both algebraic K– and L–theory will be applied to investigate the signatures of fibre bundles. The absolute torsion of a finite contractible chain complex of finitely generated based R–modules C is defined by
.C / .d Codd Ceven/ K1.R/: D C W ! 2 for a chain contraction ; it is independent of the choice of . The algebraic mapping cone of a chain equivalence of finite chain complexes of finitely generated based
Published: 16 March 2007 DOI: 10.2140/gt.2007.11.215 216 Andrew Korzeniewski
R–modules f C D is a contractible chain complex C.f /. The naive absolute W ! torsion .C.f // K1.R/ only has good additive and composition formulae modulo 2 Im.K1.ޚ/ K1.R//. Likewise, the naive definition of the torsion of an n–dimensional ! symmetric Poincare´ complex .C; / (see Ranicki [6]) with C a based f.g. free R– module chain complex
n .C; / .C. 0 C C // D W ! only has good cobordism and additivity properties in Ke1.R/. The Tate Z2 –cohomology class n .C; / Hb .Z2 K1.R// 2 I may not be defined, and even if defined may not be a cobordism invariant. In [8] Andrew Ranicki developed a theory of absolute torsion for chain equivalences of round chain complexes, that is chain complexes C satisfying .C / 0. This absolute D torsion has a good composition formula but it is not additive, and for round Poincare´ complexes .C; 0/ is not a cobordism invariant, contrary to the assertions of Ranicki [9, (7.21), (7.22)]. There are two main aims of this paper, firstly to develop a more satisfactory definition of the absolute torsion of a chain equivalence with good additive and composition formulae and secondly to define an absolute torsion invariant of Poincare´ complexes which behaves predictably under cobordism. Section 1 is devoted to the first of these aims. Following [8] we work in the more general context of an additive category ށ. The chief novelty here is the introduction of a signed chain complex; this is a pair .C; C / where C is a finite chain complex and C is a “sign” term living in K1.ށ/, which will be made precise in Section 1. We give definitions for the sum and suspension of two signed chain complexes, and we define the absolute torsion of a chain equivalence of signed chain complexes NEW.f C D/ Kiso.ށ/: W ! 2 1 This gives us a definition of absolute torsion with good additive and composition formulae at the cost of making the definition more complicated by adding sign terms to the chain complexes. This definition is similar to the one given in [8], indeed if the chain complexes C and D are round and C D 0 then the definition of the D D absolute torsion of a chain equivalence f C D is precisely that given in [8]. When W ! working over a ring R the absolute torsion defined here reduces to the usual torsion in K .R/. z1 In Section 3 we work over a category with involution and define the dual of a signed chain complex. We can then define in Section 4 the absolute torsion of a symmetric
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n Poincare´ complex to be the absolute torsion of the chain equivalence 0 C C . W ! This new invariant is shown to be additive and to have good behaviour under round algebraic cobordism. Although we have to choose a sign C in order to define the absolute torsion of 0 , we show that the absolute torsion is independent of this choice. In Section 6 we state a product formula for the absolute torsion and prove it for group rings. It should be noted that the definitions in “Round L–Theory” [2] used the absolute torsion invariant of [8], [9], which is not a cobordism invariant. In Section 7 we show that the absolute torsion defined here may be used as the “correct” definition; in this case the statements in [2] are correct. In Section 8 we investigate the absolute torsion of manifolds. This invariant is only n defined when we pass to the reduced group Hb .Z2 K1.ZŒ1M // and we provide I some examples. In Section 9 the “sign” term of the absolute torsion of a manifold is identified with more traditional invariants of a manifold such as the signature, the Euler characteristic and the semi-characteristic. The forthcoming paper [1] will make extensive use of the invariants and techniques developed here. Section 2 and Section 5 develop the notion of the signed derived category which will be required by [1]. I would like to thank my supervisor Andrew Ranicki for his help and encouragement during the writing of this paper. I would also like to thank Ian Hambleton for many useful conversations and for carefully checking the computations.
1 Absolute torsion of contractible complexes and chain equiv- alences
In this section we introduce the absolute torsion of contractible complexes and chain equivalences and derive their basic properties. This closely follows [8] but without the assumption that the complexes are round ( .C / 0 K0.ށ/); we also develop the D 2 theory in the context of signed chain complexes which we will define in this section. Let ށ be an additive category. Following [8] we make the following definition.
Definition 1 (1) The class group K0.ށ/ has one generator ŒM for each object in ށ and relations: (a) ŒM ŒM if M is isomorphic to ŒM D 0 0 (b) ŒM N ŒM ŒN for objects M; N in ށ ˚ D C iso iso (2) The isomorphism torsion group K1 .ށ/ has one generator .f / for each isomorphism f M N in ށ, and relations: W !
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(a) iso.gf / iso.f / iso.g/ for isomorphisms f M N , g N P D C W ! W ! (b) iso.f f / iso.f / iso.f / for isomorphisms f M N and ˚ 0 D C 0 W ! f M N 0W 0 ! 0 1.1 Sign terms
iso The traditional torsion invariants are considered to lie in Ke1 .ށ/, a particular quotient of Kiso.ށ/ (defined below) in which the torsion of maps such as 0 1 C D D C 1 1 0 W ˚ ! ˚ are trivial. In absolute torsion we must consider such rearrangement maps; to this end we recall from [8] the following notation:
Definition 2 Let C; D be finite chain complexes in ށ.
(1) The suspension of C is the chain complex SC such that SCr Cr 1 . D (2) The sign of two objects M; N ށ is the element 2 Á Á .X; Y / iso 0 1N M N N M Kiso.ށ/: WD 1M 0 W ˚ ! ˚ 2 1 The sign only depends on the stable isomorphism classes of M and N and satisfies (a) .M M ; N / .M; N / .M ; N /, ˚ 0 D C 0 (b) .M; N / .N; M /, D (c) .M; M / iso. 1 M M /, D W ! (d) 2.M; M / 0. D We may extend to a morphism of abelian groups
iso K0.ށ/ K0.ށ/ K .ށ/ .ŒM ; ŒN / .M; N /: W ˝ ! 1 I 7! iso (3) The reduced isomorphism torsion group Ke1 .ށ/ is the quotient
iso iso iso Ke .ށ/ K .ށ/=Im. K0.ށ/ K0.ށ/ K .ށ//: 1 WD 1 W ˝ ! 1 (4) The intertwining of C and D is the element defined by
X iso ˇ.C; D/ ..C2i; D2j / .C2i 1; D2j 1// K1 .ށ/: WD C C 2 i>j
This depends only on the isomorphism classes of the chain complexes C and D.
Example 3 The reader may find it useful to keep the following example in mind, as it is the most frequently occurring context.
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Let R be an associative ring with 1 such that rankR.M / is well-defined for f.g free modules M . We define ށ.R/ to be the category of based f.g. R–modules. In this case the map K0.ށ.R// Z given by M dim M is an isomorphism. We have a ! 7! forgetful functor iso iso K .ށ.R// K1.R/ .f / .f /; 1 ! I 7! iso mapping elements of K1 .ށ.R// to the more familiar K1.R/ in the obvious way. In particular
Im. K0.ށ.R// K0.ށ.R// K1.R// . 1/ Im.K1.Z/ K1.R// W ˝ ! D f ˙ g D ! justifying the terminology of a “sign” term; the map is given explicitly for modules M and N by .M; N / rankR.M /rankR.N / . 1/: D We will make use of the notation
Ceven C0 C2 C4 D ˚ ˚ ˚ Codd C1 C3 C5 D ˚ ˚ ˚ and as usual we define the Euler characteristic .C / as
.C / ŒCeven ŒCodd K0.ށ/: D 2 We also recall from [8] (Proposition 3.4 and the remark above Proposition 3.3) the following relationships between the “sign” terms.
