Simulia Virtual Surgery

Introduction

Surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory, without losing sight of the geometric motivation.

0.1 Historical Background

A closed m-dimensional topological manifold M has Poincar´e duality

Isomorphisms

Hm−∗(M) ∼= H∗(M) . In order for a space X to be homotopy equivalent to an m-dimensional manifold it is thus necessary (but not in general sufficient) for X to be an m-dimensional Poincar´e duality space, with Hm−∗(X) ∼= H∗(X). The topological structure set ST OP (X) is defined to be the set of equivalence

Classes Of Pairs

(m-dimensional manifold M , homotopy equivalence h : M →X)

(M, H) ∼(M ′, H′) If There Exists A Homeomorphism

f : M →M ′ such that h′f ≃h : M →X. The basic problem of surgery theory is to decide if a Poincar´e complex X is homotopy equivalent to a manifold (i.e. if ST OP (X) is non-empty), and if so to compute ST OP (X) in terms of the algebraic topology of X.

Surgery theory was first developed for differentiable manifolds, and then extended to PL and topological manifolds. The classic Browder–Novikov–Sullivan–Wall obstruction theory for de- ciding if a Poincar´e complex X is homotopy equivalent to a manifold has

Andrew Ranicki

(i) the primary topological K-theory obstruction νX ∈[X, B(G/T OP)] to a T OP reduction eνX : X →BT OP of the Spivak normal fibration νX : X →BG, which vanishes if and only if there exists a manifold M with a normal map (f, b) : M →X, that is a degree 1 map f : M →X

Σ∗(F, B) ∈Lm(Z[Π1(X)])

in the surgery obstruction group of Wall , which is defined if the obstruction in (i) vanishes, and which depends on the choice of T OP reduction eνX, or equivalently on the bordism class of the normal map (f, b) : M →X. The surgery obstruction is such that σ∗(f, b) = 0 if (and for m ≥5 only if) (f, b) is normal bordant to a homotopy equivalence.

There exists a T OP reduction eνX of νX for which the corresponding normal map (f, b) : M →X has zero surgery obstruction if (and for m ≥5 only if) the structure set ST OP (X) is non-empty. A relative version of the theory gives a two-stage obstruction for deciding if a homotopy equivalence M →X from a manifold M is homotopic to a homeomorphism, which is traditionally formulated as the surgery exact sequence . . . →Lm+1(Z[π1(X)]) →ST OP (X) →[X, G/T OP] →Lm(Z[π1(X)]) .

See the paper by Browder for an account of the original Sullivan-Wall surgery exact sequence in the differentiable category in the case when X

Has The Homotopy Type Of A Differentiable Manifold

. . . →Lm+1(Z[π1(X)]) →SO(X) →[X, G/O] →Lm(Z[π1(X)]) . The algebraic L-groups L∗(Λ) of a ring with involution Λ are defined using quadratic forms over Λ and their automorphisms, and are 4-periodic Lm(Λ) = Lm+4(Λ) .

The surgery classification of exotic spheres of Kervaire and Milnor in- cluded the first computation of the L-groups, namely

0

if m ≡3 (mod 4) . The relationship between topological and PL manifolds was investigated using surgery methods in the 1960’s by Novikov, Casson, Sullivan, Kirby and Siebenmann (cf. Ranicki ), culminating in a disproof of the manifold Hauptvermutung : there exist homeomorphisms of PL manifolds which are not homotopic to PL homeomorphisms, and in fact there exist

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topological manifolds without PL structure. The surgery exact sequence for the PL manifold structure set SP L(M) for a PL manifold M was re- lated to the exact sequence for ST OP (M) by a commutative braid of exact

With

π∗(G/T OP) = L∗(Z) . Quinn gave a geometric construction of a spectrum of simplicial sets

A : H∗(X; L•(Z)) →L∗(Z[Π1(X)])

and for a manifold X the surgery obstruction function is given by

A

→Lm(Z[π1(X)]) . The surgery classifying spectra L•(Λ) and the assembly map A were con- structed algebraically in Ranicki for any ring with involution Λ, using quadratic Poincar´e complex n-ads over Λ. The spectrum L•(Z) is appropri- ate for the surgery classification of homology manifold structures (Bryant, Ferry, Mio and Weinberger ); for topological manifolds it is necessary to work with the 1-connective spectrum L• = L•(Z)⟨1⟩, such that Ln is n-connected with L0 ≃G/T OP.