Lemma 4 Let C; C 0; D; D0 be finite chain complexes over ށ. Then iso iso (1) ˇ.C; D/ ..C D/even Ceven Deven/ ..C D/odd Codd Dodd/ D ˚ ! ˚ ˚ ! ˚ (2) ˇ.C C ; D/ ˇ.C; D/ ˇ.C ; D/ ˚ 0 D C 0 (3) ˇ.C; D D / ˇ.C; D/ ˇ.C; D / ˚ 0 D C 0 P r (4) ˇ.C; D/ ˇ.D; C / . / .Cr ; Dr / .Ceven; Deven/ .Codd; Dodd/ C D (5) ˇ.SC; SD/ ˇ.C; D/ D (6) ˇ.SC; C / .Codd; Ceven/. D 1.2 Signed chain complexes
In order to make to formulae in this paper more concise we introduce the concept of a signed chain complex; this is a chain complex with an associated element in iso Im. K0.ށ/ K0.ށ/ K .ށ// which we refer to as the sign of the complex. We W ˝ ! 1 use this element in the definition of the absolute torsion invariants.
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Definition 5 (1) A signed chain complex is a pair .C; C / where C is a finite chain complex in ށ and C an element of iso Im. K0.ށ/ K0.ށ/ K .ށ// W ˝ ! 1 We will usually suppress mention of C denoting such complexes as C .
(2) Given a signed chain complex .C; C / we give the suspension of C , SC the sign SC C : D (3) We define the sum signed chain complex of two signed chain complexes .C; C /, .D; D / as .C D; C D / where C D is the usual based sum of two chain ˚ ˚ ˚ complexes and C D defined by ˚ C D C D ˇ.C; D/ .Codd; .D// ˚ D C C (it is easily shown that .C D/ E C .D E/ ). ˚ ˚ D ˚ ˚ 1.3 The absolute torsion of isomorphisms
We now define the absolute torsion of a collection of isomorphisms fr Cr Dr f W ! g between two signed chain complexes. Note that the map f need not be a chain isomor- phism (ie f dC dD f need not hold). In the case where f is a chain isomorphism the D torsion invariant defined here will coincide with the definition of the absolute torsion of a chain equivalence given later.
Definition 6 The absolute torsion of a collection of isomorphisms fr Cr Dr f W ! g between the chain groups of signed chain complexes C and D is defined as
NEW X1 r iso iso iso .f / . / .fr Cr Dr / C D K1 .ށ/: D r W ! C 2 D 1 Lemma 7 We have the following properties of the absolute torsion of isomorphisms:
(1) The absolute torsion of isomorphisms is logarithmic, that is for isomorphisms f C D and g D E. W ! W ! NEW.gf / NEW.f / NEW.g/ iso D iso C iso (2) The absolute torsion of isomorphisms is additive, that is for isomorphisms f C D and f C D W ! 0W 0 ! 0 NEW.f f / NEW.f / NEW.f / iso ˚ 0 D iso C iso 0
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(3) The absolute torsion of the rearrangement isomorphism 0 1 C D D C 1 0 W ˚ ! ˚ is . .C /; .D// Kiso.ށ/. 2 1 (4) The absolute torsion of the isomorphism 0 1 S.C D/ SC SD 1 0 W ˚ ! ˚ is . .D/; .C // Kiso.ށ/. 2 1 Proof Parts 1 and 2 follow straight from the definitions, keeping in mind the fact that the sign terms and ˇ depend only on the isomorphism classes of their respective inputs. For part 3 we apply Lemma 4 part (4) to get
NEW X1 r iso .C D D C / . / .Cr ; Dr / C D D C ˚ ! ˚ D ˚ C ˚ r 0 D X1 r . / .Cr ; Dr / ˇ.C; D/ ˇ.D; C / D C r 0 D .Codd; .D// .Dodd; .C // C .Ceven; Deven/ .Codd; Dodd/ D .Codd; .D// .Dodd; .C // C . .C /; .D//: D For part 4
NEW iso .S.C D/ SC SD/ SC SD S.C D/ ˚ ! ˚ D ˚ ˚ ˇ.SC; SD/ .Ceven; .SD// D C ˇ.C; D/ .Codd; .D// C . .D/; .C //: D using Lemma 4 part (5).
1.4 The absolute torsion of contractible complexes and short exact se- quences
We recall from [8] the following: Given a finite contractible chain complex over ށ
C Cn C0 W ! !
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and a chain contraction Cr Cr 1 we may form the isomorphism W ! C 0d 0 0 1 B d 0 C d B C Codd C1 C3 C5 Ceven C0 C2 C4 : C D B0 d C W D ˚ ˚ ˚ ! D ˚ ˚ ˚ @ : : : : A : : : ::
The element iso.d / Kiso.ށ/ is independent of the choice of and is denoted C 2 1 .C / (following [8, Section 3]). We define the absolute torsion of a contractible signed chain complex C as
NEW iso .C / .C / C K .ށ/: D C 2 1 Following [8] we give ށ the structure of an exact category by declaring that a sequence
i j 0 M M M 0 ! ! 00 ! 0 ! is exact if there exists a splitting morphism k M M such that j k 1 M M W 0 ! 00 D W 0 ! 0 and the map .i k/ M M M W ˚ 0 ! 00 is an isomorphism. Given a short exact sequence
i j 0 C C C 0 ! ! 00 ! 0 ! of signed chain complexes over ށ, we may find a sequence of splitting morphisms k C C r 0 such that j k 1 C C .r 0/ and each .i k/ Cr C f W r0 ! r00j g D W r0 ! r0 W ˚ r0 ! C .r 0/ is an isomorphism. The torsion of this collection of isomorphisms r00 NEW ..i k/ Cr C C / iso W ˚ r0 ! r00 is independent of the choice of the kr , so we may define the absolute torsion of a short exact sequence as
NEW NEW .C; C ; C i; j / ..i k/ Cr C C /: 00 0I D iso W ˚ r0 ! r00 Lemma 8 We have the following properties of the absolute torsion of signed con- tractible complexes:
(1) Suppose we have a short exact sequence
i j 0 C C C 0 ! ! 00 ! 0 !
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of contractible signed complexes. Then
NEW.C / NEW.C / NEW.C / NEW.C; C ; C i; j /: 00 D C 0 C 00 0I
(2) Let C , C 0 be contractible signed complexes. Then NEW.C C / NEW.C / NEW.C /: ˚ 0 D C 0
Proof
(1) From [8, Proposition 3.3] we have that
X1 r iso .C 00/ .C / .C 0/ . / ..i k/ Cr Cr0 C 00/ ˇ.C; C 0/ D C C r W ˚ ! C D 1 for some choice of splitting morphisms k C C r 0 . By the definition f W r0 ! r00j g of the absolute torsion of a short exact sequence and the definition of the sum torsion (noting that contractible complexes have .C / 0 K0.ށ/) we get D 2
NEW X1 r iso .C; C 00; C 0 i; j / . / ..i k/ Cr Cr0 C 00/ I D r W ˚ ! D 1 ˇ.C; C 0/ C C C : C 0 C 00 By comparing these two formulae and the definition of the absolute torsion of a contractible signed complex, the result follows. (2) Apply the above to C C C . 00 D ˚ 0
1.5 The absolute torsion of chain equivalences
Sign Convention 9 We define the algebraic mapping cone of a chain map f C D W ! as follows: Â r 1 Ã dD . / C f dC.f / C.f /r Dr Cr 1 C.f /r 1 Dr 1 Cr 2 D 0 dC W D ˚ ! D ˚ We make C.f / into a signed complex by setting
C.f / D SC : D ˚
Lemma 10 The absolute torsion of a chain isomorphism f C D of signed chain W ! complexes satisfies NEW.f / NEW.C.f //: iso D
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Proof In the case of an isomorphism we may choose the chain contraction for C.f / to be  0 0 à C.f / r 1 C.f /r C.f /r 1: D . / f 0 W ! C We have a commutative diagram
.dC.f / C.f // .D1 C0/ .D3 C2/ C / D0 .D2 C1/ .D4 C3/ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚
0 1 f dD 0 ::: 1 B 0 f dC :::C B C B 0 0 f : : :C @ : : : : A : : : :: C0 D1 C2 D3 / D0 C1 D2 C3: ˚ ˚ ˚ ˚ ˚ ˚ ˚ The torsion of the upper map is iso.C.f //, the torsion of the lower isomorphism P r iso is r1 . / .fr Cr Dr / .Codd; Codd/ and the difference between the D 1 W ! PC r torsions of the downward maps is r1 . / .Cr ; Cr 1/ (using the fact that Cr D 1 Š Dr ). Hence NEW.C.f // iso.C.f // D C C.f / X1 r iso X1 r . / .fr C r Dr / . / .Cr ; Cr 1/ D r W ! r D 1 D 1 ˇ.C; SC / .Codd; .SC // .Codd; Codd/ C D C C C X1 r iso . / .fr Cr Dr / C D D r W ! C D 1 NEW.f / D iso (using the formulae of Lemma 4).