A

−→Lm(Z[π1(X)]) →. . . . The algebraic surgery theory of Ranicki , provided one-stage ob-

Structions :

(i) An m-dimensional Poincar´e duality space X has a total surgery ob- struction s(X) ∈Sm(X) such that s(X) = 0 if (and for m ≥5 only if) X is homotopy equivalent to a manifold.

(ii) A homotopy equivalence of m-dimensional manifolds h : M ′ →M has a total surgery obstruction s(h) ∈Sm+1(M) such that s(h) = 0 if (and for m ≥5 only if) h is homotopic to a homeomorphism.

Moreover, if X is an m-dimensional manifold and m ≥5 the geometric surgery exact sequence is isomorphic to the algebraic surgery exact se-

Quence

. . .

/ Lm(Z[Π1(X)])

. . .

With

ST OP (X) →Sm+1(X) ; (M, h : M →X) 7−→s(h) . Given a normal map (f, b) : M m →X it is possible to kill an element x ∈πr(f) by surgery if and only if x can be represented by an embedding Sr−1 × Dn−r+1֒ →M with a null-homotopy in X, in which case the trace

Of The Surgery Is A Normal Bordism

((g, c); (f, b), (f ′, b′)) : (N; M, M ′) →X × ([0, 1]; {0}, {1})

N M+1 = M × I ∪Dr × Dm−R+1 ,

M ′m = (M\Sr−1 × Dm−r+1) ∪Dr × Sm−r . The normal map (f ′, b′) : M ′ →X is the geometric effect of the surgery on (f, b). Surgery theory investigates the extent to which a normal map can be made bordant to a homotopy equivalence by killing as much of π∗(f) as pos- sible. The original definition of the non-simply-connected surgery obstruc- tion σ∗(f, b) ∈Lm(Z[π1(X)]) (Wall ) was obtained after preliminary surgeries below the middle dimension, to kill the relative homotopy groups

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πr(f) for 2r ≤m. It could thus be assumed that (f, b) : M →X is [m/2]- connected, with πr(f) = 0 for 2r ≤m, and σ∗(f, b) was defined using the Poincar´e duality structure on the middle-dimensional homotopy kernel(s).

The surgery obstruction theory is much easier in the even-dimensional case m = 2n when πr(f) can be non-zero at most for r = m + 1 than in the odd-dimensional case m = 2n+1 when πr(f) can be non-zero for r = m+1 and r = m + 2.

Wall [28,§18G] asked for a chain complex formulation of surgery, in which the obstruction groups Lm(Λ) would appear as the cobordism groups of chain complexes with m-dimensional quadratic Poincar´e duality, by anal- ogy with the cobordism groups of manifolds Ω∗. Mishchenko initiated such a theory of “m-dimensional symmetric Poincar´e complexes” (C, φ) with C an m-dimensional f. g. free Λ-module chain complex

−−→C0

and φ a quadratic structure inducing m-dimensional Poincar´e duality iso- morphisms φ0 : H∗(C) →Hm−∗(C). The cobordism groups Lm(Λ) (which are covariant in Λ) are such that for any m-dimensional geometric Poincar´e complex X there is defined a symmetric signature invariant

Σ∗(X) = (C( E

X), φ) ∈Lm(Z[π1(X)]) . The corresponding quadratic theory was developed in Ranicki ; the m-dimensional quadratic L-groups Lm(Λ) for any m ≥0 were obtained as the groups of equivalence classes of “m-dimensional quadratic Poincar´e complexes” (C, ψ). The surgery obstruction of an m-dimensional normal map (f, b) : M m →X was expressed as a cobordism class

X) .

The symmetrization maps 1 + T : L∗(Λ) →L∗(Λ) are isomorphisms mod- ulo 8-torsion, and the symmetrization of the surgery obstruction is the

Difference Of The Symmetric Signatures

(1 + T )σ∗(f, b) = σ∗(M) −σ∗(X) ∈Lm(Z[π1(X)]) . However, the theory of is fairly elaborate. The algebra of and is used in these notes to simplify the original theory of Wall in the odd-dimensional case, without invoking the full theory of .