We can now give a definition of the absolute torsion of a chain equivalence f C D W ! which coincides with the previous definition in the case when f is a chain isomorphism.
Definition 11 We define the absolute torsion of a chain equivalence of signed chain complexes f C D as W ! NEW.f / NEW.C.f // Kiso.ށ/: D 2 1 In the case where f is a chain isomorphism the above lemma shows that this definition of the torsion agrees with that given in Definition 6.
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The following lemma explicitly describes how to compute the absolute torsion of a chain equivalence. It is intended primarily to allow readers already familiar with the absolute torsion of [8] to form a direct comparison with the new definition.
Lemma 12 The absolute torsion of a chain equivalence of chain complexes with torsion f C D is W ! NEW .f / .C.f // ˇ.D; SC / .Dodd; .C // D C K1.ށ/ D C 2 (cf definition of torsion [8, pp 223 and 226]. The two definitions coincide if both .C / and .D/ are even and C D ). D Proof The proof is simply a matter of unravelling definitions.
We have the following properties of the absolute torsion of chain equivalences.
Proposition 13 (1) Let f C D and g D E be chain equivalences of signed chain complexes W ! W ! in ށ, then NEW.gf / NEW.f / NEW.g/ Kiso.ށ/: D C 2 1 (2) Suppose f C D is a chain equivalence of contractible signed chain com- W ! plexes. Then NEW.f / NEW.D/ NEW.C / Kiso.ށ/: D 2 1 (3) The absolute torsion NEW.f / is a chain homotopy invariant of f . (4) Suppose we have a commutative diagram of chain maps as follows where the rows are exact and the vertical maps are chain equivalences
i j 0 / A / B / C / 0
a b c j i0 0 0 / A0 / B0 / C 0 / 0: Then NEW.b/ NEW.a/ NEW.c/ NEW.A; B; C i; j / D C I NEW.A ; B ; C i ; j / Kiso.ށ/: C 0 0 0I 0 0 2 1 (5) The torsion of a sum f f C C D D is given by ˚ 0W ˚ 0 ! ˚ 0 NEW.f f / NEW.f / NEW.f / Kiso.ށ/: ˚ 0 D C 0 2 1
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(6) Suppose we have a short exact sequence
f g 0 A B C 0 ! ! ! ! where C is a contractible complex and f is a chain equivalence. Then
NEW.f / NEW.A; B; C f; g/ NEW.C /: D I C
Proof The proofs of these follow those in [8, Proposition 4.2 and Proposition 4.4] modified where appropriate.
(1) We denote by C the chain complex defined by
dC dC Cr Cr 1 Cr 1 Cr : D W D C ! D We define a chain map
h C.g/ C.f / W ! by
0 1 0 0 C.g/r Er 1 Dr C.f /r Dr Cr 1: W D C ˚ ! D ˚ The algebraic mapping cone C.h/ fits into the short exact sequences
i j (1–1) 0 C.f / C.h/ C.g/ 0 ! ! ! ! i0 j 0 (1–2) 0 C.gf / C.h/ C. 1D D D/ 0 ! ! ! W ! ! where
1 i C.f /r C.h/r C.f /r C.g/r D 0 W ! D ˚ j . 0 1 / C.h/r C.f /r C.g/r C.g/r D W D ˚ ! 0 0 ! 0 1 i 0 1 0 C.gf /r Er SCr C.h/r Dr SCr Er SDr D 0 f W D ˚ ! D ˚ ˚ ˚ 1 0 0 0 Á j 0 0 f 0 1 C.h/r Dr SCr Er SDr C. 1D /r Dr SDr : D W D ˚ ˚ ˚ ! D ˚ Applying Lemma 8 part (1) to the first short exact sequence (1–1) we have
(1–3) NEW.h/ NEW.f / NEW.g/: D C
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Notice that 0 0 1 0 ! ! NEW NEW 0 1 0 0 iso ..i 0k0// iso 1 0 0 0 C.gf /r C. 1D /r C.h/ D 0 f 0 1 W ˚ ! NEW.D SC SC D/ D iso ˚ ! ˚ NEW.E SC SC E/ C iso ˚ ! ˚ NEW.D E E D/ C iso ˚ ! ˚ . .D/; .D// D (using the results of Lemma 7, the fact that .C / .D/ .E/ and that f NEWD D NEW has no effect on the torsion). We also see that .C. 1D // . 1D / D iso D . .D/; .D//. Applying these two expressions and Lemma 8 part (1) to the second exact sequence (1–2) we see that NEW.gf / NEW.h/ D and comparison with (1–3) yields the result. 0 (2) By construction we have C.0 D/ D and hence ! D 0 NEW.0 D/ NEW.D/: ! D Applying this and the composition formula (part 1) to the composition
0 f 0 C D ! ! yields the result. (3) A chain homotopy g f f C D W ' 0W ! gives rise to an isomorphism
r Á 1 . / g C.f / D SC C.f / D SC 0 1 W D ˚ ! 0 D ˚ which has trivial torsion. Using part 2 0 NEW.C.f // NEW.C.f // D 0 the result follows.
(4) We choose splitting morphisms k Cr Br r 0 and k C B r 0 . f W ! j g f 0W r0 ! r0 j g We have the following short exact sequence of mapping cones: Â Ã Á j 0 0 i0 0 0 i 0 j 0 C.a/ C.b/ C.c/ 0 ! ! ! !
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We note that NEW C.a/; C.b/; C.c/ i0 i0 I i I i NEW i0 0 k0 0 A SA C SC B B D iso 0 i 0 k W 0 ˚ ˚ 0 ˚ ! 0 ˚ NEW i0 k0 0 0 A C SA SC B B D iso 0 0 i k W 0 ˚ 0 ˚ ˚ ! 0 ˚ NEW.SA C C SA/ C iso ˚ 0 ! 0 ˚ NEW..i k/ SA SC SB/ NEW..i k // . .SA/; .C // D iso W ˚ ! C iso 0 0 C 0 NEW..i k/ S.A C / SB/ NEW..i k // D iso W ˚ ! C iso 0 0 NEW.SA SC S.A C // . .C /; .A// C iso ˚ ! ˚ C NEW..i k/ A C B/ NEW..i k // D iso W ˚ ! C iso 0 0 . .C /; .A// . .C /; .A// C NEW..i k/ A C B/ NEW..i k // D iso W ˚ ! C iso 0 0 NEW.A ; B ; C i ; j / NEW.A; B; C i; j /: D 0 0 0I 0 0 I The result now follows from applying Lemma 8 part (1) to the short exact sequence above. (5) Applying the result for a commutative diagram of short exact sequences (part (4)) with a f C D, c f C D and b f f C C D D D W ! D 0W 0 ! 0 D ˚ 0W ˚ 0 ! ˚ 0 yields the result. (6) We have a commutative diagram with short exact rows:
f g 0 / A / B / C / 0
f 1 1 0 / B / B / 0 / 0 The result follows by applying part (4) to the above diagram.