0.2 What Is In These Notes

These notes give an elementary account of the construction of the L- groups L∗and the surgery obstruction σ∗for differentiable manifolds. For

Andrew Ranicki

the more computational aspects of the L-groups see the papers by Ham- bleton and Taylor and Stark . The even-dimensional L-groups L2n(Λ) are the Witt groups of nonsin- gular (−1)n-quadratic forms over Λ.

It Is Relatively Easy To Pass From

an n-connected 2n-dimensional normal map (f, b) : M 2n →X to a (−1)n- quadratic form representing σ∗(f, b), and to see how the form changes under a surgery on (f, b). This will be done in §§1–5 of these notes.

The odd-dimensional L-groups L2n+1(Λ) are the stable automorphism groups of nonsingular (−1)n-quadratic forms over Λ. It is relatively hard to pass from an n-connected (2n + 1)-dimensional normal map (f, b) : M 2n+1 →X to an automorphism of a (−1)n-quadratic form representing σ∗(f, b), and even harder to follow through in algebra the effect of a surgery on (f, b). Novikov suggested the reformulation of the odd-dimensional theory in terms of the language of hamiltonian physics, and to replace the automorphisms by ordered pairs of lagrangians (= maximal isotropic sub- spaces). This reformulation was carried out in Ranicki , where such pairs were called ‘formations’, but it was still hard to follow the algebraic effects of individual surgeries. This became easier after the further refor- mulation of Ranicki in terms of chain complexes with Poincar´e duality – see §§8,9 for a description of how the kernel formation changes under a surgery on (f, b).

The original definition of L∗(Λ) in Wall was for the category of based f. g. free Λ-modules and simple isomorphisms, for surgery up to simple homotopy equivalence. Here, f. g. stands for finitely generated and simple means that the Whitehead torsion is trivial, as in the hypothesis of the s- cobordism theorem. These notes will only deal with free L-groups L∗(Λ) =

Lh

∗(Λ), the obstruction groups for surgery up to homotopy equivalence. The algebraic theory of ǫ-quadratic forms (K, λ, µ) over a ring Λ with an involution Λ →Λ; a 7→¯a is developed in §§1,2, with ǫ = ±1 and

An Ǫ-Quadratic Refinement Of Λ Such That

λ(x, x) = µ(x) + ǫµ(x) ∈Λ (x ∈K) .

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For an n-connected 2n-dimensional normal map (f, b) : M 2n →X geomet- ric (intersection, self-intersection) numbers define a (−1)n-quadratic form (Kn(M), λ, µ) on the kernel stably f. g. free Z[π1(X)]-module

X The Pullback Of E

X to M. The hyperbolic ǫ-quadratic form on a f. g. free Λ-module Λk

Λ : Λ2K × Λ2K →Λ ;

((a1, a2, . . . , a2k), (b1, b2, . . . , b2k)) 7−→

I=1

a2i−1¯a2i . The even-dimensional L-group L2n(Λ) is defined in §3 to be the abelian group of stable isomorphism classes of nonsingular (−1)n-quadratic forms on (stably) f. g.

free Λ-modules, where stabilization is with respect to the hyperbolic forms H(−1)n(Λk).

A Nonsingular (−1)N-Quadratic Form

(K, λ, µ) represents 0 in L2n(Λ) if and only if there exists an isomorphism

(K, Λ, Μ) ⊕H(−1)N(Λk) ∼= H(−1)N(Λk′)

for some integers k, k′ ≥0. The surgery obstruction of an n-connected 2n-dimensional normal map (f, b) : M 2n →X is defined by σ∗(f, b) = (Kn(M), λ, µ) ∈L2n(Z[π1(X)]) .

The algebraic effect of a geometric surgery on an n-connected 2n- dimensional normal map (f, b) is given in §5. Assuming that the result of the surgery is still n-connected, the effect on the kernel form of a surgery on Sn−1 × Dn+1֒ →M (resp. Sn × Dn֒ →M) is to add on (resp. split off) a hyperbolic (−1)n-quadratic form H(−1)n(Z[π1(X)]).

§6 introduces the notion of a “(2n + 1)-complex” (C, ψ), which is a f. g.

D

−−→Cn →0 →. . . together with a quadratic structure ψ inducing Poincar´e duality isomor- phisms (1 + T )ψ : H2n+1−∗(C) →H∗(C).

(This Is Just A (2N + 1)-

dimensional quadratic Poincar´e complex (C, ψ) in the sense of , with Cr = 0 for r̸ = n, n+1.) An n-connected (2n+1)-dimensional normal map

Andrew Ranicki

(f, b) : M 2n+1 →X determines a kernel (2n+ 1)-complex (C, ψ) (or rather a homotopy equivalence class of such complexes) with

X)) .

The cobordism of (2n+1)-complexes is defined in §7. The odd-dimensional L-group L2n+1(Λ) is defined in §8 as the cobordism group of (2n + 1)- complexes. The surgery obstruction of an n-connected normal map (f, b) : M 2n+1 →X is the cobordism class of the kernel complex σ∗(f, b) = (C, ψ) ∈L2n+1(Z[π1(X)]) .

The odd-dimensional L-group L2n+1(Λ) was originally defined in as a potentially non-abelian quotient of the stable unitary group of the matrices of automorphisms of hyperbolic (−1)n-quadratic forms over Λ

Autλh(−1)N(Λk)

and EU(−1)n(Λ) ⊳U(−1)n(Λ) the normal subgroup generated by the ele-



for any invertible matrix α, and any square matrix β. The group L2n+1(Λ)

Is Abelian, Since

[U(−1)n(Λ), U(−1)n(Λ)] ⊆EU(−1)n(Λ) . The surgery obstruction σ∗(f, b) ∈L2n+1(Z[π1(X)]) of an n-connected (2n + 1)-dimensional normal map (f, b) : M 2n+1 →X is represented by an automorphism of a hyperbolic (−1)n-quadratic form obtained from a high-dimensional generalization of the Heegaard decompositions of 3- dimensional manifolds as twisted doubles.

§8, §9 and §10 describe three equivalent ways of defining L2n+1(Λ), using unitary matrices, formations and chain complexes. In each case it is nec- essary to make some choices in passing from the geometry to the algebra, and to verify that the equivalence class in the L-group is independent of the choices.

The definition of L2n+1(Λ) using complexes given in §8 is a special case of the general theory of chain complexes with Poincar´e duality of Ranicki

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(given geometrically by taking product with C P 2, as in Chapter 9 of Wall ) was proved in using an algebraic analogue of surgery below the middle dimension: it is possible to represent every element of Lm(Λ) by a quadratic Poincar´e complex (C, ψ) which is “highly-connected”, meaning

R̸ = N, N + 1

if m = 2n + 1 . In these notes only the highly-connected (2n + 1)-dimensional quadratic Poincar´e complexes are considered, namely the “(2n+1)-complexes” of §6.

I am grateful to the referee for suggesting several improvements.

§1. Duality

§1 considers rings Λ equipped with an “involution” reversing the order of multiplication. An involution allows right Λ-modules to be regarded as left Λ-modules, especially the right Λ-modules which arise as the duals of left Λ- modules. In particular, the group ring Z[π1(M)] of the fundamental group π1(M) of a manifold M has an involution, which allows the Poincar´e duality

Of The Universal Cover F

M to be regarded as Z[π1(M)]-module isomorphisms.

X ; (G, X) 7−→Gx

induces a left action of the group ring Z[π] on the homology of e

X)

be the compactly supported cohomology groups, regarded as left Z[π]-

G∈Π

ngxg−1 .

X.) Cap Product

with any homology class [X] ∈Hm(X) defines Z[π]-module morphisms

X) .

Definition 1.1 An oriented m-dimensional geometric Poincar´e complex (Wall ) is a finite CW complex X with a fundamental class [X] ∈ Hm(X) such that cap product defines Z[π1(X)]-module isomorphisms

With E

X the universal cover of X. See 1.14 below for the general definition of a geometric Poincar´e complex, including the nonorientable case.

Example 1.2 A compact oriented m-dimensional manifold is an oriented m-dimensional geometric Poincar´e complex. In order to also deal with nonorientable manifolds and Poincar´e com-

Plexes It Is Convenient To Have An Involution:

Definition 1.3 Let Λ be an associative ring with 1. An involution on Λ

Satisfying

(a + b) = a + b , (ab) = b.a , a = a , 1 = 1 ∈Λ . Example 1.4 A commutative ring Λ admits the identity involution Λ →Λ ; a 7−→a = a .