1.6 Applications to topology and examples of use
Let X be a connected finite CW –complex. We may form the cellular chain complex of the universal cover of X as a complex C.X / over the fundamental group ring z ZŒ X ; we may further make C.X / into a signed complex with an arbitrary choice of 1 z C.X / . For a cellular homotopy equivalence f X X we have an associated chain z W ! equivalence f C.X / C.X /; we can make C.X / into a signed chain complex by W z ! z z choosing some C.X / and defining the torsion of f to be z NEW NEW .f / .f C.X / C.X // K1.ZŒ1X / WD W z ! z 2
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which is independent of the choice of C.X / . We now give some examples. z (1) The torsion of the identity map of any connected CW –complex is trivial. (2) Let X ރސ2 ; we choose homogeneous coordinates .x y z/ and we give X D W W a CW –structure as follows: 0–cell .1 0 0/ W W 2–cell .z1 1 0/ W W 4–cell .z1 z2 1/ W W Let f ރސ2 ރސ2 be the cellular self-homeomorphism given by complex W ! conjugation in all three coordinates, that is,
f .x y z/ .x y z/: W W W 7! x W x W x This map preserves the orientation of the 0-cell and 4-cell, and it reverses the orientation of the 2-cell. Hence NEW.f / . 1/. In Corollary 38 we show D that for any orientation preserving self-homeomorphism g of a simply connected manifold of dimension 4k 2, that NEW.g/ 0. This example shows that C D for self-homeomorphism f of a 4k –dimensional manifold it is possible for NEW.f / 0: ¤
2 The signed derived category
The forthcoming paper [1] will require the use of the signed derived category ޓބ.ށ/. In this section we define ޓބ.ށ/ and prove some basic properties.
Definition 14 The signed derived category ޓބ.ށ/ is the category with objects finite signed chain complexes in ށ and morphisms chain homotopy classes of chain maps between such complexes.
Proposition 15
(i) The Euler characteristic defines a surjection
X1 r K0.ޓބ.ށ// K0.ށ/ ŒC; C .C / . / ŒCr : W ! I 7! D r D 1 (ii) Isomorphism torsion defines a forgetful map
iso iso iso NEW NEW i K1 .ޓބ.ށ// K1 .ށ/ .f / Œ .f / .f / W ! I 7! D
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which is a surjection split by the injection Kiso.ށ/ Kiso.ޓބ.ށ// iso.f A B/ iso.f .A; 0/ .B; 0//: 1 ! 1 I W ! 7! W ! where .A; 0/ (respectively .B; 0/) is the signed chain complex which is A (respectively B ) in dimension 0, trivial elsewhere and with trivial sign. (iii) The diagram
iso K0.ޓބ.ށ// K0.ޓބ.ށ// / K .ޓބ.ށ// ˝ 1 i ˝ Š iso K0.ށ/ K0.ށ/ / K .ށ/ ˝ 1
commutes, that is the sign of objects .C; C /, .D; D / in ޓބ.ށ/ has image iso i ..C; C /; .D; D // . .C /; .D// K1 .ށ/: D 2 Proof (i) A short exact sequence 0 C D E 0 of finite chain complexes in ށ ! ! ! ! determines a relation
ŒC; C ŒD; D ŒE; E 0 K0.ޓބ.ށ// C D 2 for any signs C ; D ; E . (ii) By construction. (iii) The sign
NEW 0 1 iso ..C;C /;.D;D// .C;C / .D;D/ .D;D/ .C;C // K .ޓބ.ށ/ D 1 0 W ˚ ! ˚ 2 1 has image
NEW 0 1 i ..C; C /; .D; D // 1 0 C D D C D W ˚ ! ˚ . .C /; .D// Kiso.ށ/: D 2 1 by Lemma 7 part 3.
3 Duality properties of absolute torsion
In this section we extend the notion of absolute torsion to encompass dual objects and dual maps. We now work over an additive category with involution (defined below) n and introduce the notion of a dual signed complex C (also defined below). We prove the following results.
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Proposition 16
(1) Let C be a contractible signed complex. Then
NEW.C n / . /n 1 NEW.C / Kiso.ށ/: D C 2 1 (2) Let f C D be a chain equivalence of signed chain complexes. Then W ! NEW.f n C n Dn / . /n NEW.f / Kiso.ށ/: W ! D 2 1 (3) Let f C n D be a chain equivalence of signed chain complexes. Then the W ! chain equivalence Tf Dn C (defined below) satisfies W ! NEW.Tf Dn C / . /n NEW.f / n .n 1/. .C /; .C // Kiso.ށ/: W ! D C 2 C 2 1 The rest of this section will be concerned with defining these concepts and proving Proposition 16. Following Ranicki [9] we define an involution on an additive category ށ to be a contravariant functor
ށ ށ M M ; .f M N / .f N M / W ! I ! W ! ! W ! together with a natural equivalence
e idށ ށ ށ M .e.M / M M / W ! W ! I ! W ! such that for any object M of ށ
e.M / .e.M / 1/ M M : D W ! iso An involution on ށ induces an involution on K0.ށ/ and K1 .ށ/ in the obvious way. We can easily deduce the following properties of the sign terms with respect to the involution
(1) .C n / . /n .C / D (2) .A ; B / .B; A/ D (3) ˇ.C n ; Dn / . /nˇ.D; C / . D Throughout the rest of this chapter ށ is an additive category with involution.
Sign Convention 17 Given an n–dimensional chain complex
dC dC dC dC C Cn Cn 1 Cn 2 C0 W ! ! ! !
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dC n dC n dC n C n C 0 C 1 C n W ! ! ! n r with the r th object given by C Cn r and the differentials given by D r n r n r 1 dC n . / d C C C : D C W ! We define the sign term
X r r iso ˛n.C / .C ; C /: K .ށ/: D 2 1 r n 2;n 3.mod4/ Á C C Given a signed chain complex .C; C / we define the dual signed chain complex n .C ; n / by C n 1 n 1 iso C n . / C . / C ˇ.C; C / ˛n.C / K .ށ/: WD C C C 2 1 We now compute the absolute torsion of various useful chain equivalences.
Lemma 18 Let C and D be signed chain complexes over ށ. (1) For chain isomorphisms f C D we have W ! NEW.f n Dn C n / . /n NEW.f / : W ! D ( NEW n n n 0 for n even (2) ..C D/ C D / ˚ ! ˚ D . .C /; .D// for n odd. NEW n n 1 (3) .1 C .SC / C / 0. NEW W r !n 1 n D (4) .. 1/ Cr C S.C /r / 0. NEW .nW 1/r n! n D n (5) .. 1/ .C / Cr / .n 1/. .C /; .C // . C W r ! D 2 C Proof Part 1 follows straight from the definitions. For part 2, NEW..C D/n C n Dn / ˚ ! ˚ n n n C D .C D/ D ˚n n ˚ n . .C /; .D /even/ . / . .C /; Deven/ : D C The result follows after considering the odd and even cases. Part 3 follows straight from the definitions. For part 4, NEW r n 1 n .. 1/ Cr C S.C /r / W ! P n n 1 .C ; C / S.C / C r n.mod2/ r r D C CP Á ˛n 1.C / ˛n.C / r n.mod2/ .Cr ; Cr / D C C C 0: Á D
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For part 5,
NEW .n 1/r n n .. 1/ .C / Cr / C W r ! .n 1/r n n .. / C C / C .C / C r r D n nC W P! ˛n.C / . / ˛n.C / .n 1/ rodd .Cr ; Cr / D P C C C r n 2;n 3.mod4/..Cr ; Cr / .Cn r ; Cn r // D Á C C P C .n 1/ r .Cr ; Cr / ( C C odd 0 for n 0 or n 3 Á Á D . .C /; .C // for n 1 or n 2 Á Á n .n 1/. .C /; .C //: D 2 C This proves the Lemma.