Definition 1.5 Given a group π and a group morphism

W : Π →Z2 = {±1}

define the w-twisted involution on the integral group ring Λ = Z[π]

G∈Π

w(g)ngg−1 (ng ∈Z) .

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In the topological application π is the fundamental group of a space with w : π →Z2 an orientation character. In the oriented case w(g) = +1 for all g ∈π.

Example 1.6 Complex conjugation defines an involution on the ring of

Complex Numbers Λ = C

C →C ; z = a + ib 7−→z = a −ib . A “hermitian” form is a symmetric form on a (finite-dimensional) vector space over C with respect to this involution.

The Study Of Forms Over

rings with involution is sometimes called “hermitian K-theory”, although “algebraic L-theory” seems preferable. The dual of a left Λ-module K is the right Λ-module

With

K∗× Λ →K∗; (f, a) 7−→(x 7−→f(x).a) . An involution Λ →Λ; a 7−→¯a determines an isomorphism of categories

→{Left Λ-Modules} ; L 7−→Lop ,

with Lop the left Λ-module with the same additive group as the right Λ-

Module L And Λ Acting By

Λ × Lop →Lop ; (a, x) 7−→xa . From now on we shall work with a ring Λ which is equipped with a particular choice of involution Λ →Λ. Also, Λ-modules will always be understood to be left Λ-modules.

For any Λ-module K the Λ-module (K∗)op is written as K∗. Here is the

Definition Of K∗All At Once:

Definition 1.7 The dual of a Λ-module K is the Λ-module

Λ × K∗→K∗; (A, F) 7−→(X 7−→F(X).A)

for all a ∈Λ, f ∈K∗, x ∈K.

Andrew Ranicki

Definition 1.8 The dual of a Λ-module morphism f : K →L is the

.

Thus Duality Is A Contravariant Functor

∗: {Λ-modules} →{Λ-modules} ; K 7−→K∗. Definition 1.9 For any Λ-module K define the Λ-module morphism eK : K →K∗∗; x 7−→(f 7−→f(x)) .

The morphism eK is natural in the sense that for any Λ-module mor- phism f : K →L there is defined a commutative diagram

/ L∗∗

Definition 1.10 (i) A Λ-module K is f. g. projective if there exists a Λ-module L such that K ⊕L is isomorphic to the f. g. free Λ-module Λn, for some n ≥0.

(ii) A Λ-module K is stably f. g. free if K ⊕Λm is isomorphic to Λn, for some m, n ≥0. In particular, f. g. free Λ-modules are stably f. g. free, and stably f. g.

free Λ-modules are f. g. projective. Proposition 1.11 The dual of a f. g. projective Λ-module K is a f. g. projective Λ-module K∗, and eK : K →K∗∗is an isomorphism. Moreover, if K is stably f. g. free then so is K∗.

Proof : For any Λ-modules K, L there are evident identifications

Ek⊕L = Ek ⊕El : K ⊕L →(K ⊕L)∗∗= K∗∗⊕L∗∗,

so it suffices to consider the special case K = Λ. The Λ-module isomor-

→Λ∗; A 7−→(B 7−→Ba) ,

can be used to construct an explicit inverse for eΛ (eΛ)−1 : Λ∗∗→Λ ; g 7−→g(f(1)) .

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In dealing with f. g. projective Λ-modules K use the natural isomorphism eK : K ∼= K∗∗to identify K∗∗= K . For any morphism f : K →L of f. g. projective Λ-modules there is a corresponding identification f ∗∗= f : K∗∗= K →L∗∗= L .

Remark 1.12 The additive group HomΛ(Λm, Λn) of the morphisms Λm →Λn between f. g. free Λ-modules Λm, Λn may be identified with the additive group Mm,n(Λ) of m × n matrices (aij)1≤i≤m,1≤j≤n with en-

I=1

xiain)) .

Corresponds To The Multiplication Of Matrices

Mm,n(Λ) × Mn,p(Λ) →Mm,p(Λ) ; ((aij), (bjk)) 7−→(cik)

J=1

aijbjk (1 ≤i ≤m, 1 ≤k ≤p) ) .

Λm ∼=

→(Λm)∗; (x1, x2, . . . , xm) 7−→((y1, y2, . . . , ym) 7−→

To Identify

(Λm)∗= Λm .

F 7−→F ∗

can be identified with the isomorphism defined by conjugate transposition

∼=

→Mn,m(Λ) ; α = (aij) 7−→α∗= (bji) , bji = aij .



: Λ ⊕Λ →Λ ⊕Λ ; (x, y) 7−→(xa + yb, xc + yd) .

: (Λ ⊕Λ)∗= Λ ⊕Λ →(Λ ⊕Λ)∗= Λ ⊕Λ ;

(x, y) 7−→(xa + yc, xb + yd) . The dual of a chain complex of modules over a ring with involution Λ

/ Cr−1

/ . . .

D∗/ Cr+1

/ . . .

With

Cr = (Cr)∗= HomΛ(Cr, Λ) . Definition 1.14 An m-dimensional geometric Poincar´e complex (Wall ) is a finite CW complex X with an orientation character w(X) : π1(X) →Z2 and a w(X)-twisted fundamental class [X] ∈Hm(X; Zw(X)) such that cap product defines Z[π1(X)]-module isomorphisms

With E

X the universal cover of X. The w(X)-twisted cohomology groups

With C( E

X) the cellular Z[π1(X)]-module chain complex, using the w(X)- twisted involution on Z[π1(X)] (1.5) to define the left Z[π1(X)]-module

X)∗= Homz[Π1(X)](C( E

X), Z[π1(X)]) . The orientation character w(X) : π1(X) →Z2 sends a loop g : S1 →X to w(g) = +1 (resp. = −1) if g is orientation-preserving (resp. orientation- reversing).

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An oriented Poincar´e complex X (1.1) is just a Poincar´e complex (1.14) with w(X) = +1. Example 1.15 A compact m-dimensional manifold is an m-dimensional geometric Poincar´e complex.

§2. Quadratic Forms

In the first instance suppose that the ground ring Λ is commutative, with the identity involution ¯a = a (1.4). A symmetric form (K, λ) over Λ is a

Λ(X, Y + Z) = Λ(X, Y) + Λ(X, Z) ,

λ(x, y) = λ(y, x) ∈Λ . A quadratic form (K, λ, µ) over Λ is a symmetric form (K, λ) together with

Μ(X + Y) = Μ(X) + Μ(Y) + Λ(X, Y) ,

µ(ax) = a2µ(x) ∈Q+1(Λ) .

In Particular, For Every X ∈K

2µ(x) = λ(x, x) ∈Q+1(Λ) = Λ . If 2 ∈Λ is invertible (e.g. if Λ is a field of characteristic̸ = 2, such as R, C, Q) there is no difference between symmetric and quadratic forms, with µ determined by λ according to µ(x) = λ(x, x)/2.

A symplectic form (K, λ) over a commutative ring Λ is a Λ-module K together with a bilinear pairing λ : K ×K →Λ such that for all x, y, z ∈K

Λ(X, Y + Z) = Λ(X, Y) + Λ(X, Z) ,

λ(x, y) = −λ(y, x) ∈Λ . A (−1)-quadratic form (K, λ, µ) over Λ is a symplectic form (K, λ) together

Μ(X + Y) = Μ(X) + Μ(Y) + Λ(X, Y) ,

µ(ax) = a2µ(x) ∈Q−1(Λ) .

In Particular, For Every X ∈K

2µ(x) = λ(x, x) ∈Q−1(Λ) = {a ∈Λ | 2a = 0} . If 2 ∈Λ is invertible then Q−1(Λ) = 0 and there is no difference between symplectic and (−1)-quadratic forms, with µ = 0.

In the applications of forms to surgery theory it is necessary to work with quadratic and (−1)-quadratic forms over noncommutative group rings with the involution as in 1.5. §2 develops the general theory of forms over rings with involution, taking account of these differences.

Let X be an m-dimensional geometric Poincar´e complex with universal

Cover E

X and fundamental group ring Λ = Z[π1(X)], with the w(X)-twisted

Λ(Y, X) = (−1)R(M−R)Λ(X, Y)

with Λ →Λ; a 7−→¯a the involution of 1.5. If M is an m-dimensional manifold with fundamental group ring Λ =

M) →Λ Can Be Interpreted

geometrically using the geometric intersection numbers of cycles. For any

Ngg ∈Λ

with ng ∈Z the algebraic number of intersections in f

M Of X And Gy. In

particular, for m = 2n there is defined a (−1)n-symmetric pairing

M) →Λ

which is relevant to surgery in the middle dimension n.