Lemma 19 The torsion of a contractible signed chain complex C in ށ satisfies
NEW.C n / . /n 1 NEW.C / Kiso.ށ/: D C 2 1
n n Proof We denote by C the chain complex with .C /r .Cn r / and x x D n r n r 1 dC n dC C C C : x D W x ! x We have an isomorphism f C n C n given by f 1 if r n 2; n 3.mod4/ W r ! xr D Á C C and f 1 otherwise. By considering the torsion of this isomorphism we have D n n (3–1) .C / .C / ˛n.C /: D x C Let neven be the greatest even integer n, similarly nodd . For any chain contraction for C we have the following commutativeÄ diagram:
0 d 0 0 1 d 0 @ 0 d A : : : : : : : :: C neven C 0 / C nodd C 1 ˚ ˚ ˚ ˚
0 1 1 0 1 1 1 1 @ .. A @ .. A . 0 d 0 1 . 1 1 0 d 0 0 d @ A : : : :: : : : : C 0 C neven / C 1 ::: C nodd ˚ ˚ ˚ ˚
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The torsion of the lower map in this diagram is .C / ; the torsion of the uppermost map is . /n 1 .C n /. So, by first considering the torsions of the maps in the above C x diagram we have X X Á .C n / . /n 1 .C / . /n 1 .C i; C j / .C i; C j / x D C C C i>j i;jeven i>j i;jodd I I . /n 1 .C / . /nˇ.C; C / : D C C Hence by (3–1) n n 1 n .C / . / .C / . / ˇ.C; C / ˛n.C /: D C C C Using the definition of the dual signed chain complex we have NEW.C n / . /n 1 NEW.C / : D C Lemma 20 Let C; D be n–dimensional signed chain complexes in ށ and f C D W ! a chain equivalence. Then NEW.f n Dn C n / . /n NEW.f / Kiso.ށ/: W ! D 2 1
Proof We have an isomorphism of chain complexes C.f n / C.f /n 1 given W ! C by
0 . /n r Á n n r n r 1 1 0 n 1 n r 1 n r C.f /r C D C.f / D C : D ˚ C ! rC D C ˚ The torsion of the map is given by NEW./ NEW.. /n r S.Dn / Dn 1 / D W ! C NEW.C n .SC /n 1 / C ! C NEW..SC /n 1 Dn 1 .SC D/n 1 / C C ˚ C ! ˚ C . /n 1 NEW.D SC SC D/ C C ˚ ! ˚ n. .D/; .D// .n 1/. .D/; .D// D C C . .D/; .D// C 0; D and the result follows since NEW.C.f /n 1 / . /n NEW.f / : C D We define the duality isomorphism T as p q pq T Homށ.C ; Dq/ Homށ.D ; Cp/ . / : W ! I !
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Lemma 21 Let C; D be n–dimensional signed chain complexes in ށ and f C n W ! D a chain equivalence. Then NEW.Tf Dn C / . /n NEW.f / n .n 1/. .C /; .C // Kiso.ށ/: W ! D C 2 C 2 1 Proof Using Lemma 18 and Lemma 20 we have NEW.Tf Dn C / NEW.f Dn .C n /n / W ! D W ! NEW.. 1/.n 1/r .C n /n C / C C ! . /n NEW.f / n .n 1/. .C /; .C // D C 2 C as required.
Together the above three lemmas prove Proposition 16.
4 Torsion of Poincare´ complexes
We now move on to consider symmetric Poincare´ complexes. These are algebraic objects which encapsulate the properties of Poincare´ duality spaces (see Ranicki [6; 7] for a more complete discussion). We will restrict ourselves to considering symmetric Poincare´ complexes over a ring R, that is we work over ށ ށ.R/ and we will consider D the torsion invariants to lie in the more familiar K1.R/. We will define the notion of the absolute torsion of a symmetric Poincare´ complex to be, essentially, the torsion of the Poincare´ duality chain equivalence. In the case of compact oriented manifolds M n with CW –structure this is the torsion of the map ŒM n C.M /n C.M /: \ W ! In this section we develop the theory from this algebraic viewpoint; it will be applied to geometric objects in a later section. We recall from Ranicki [6; 5] the following definition.
Definition 22
(1) An n–dimensional symmetric complex .C; 0/ is a chain complex C in ށ.R/, together with a collection of morphisms n r s s C Cr s 0 D f W C ! j g such that r n s 1 s n r s 1 dC s . / sdC . / C C . s 1 . / T s 1/ 0 C C Cr C C C D W !
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n where s 0, and 1 0. Hence 0 C C is a chain map and 1 is a D W ! chain homotopy 1 0 T 0 . W ' (2) The complex is said to be Poincare´ if 0 is a chain equivalence. (3) The complex is said to be round if C is round.
(4) A morphism between n-dimensional symmetric complexes .C; / and .C 0; 0/ n 1 s r consists of a chain map f C C and morphisms s C C C C s 0 W ! 0 W 0 ! r0 such that r n s s n r s s0 f sf dC s . / sdC . / C .s 1 . / T s 1/ C 0 C Cr0 D 0 C 0 C C W ! (in particular f 0f ). Such a morphism is said to be a homotopy equiva- 0 ' lence if f is a chain equivalence. n (5) A symmetric complex .C; / is said to be connected if H0. 0 C C / 0. W ! D (6) The boundary .@C; @ / of a connected n–dimensional symmetric complex .C; / is the .n 1/–dimensional symmetric Poincare´ complex defined by r dC . / 0 Á n r n 1 r d@C r @Cr Cr 1 C @Cr 1 Cr C C 0 . / d D C W D C ˚ ! D ˚ n r 1 r n Á . / T 1 . / n r 1 n r n r @ 0 @C C Cr 1 @Cr Cr 1 C D 1 0 W D ˚ C ! D C ˚ n r 1 r n Á . / T s 1 . / n r s 1 n s r @ s C @C C C C Cr s 1 D 0 0 W D ˚ C ! n r @Cr Cr 1 C : D C ˚ (7) A signed symmetric (Poincare)´ complex is a symmetric (Poincare)´ complex .C; 0/ where in addition C is a signed chain complex.
Example 23 An n–dimension manifold M with universal covering Mf determines a symmetric Poincare´ complex .C.Mf/; / in ށ.ZŒ1M / with n 0 ŒM C.Mf/ C.Mf/: D \ W ! Lemma 24 The boundary .@C; @ / of any signed n–dimensional symmetric complex .C; / satisfies
NEW n 1 n .@ 0 .@C / @C / .n 1/. .C /; .C // K1.R/: W ! D 2 C 2 Proof The map  n r 1 rn à . / T 1 . / n r 1 @ 0 @C @Cr D 1 0 W ! is an isomorphism.
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We have  rn à à NEW NEW 0 . / n n 1 n .@ 0/ .C C / C C D 1 0 W ˚ ! ˚ NEW..C C n /n 1 .C /n 1 .C n /n 1 / D ˚ ! ˚ NEW..C n /n 1 .C n 1 /n 1 / C ! NEW.. /nr .C n 1 /n 1 C / C W ! NEW..C /n 1 C n / C ! NEW.C n C C C n / D ˚ ! ˚ n .n 1/. .C /; .C // D 2 C using the results of Lemma 18.
We can now define a new absolute torsion invariant of Poincare´ complexes which is additive and a cobordism invariant.
Definition 25 We define the absolute torsion of a signed Poincare´ complex .C; / as
NEW NEW .C; / . 0/ K1.R/: D 2
Proposition 26 Let .C; / and .C 0; 0/ be signed n–dimensional Poincaré complexes. Then:
(1) Additivity:
NEW NEW NEW .C C ; / .C; / .C / K1.R/: ˚ 0 ˚ 0 D C 0 0 2 (2) Duality:
NEW n NEW n .C; / . / .C; / .n 1/. .C /; .C // K1.R/ D C 2 C 2 (n.b. the above sign term disappears in the case where anti-symmetric forms over the ring R necessarily have even rank; this is the case for R Z or R Q but D D not R ރ). D (3) Homotopy invariance: Suppose .f; s/ is a homotopy equivalence from .C; / to .C 0; 0/. Then
NEW NEW n .C ; / .C; / .f / . / .f / K1.R/: 0 0 D C C 2
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(4) Cobordism Invariance: Suppose that .C; / is homotopy equivalent to the bound- ary of some .n 1/–dimensional symmetric complex with torsion .D; D /. C Then NEW .C; / . /n 1 NEW.C @D/ NEW.C @D/ D C ! ! 1 .n 1/.n 2/. .D/; .D// K1.R/: C 2 C C 2 (5) Orientation change: NEW NEW .C; / .C; / . .C /; .C // K1.R/: D C 2 (6) The absolute torsion of a signed Poincaré complex is independent of the choice of sign C .
Proof (1) A symmetric Poincare´ complex of odd dimension satisfies .C / 0, hence the n n n D map .C C / C C has trivial absolute torsion. Additivity now ˚ 0 ! ˚ 0 follows from the additivity of chain equivalences.
(2) We know that 0 is homotopic to T 0 ; duality now follows by applying Lemma 21.
(3) We have that f 0f and hence 0 ' NEW NEW NEW n NEW . / .f / . 0/ . / .f / : 0 D C C (4) This follows from Lemma 24 and homotopy invariance. NEW NEW NEW NEW (5) We have that . 0/ . 0/ . 1 C C / . 0/ D C W ! D C .C / . 1/. NEW (6) A change in leads to a corresponding change in n so . / is C c 0 unchanged.
5 The signed Poincare´ derived category with involution
In this section we will add an involution to a particular subcategory of the signed derived category. Let ޓސބn.ށ.R// denote the category whose object are signed n– dimensional chain complexes C in ށ.R/ which are isomorphic to their dual complexes C n and .C / 0 if n is odd. Then we have an involution D C C n W 7! .f C D/ .f n Dn C n / W W ! 7! W !
Geometry & Topology, Volume 11 (2007) Absolute Whitehead torsion 239 with the natural equivalence e.C / given by
e.C / . /.n 1/r C .C n /n : D C W ! We call this category the signed Poincare´ derived category with n–involution. In order to show that this is a covariant functor of additive categories we must show that .A B/ A B . However, the condition that .C / is odd if n is odd implies ˚ D ˚ that the torsion of the rearrangement map .C D/n C n Dn is trivial (see ˚ ! ˚ Lemma 18 part (2)) and the functor is additive. As in the case of the signed derived category we have a map
iso iso iso NEW i K1 .ޓސބn.ށ.R/// K1 .ށ.R// .f / .f /: W ! I 7! The behaviour of i under the involution on ޓސބn.ށ.R// is given by n i .f / . / i .f /: D
6 A product formula
In this section we will quote a formula for the absolute torsion of a product of symmetric Poincare´ complexes and prove it in a special case.
All tensor products will be over the integers Z. For rings R and R0 we have nat- ural inclusion maps K1.R/ K1.R R / and K1.R / K1.R R /. For an ! ˝ 0 0 ! ˝ 0 isomorphism f A B in ށ.R/ and a module M in ށ.R / we have W ! 0
.f 1 A M B M / rankR .M / .f / K1.R R0/ ˝ W ˝ ! ˝ D 0 2 ˝ .1 f M A M B/ rankR .M / .f / K1.R R0/: ˝ W ˝ ! ˝ D 0 2 ˝ We recall from [6] the definition of the tensor product of symmetric Poincare´ complexes.
Definition 27
(1) The tensor product C D of a chain complex C in ށ.R/ and a chain complex ˝ D in ށ.R / is the chain complex in ށ.R R / 0 ˝ 0 X1 dC D .C D/r Cs Dr s .C D/r 1 ˝ W ˝ D s ˝ ! ˝ I D 1 r s x y x dD .y/ . / dC .x/ y: ˝ 7! ˝ C ˝
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(2) The tensor product .C D; / of an n–dimensional symmetric Poincare´ ˝ ˝ complex .C; / with an m–dimensional symmetric Poincare´ complex .D; / is an .n m/–dimensional Poincare´ complex defined by C s X .n r/s n m . /s . 1/ C r T s r .C D/ C C D: ˝ D r ˝ W ˝ ! ˝ D 1
Proposition 28 Let .C; / and .C 0; 0/ be symmetric Poincaré complexes over a rings with involution R and R0 respectively. Then NEW.C C ; / .C / NEW.C ; / .C / NEW.C; /: ˝ 0 ˝ 0 D 0 0 C 0 The proof of this result requires the theory of signed complexes to be extended to tensor products; this theory is developed in [1] using the theory of signed filtered complexes. We will use the following “ad hoc” methods to prove the product formula under the following conditions.
Assumption 29 The rings R and R are such that .4k 2/–dimensional symmetric 0 C forms necessarily have even rank (eg group rings).
The ring R ރ is an example which does not satisfy this assumption. We say a D module M in ށ.R/ is even if rankR.M / is even; similarly we say a chain complex C in ށ.R/ is even if Cr is even for all r .
Definition 30 Let .C; C / and .D; D / be even signed complexes. We define the signed complex tensor product by
.C D; C D / .C D; 0/: ˝ ˝ D ˝ Lemma 31
(1) Let C , C 0 be signed, even complexes in ށ.R/ and D a signed even complex in ށ.R0/. Then NEW..C C / D .C D/ .C D// 0 ˚ 0 ˝ ! ˝ ˚ 0 ˝ D NEW.D .C C / .D C / .D C // 0: ˝ ˚ 0 ! ˝ ˚ ˝ 0 D (2) Let C be a signed contractible even complex in ށ.R/ and D a signed even complex in ށ.R0/. Then NEW.C D/ NEW.D C / .D/ NEW.C /: ˝ D ˝ D
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(3) Let f C C be a chain equivalence of even complexes in ށ.R/ and D a W ! 0 signed even complex in ށ.R0/. Then NEW.f 1 C D C D/ NEW.1 f D C D C / ˝ W ˝ ! 0 ˝ D ˝ W ˝ ! ˝ 0 .D/ NEW.f /: D
Proof For even modules M; N the map .M; N / 0 K1.R/ and . 1 M D 2 W ! M / 0, so the torsion of rearrangement maps is always zero and C D D C D ˚ D C for even chain complexes C , D. Part 1 follows straight from these facts. For part 2, let C C 1 be a chain contraction of C , then .dC / 1 is a chain contraction W ! C C ˝ of C D. We have a commutative diagram ˝
.dC / 1 .C D/odd C ˝ / .C D/even ˝ ˝
.Ceven Dodd/ .Codd Deven/ / .Codd Dodd/ .Ceven Deven/ ˝ ˚ ˚ ˝ ˚ ˚ with the bottom map given by .dC / 1 .dC / 1. The torsion of the top C ˝ ˚ C ˝ map is .C D/, the torsion of the bottom map is .D/ .C /. Part 2 follows from the ˝ fact that the torsions of the left and right maps are zero, since they are rearrangements. For part 3 we have that
NEW.C.f 1/ C.f / D/ 0 ˝ ! ˝ D since it is a rearrangement map and the result now follows straight from part 2.
Proposition 32 Let .C; C / and .D; D / be Poincaré complexes over ring R and R0 respectively, where R and R0 satisfy Assumption 29. Then NEW.C D; C D / .D/ NEW.C; C / .C / NEW.D; D / ˝ ˝ D C K1.R R /: 2 ˝ 0
Proof Given any f.g. based chain complex C we may form a direct sum with a contractible chain complex to form a new chain complex which is of even rank in every dimension except one. Hence given a Poincare´ pair .C; / we may form a new C Poincare´ pair .C 0; 0 / which is even in every dimension by
(1) Forming the direct sum with a contractible complex (letting C 0 vanish on this contractible complex) such that C 0 is even in every dimension except possibly the middle.