An Element

x ∈πn(M) can be killed by surgery if and only if it is represented by

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an embedding Sn × Dn֒ →M 2n. The condition that the Hurewicz image

X ∈Hn(F

M) be such that λ(x, x) = 0 ∈Λ is necessary but not sufficient to kill x ∈πn(M) by surgery. The theory of forms developed in §2 is required for an algebraic formulation of the necessary and sufficient condition for an element in the kernel Kn(M) of an n-connected 2n-dimensional normal map (f, b) : M →X to be killed by surgery, assuming n ≥3.

As in §1 let Λ be a ring with involution, not necessarily a group ring. Definition 2.1 A sesquilinear pairing (K, L, λ) on Λ-modules K, L is a

Such That For All W, X ∈K, Y, Z ∈L, A, B ∈Λ

(i) λ(w + x, y + z) = λ(w, y) + λ(w, z) + λ(x, y) + λ(x, z) ∈Λ , (ii) λ(ax, by) = bλ(x, y)a ∈Λ .

The Dual (Or Transpose) Sesquilinear Pairing Is

T λ : L × K →Λ ; (y, x) 7−→T λ(y, x) = λ(x, y) . Definition 2.2 Given Λ-modules K, L let S(K, L) be the additive group of sesquilinear pairings λ : K ×L →Λ. Transposition defines an isomorphism

→S(K, L) .

Proposition 2.3 For any Λ-modules K, L there is a natural isomorphism

→Homλ(K, L∗) ;

(λ : K × L →Λ) 7−→(λ : K →L∗; x 7−→( y 7−→λ(x, y) )) . For f. g. projective K, L the transposition isomorphism T : S(K, L) ∼=

→Homλ(L, K∗) ;

(λ : K →L∗) 7−→(λ∗: L →K∗; y 7−→(x 7−→λ(y, x))) .

Qǫ(K) = Coker(1 −Tǫ : Homλ(K, K∗) →Homλ(K, K∗))

for any f. g. projective Λ-modules K, L. Remark 2.4 For f. g. free Λ-modules Λm, Λn it is possible to identify S(Λm, Λn) with the additive group Mm,n(Λ) of m × n matrices (aij) with

J=1

yjaijxi . The transposition isomorphism T : S(Λm, Λn) ∼= S(Λn, Λm) corresponds to the isomorphism defined by conjugate transposition of matrices

∼=

→Mn,m(Λ) ; (aij) 7−→(bji) , bji = aij . The group S(K, L) is particularly significant in the case K = L :

S(K) = S(K, K)

be the abelian group of sesquilinear pairings λ : K × K →Λ. (ii) The ǫ-transposition involution is given for ǫ = ±1 by

Such That

Tǫλ(x, y) = ǫλ(y, x) ∈Λ , (Tǫ)2 = id : S(K) →S(K) . Definition 2.6 The ǫ-symmetric group of a Λ-module K is the additive

Group

Qǫ(K) = ker(1 −Tǫ : S(K) →S(K)) .

The Ǫ-Quadratic Group Of K Is The Additive Group

Qǫ(K) = coker(1 −Tǫ : S(K) →S(K)) .

The Ǫ-Symmetrization Morphism Is Given By

1 + Tǫ : Qǫ(K) →Qǫ(K) ; ψ 7−→ψ + Tǫψ .

19

For ǫ = +1 it is customary to refer to ǫ-symmetric and ǫ-quadratic objects as symmetric and quadratic, as in the commutative case. For K = Λ there is an isomorphism of additive groups with involution

Qǫ(Λ) = Λ/{A −Ǫa | A ∈Λ} ,

1 + Tǫ : Qǫ(Λ) →Qǫ(Λ) ; a 7−→a + ǫa . Example 2.7 Let Λ = Z.

1 + T+ = 2 : Q+1(Z) = Z →Q+1(Z) = Z .