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(2) If the complex is odd (and hence of dimension 4k), forming the direct sum with the Poincare´ complex which is R in dimension 2k , vanishes otherwise and has 0 1 R R. D W !
D D We may form a similar complex .D0; 0 / from .D; /. Using Lemma 31 we see that
NEW C D NEW D NEW C . 0 0 / .C / . 0 / .D / . 0 / K1.R R /: 0 ˝ 0 D 0 0 C 0 0 2 ˝ 0 Hence
NEW.C D ; C 0 D0 / .C / NEW.D ; D0 / .D / NEW.C ; C 0 /: 0 ˝ 0 ˝ D 0 0 0 C 0 0 0 Let RC denote the chain complex which is 0 if C is even and R in dimension 2k D otherwise; similarly R0 . By direct computation (for the sake of clarity we now suppress mention of the morphisms )
NEW.RC D/ .C /. NEW.D/ .k 1/ .D// . 1// ˝ D C C NEW.C R D / .D/. NEW.C / .l 1/ .C / . 1// ˝ 0 D C C NEW.RC R D / .k l/ .C / .D/ . 1/: ˝ 0 D C The Poincare´ complex .C D / is homotopy equivalent to .C RC / .D R D /; 0 ˝ 0 ˚ ˝ ˚ 0 using the invariance of absolute torsion under homotopy equivalence, its additivity properties and the above three formulae we have
NEW NEW NEW .C D/ .C / .D/ .D/ .C / K1.R R / ˝ D C 2 ˝ 0 as required.
7 Round L–theory
n We refer the reader to [2] for the definition of the round symmetric L–groups Lr .A/. The absolute torsion defined in this paper as .C; 0/ . 0/ (here . 0/ refers D to the absolute torsion defined in [8]) is not a cobordism invariant. We can define such an invariant using the absolute torsion of a Poincare´ complex. If this invariant is substituted for .C; 0/ as defined in [2] then the results become correct.
Lemma 33 Let .C; / be a round Poincaré complex. The reduced element
NEW n .C; / Hb .Z2 K1.R// 2 I
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is independent of the choice of sign C ; moreover we have a well defined homomor- phism n n L .R/ Hb .Z2 K1.R// r ! I given by .C; / NEW.C; /. 7! n NEW Proof The element .C; / Hb .Z2 K1.R// is independent of the choice of 2 I sign by Proposition 26 part (6). The absolute torsion is additive by Proposition 26 part (1). The absolute torsion of the boundary of a round symmetric complex is trivial in n the reduced group Hb .Z1 K1.R// by Proposition 26 part (4). Hence the torsion of a I round null-cobordant complex is trivial and the map
n n L .R/ Hb .Z2 K1.R// r ! I given by .C; / NEW.C; / is well defined. 7!
8 Applications to manifolds
8.1 The absolute torsion of oriented manifolds
To any n-dimensional oriented manifold M we may associate a Poincare´ complex .C; / over the ring R ZŒ1M , well defined up to homotopy equivalence (see D [7]). By property (2) of Proposition 26 the absolute torsion of such a Poincare´ complex satisfies NEW.C; / . /n NEW.C; / D since .M / 0 (mod 2) unless n 0 (mod 4). Hence the torsion NEW.C; / may be Á nÁ considered to lie in the group Hb .Z2 K1.ZŒ1M //. By property (3) of Proposition I 26 if .C; / is homotopy equivalent to .C 0; 0/ then
NEW NEW n .C; / .C 0; 0/ Hb .Z2 K1.ZŒ1M // D 2 I hence NEW n NEW n .M / .C; / Hb .Z2 ZŒ1M / WD 2 I is well defined.
8.2 Examples of the absolute torsion of manifolds
8.2.1 The circle We may associate to the circle (S 1/ the following chain complex 1 1 over R ZŒ1.S / ZŒt; t by giving it the CW–decomposition consisting of one D D
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1–cell and one 0–cell: 1 1 1 ZŒt; t / ZŒt; t
t 1 1 1 t 1 t 1 ZŒt; t / ZŒt; t In this diagram the two modules on the right are the chain complex, the two modules on the left are the dual complex and the sideways arrows represent 0 . n NEW 1 1 Hence .S / . t/ Hb .Z2 K1.ZŒt; t /. D 2 I 8.2.2 The absolute torsion of an algebraic mapping torus The mapping torus of a map f M M is the space obtained from M I by attaching the boundaries W ! M 0 and M 1 using the map f . The following algebraic analogue is defined f g f g by Ranicki [10, Definition 24.3] (the reader should note the different sign convention used here).
Definition 34 The algebraic mapping torus of a morphism .f; / .C; / .C; / W ! from an n–dimensional symmetric Poincare´ complex .C; / over a ring R to itself is the .n 1/–dimensional symmetric complex .T .f /; / over the ring RŒz; z 1 C defined by
T .f / C.f z/ D n s . / s . / s z Á n r s 1 s n r 1 n r s 1 T .f / C C T .f /r : . / C s f . / C C T s 1 D W ! The complex is Poincare´ if the chain map f is a chain equivalence.
Lemma 35 Let .f; / .C; / .C; / be a self chain equivalence from an n– W ! dimensional symmetric Poincaré complex .C; / over a ring R to itself. Then
NEW NEW 1 .T .f /; / .f / .C / . z/ K1.RŒz; z /: D C 2 Proof We have a commutative diagram with short exact rows:
0 Á .. /r 0 / n r 1 n 1 r n r 0 / C / C.f z/ / S.C / / 0 C n z 0 0 . / 0f 0 / Cr / C.f z/r / SCr / 0 1 Á . 0 1 / 0
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The absolute torsion of the lower short exact sequence is trivial; for the top map we have Á NEW n r n 1 r n r 0 r C ; C.f z/ C ; S.C / ; . . / 0 / I 1 r Á Á NEW 0 . / .C SC /n 1 C n S.C n / D 1 0 W ˚ C ! ˚ NEW..C SC /n 1 C n 1 .SC /n 1 / D ˚ C ! C ˚ C NEW.. 1/r C n 1 S.C n // C W C ! NEW..SC /n 1 C n / C C ! NEW.S.C n / C n C n S.C n // C ˚ ! ˚ 0: D Using Proposition 13 part (4) we have
NEW NEW .T .f /; / .0/ D NEW NEW n n . z 0/ .. 1/ S. 0f / S.C / SC / D C W ! NEW.f / NEW. z C C / D C W ! . /n 1 NEW.f / NEW. z C C / D C C W ! NEW.f / .C / . z/ D C as required.
8.2.3 A specific example of a mapping torus We return to the example of the orien- tation preserving self-homeomorphism f ރސ2 ރސ2 given by complex conjugation W ! in some choice of homogeneous coordinates (see Section 1.6). We recall that the torsion NEW of this map is .f / . 1/ K1.Z/. Using Lemma 35 we compute the absolute D 2 torsion of the mapping torus of f as
NEW 3 1 .T .f // .z / K1.ZŒz; z / D 2 1 where z is a generator of 1.T .f // 1.S / Z. By contrast we may compute the D D absolute torsion of the space T .Id ރސ2 ރސ2/ S 1 ރސ2 as W ! D NEW 1 2 3 1 .S ރސ / . z / K1.ZŒz; z / D 2 hence the absolute torsion can distinguish between these two ރސ2 bundles over S 1 . A more thorough investigation into the absolute torsion of fibre bundles of compact manifolds is made in [1].
Geometry & Topology, Volume 11 (2007) 246 Andrew Korzeniewski
9 Identifying the sign term
Throughout this section we work over a group ring R ZŒ for some group (or, D more generally, any ring with involution R which admits a map R Z such that the ! composition Z R Z is the identity). We first identify the relationship between the ! ! “sign” term of the absolute torsion of a Poincare´ complex and the traditional signature and Euler characteristic and semi-characteristic invariants.
We have a canonical decomposition of K1.ZŒ/ as follows:
K1.ZŒ/ K1.ZŒ/ Z2 D z ˚ with the Z2 component the “sign” term identified by the map
i K1.ZŒ/ K1.Z/ Z2 ! D induced by the augmentation map i 1 (more generally, a map R Z gives a W ! ! map K1.R/ K1.Z/ Z2 which gives a splitting K1.R/ K1.R/ Z2 ). We wish ! D D z ˚ to determine the Z2 component in terms of more traditional invariants of Poincare´ complexes. The augmentation map may also be applied to a symmetric complex .C; / over ZŒ to form a symmetric complex over Z by forgetting the group. Functoriality of the absolute torsion tells us that this complex has the same sign term as .C; /, hence to identify the sign term it is sufficient to consider symmetric Poincare´ complexes over Z. We will require the Euler semi-characteristic 1=2.C / of Kervaire [3]
Definition 36 The Euler semi-characteristic .C / of a .2k 1/–dimensional 1=2 chain complex C over a field F is defined by
k 1 X i .C / . / rankF Hi.C / Z 1=2 D 2 i 0 D For a .2k 1/–dimensional chain complex C over Z we define .C F/ .C Z F/: 1=2 I D 1=2 ˝ Proposition 37 The absolute torsion of an n–dimensional symmetric Poincaré com- plex over Z is determined by the signature and the Euler characteristic and semi- characteristic as follows: (1) If n 4k then D NEW.C; / 1 .C / .1 2k/ .C / . 1/ D 2 C with .C / the signature of the complex.
Geometry & Topology, Volume 11 (2007) Absolute Whitehead torsion 247
(2) If n 4k 1 then NEW.C; / .C Q/. D C D 1=2 I (3) Otherwise NEW.C; / 0. D
As an example we have a simple corollary.
Corollary 38 The absolute torsion of an orientation preserving self-homeomorphism of a simply-connect manifold of dimension 4k 2 is trivial. C
Proof Let f M M be such a self-homeomorphism. We may triangulate M W ! and hence construct the algebraic mapping torus T .f / of the chain equivalence f C.M / C.M /. By Lemma 35 W ! NEW NEW 1 .T .f // .f / K1.ZŒz; z /: D 2 1 The augmentation map Z 1 induces a map of rings ZŒz; z Z. Since W ! W ! L–theory and the absolute torsion are functorial, T .f / represents an element of 4k 3 NEW NEW L C .Z/ with absolute torsion . T .f // .f / K1.Z/. However by D 2 part 3 of the above proposition NEW. T .f // 0. D
The aim of the rest of this section is to prove Proposition 37 . We recall from [7] the n computation of the symmetric L–groups Lh.Z/ of the integers Z: 8 Z.signature/ n 0.mod 4/ ˆ ˆ Á n d.C / .C Z2/ .C Q/: D 1=2 I 1=2 I For dimensions n 2; 3.mod 4/ the absolute torsion is a cobordism invariant (Proposi- Á tion 26 part (4)) so the above computation of the symmetric L–groups tells us that the absolute torsion is trivial in these cases, thus proving the third part of Proposition 37. If n 4k 1 then the absolute torsion is not a cobordism invariant; however it is a D C round cobordism invariant, so absolute torsion defines a map 4k 1 L .Z/ K1.Z/: rhC ! Geometry & Topology, Volume 11 (2007) 248 Andrew Korzeniewski Since .C / 0 for all odd-dimensional symmetric Poincare´ complexes every such D 4k 1 .4k 1/–dimensional complex represents an element in LrhC .Z/. In [2, Proposition C 4k 1 4.2] the group LrhC .Z/ is identified as 4k 1 L .Z/ Z2 Z2 C . .C Z2/; .C; Q//: rhC D ˚ I 7! 1=2 I 1=2 We now construct explicit generators of this group and compute their absolute torsions. We define the generator .G; / to have chain complex G concentrated in dimensions 2k and 2k 1 defined by C dG 0 G2k 1 Z G2k Z; D W C D ! D with the morphisms given by 2k 1 G Z G2k 1 Z 0 W D ! C D 1 0: D 1 G2k 1 Z G Z D W C D ! 2k D Geometrically .G; / is the symmetric Poincare´ complex over Z associated to the circle. NEW By direct computation, .G; Z2/ 1, .G; Q/ 1 and .G; / . 1/. 1=2 D 1=2 D D We define the generator .H; / to have chain complex H concentrated in dimensions 2k and 2k 1 defined by C dH 2 H2k 1 Z H2k Z; D W C D ! D with the morphisms given by 2k 1 H Z H2k 1 Z 2k 1 0 W 2k 1D ! C D 1 1 H C H2k 1: D 1 H Z H Z D W ! C W C D ! 2k D Geometrically .H; / is a symmetric Poincare´ complex over Z which is cobordant to the complex associated to the mapping torus of the self-diffeomorphism of CP2 given by complex conjugation. Again by direct computation, .H; Z2/ 1, .H; Q/ 0 1=2 D 1=2 D and NEW.H; / 0. By considering the absolute torsion of these two generators we D4k 1 see that the map L .Z/ K1.Z/ is given by rhC ! .C; / .C; Q/ . 1/ 7! 1=2 thus proving part two of Proposition 37. To prove part 1 of Proposition 37 we use the following lemma taken from [1]. Lemma 39 We have the following relationship between NEW and signature modulo 4 of a 4k -dimensional Poincaré complex .C; / NEW .C / 2 .C; / .2k 1/ .C / Z4 D C C 2 where the map 2 K1.Z/ Z2 Z4 takes . 1/ to 2 Z4 . W D ! 2 Geometry & Topology, Volume 11 (2007) Absolute Whitehead torsion 249 A simple rearrangement of the formula of the above lemma yields the first part of Proposition 37. References [1] I Hambleton, A Korzeniewski, A Ranicki, The signature of a fibre bundle is multi- plicative mod 4, Geom. Topol. 11 (2007) 251–314 [2] I Hambleton, A Ranicki, L Taylor, Round L–theory, J. Pure Appl. Algebra 47 (1987) 131–154 MR906966 [3] M Kervaire, Courbure integrale´ gen´ eralis´ ee´ et homotopie, Math. Ann. 131 (1956) 219–252 MR0086302 [4] G Lusztig, J Milnor, F P Peterson, Semi-characteristics and cobordism, Topology 8 (1969) 357–359 MR0246308 [5] A A Ranicki, Algebraic L–theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge University Press, Cambridge (1992) MR1211640 [6] A Ranicki, The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. .3/ 40 (1980) 87–192 MR560997 [7] A Ranicki, The algebraic theory of surgery. II. Applications to topology, Proc. London Math. Soc. .3/ 40 (1980) 193–283 MR566491 [8] A Ranicki, The algebraic theory of torsion. I. Foundations, from: “Algebraic and geometric topology (New Brunswick, N.J., 1983)”, Lecture Notes in Math. 1126, Springer, Berlin (1985) 199–237 MR802792 [9] A Ranicki, Additive L–theory, K–Theory 3 (1989) 163–195 MR1029957 [10] A Ranicki, High-dimensional knot theory, Springer Monographs in Mathematics, Springer, New York (1998) MR1713074 School of Mathematics, University of Edinburgh Edinburgh EH9 3JZ, United Kingdom [email protected] Proposed: Tom Goodwillie Received: 20th April 2005 Seconded: Wolfgang Lueck and Steve Ferry Accepted: 6th January 2007 Geometry & Topology, Volume 11 (2007)