Definition 2.8 An ǫ-symmetric form (K, λ) over Λ is a Λ-module K to- gether with an element λ ∈Qǫ(K). Thus λ is a sesquilinear pairing

Such That For All X, Y ∈K

λ(x, y) = ǫλ(y, x) ∈Λ .

K →K∗; X 7−→(Y 7−→Λ(X, Y))

which is also denoted by λ. The form is nonsingular if λ : K →K∗is an isomorphism. Unless specified otherwise, only forms (K, λ) with K a f. g. projective Λ-module will be considered.

Λ = 1 : Λ →Λ∗; A 7−→(B 7−→Ba)

is nonsingular.

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Definition 2.10 For any f. g. projective Λ-module L define the nonsin-

: L ⊕L∗→(L ⊕L∗)∗= L∗⊕L ;

(x, f) 7−→((y, g) 7−→f(y) + ǫg(x)) . Example 2.11 Let X be an m-dimensional geometric Poincar´e complex,

And Let E

X be a regular oriented covering of X with group of covering trans- lations π and orientation character w : π →Z2. An element g ∈π has w(g) = +1 (resp. −1) if and only if the covering translation g : e

X Is

orientation-preserving (resp. reversing). (i) Cap product with the fundamental class [X] ∈Hm(X; Zw) defines the

X) .

If m = 2n and X is a manifold geometric intersection numbers define a

(Ii) In General Hn( E

X) is not a f. g. projective Z[π]-module. If Hn( e X) is f. g. projective then the evaluation map is an isomorphism, and (Hn( e

X), Λ)

is a nonsingular form. Remark 2.12 (i) Let M be a 2n-dimensional manifold, with universal

M)) Can Be Killed By Surgery If And Only

if it can be represented by an embedding x : Sn × Dn֒ →M, in which

M) Is Such That Λ(X, X) = 0. However,

the condition λ(x, x) = 0 given by the symmetric structure alone is not sufficient for the existence of such an embedding – see (ii) below for an explicit example.

(ii) The intersection form over Z for M 2n = Sn ×Sn is the hyperbolic form

(2.10)

(Hn(Sn × Sn), λ) = H(−1)n(Z) .

Λ(X, X) = Χ(Sn) = 1 + (−1)N ∈Z ,

so that λ(x, x) = 0 for odd n. The diagonal embedding ∆: Sn֒ →Sn × Sn has normal bundle ν∆= τSn : Sn →BO(n), which is non-trivial for

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n̸ = 1, 3, 7, so that it is not possible to kill x = ∆∗[Sn] ∈Hn(Sn × Sn) by surgery in these dimensions. Definition 2.13 An ǫ-quadratic form (K, λ, µ) over Λ is an ǫ-symmetric

(I) Μ(X + Y) −Μ(X) −Μ(Y) = Λ(X, Y) ∈Qǫ(Λ) ,

(ii) µ(x) + ǫµ(x) = λ(x, x) ∈im(1 + Tǫ : Qǫ(Λ) →Qǫ(Λ)) , (iii) µ(ax) = aµ(x)a ∈Qǫ(Λ) . Definition 2.14 For any f. g. projective Λ-module L define the nonsin-

(X, F) 7−→((Y, G) 7−→F(Y) + Ǫg(X)) ,

µ : L ⊕L∗→Qǫ(Λ) ; (x, f) 7−→f(x) . (L ⊕L∗, λ) is the hyperbolic ǫ-symmetric form Hǫ(L) of 2.10. Example 2.15 (Wall [28, Chapter 5]) An n-connected normal map (f, b) : M 2n →X from a 2n-dimensional manifold with boundary (M, ∂M) to a geometric Poincar´e pair (X, ∂X) with ∂f = f| : ∂M →∂X a homotopy equivalence determines a (−1)n-quadratic form (Kn(M), λ, µ) over Λ = Z[π1(X)] with the w(X)-twisted involution (1.5), with

X A Π1(X)-Equivariant

lift of f to the universal covers.

Note That Kn(M) = 0 If (And For

n ≥2 only if) f : M →X is a homotopy equivalence, by the theorem of J.H.C. Whitehead. (i) The pairing λ : Kn(M) × Kn(M) →Λ is defined by geometric intersec- tion numbers, as follows. Every element x ∈Kn(M) is represented by an X-nullhomotopic framed immersion g : Sn ↬M with a choice of path in g(Sn) ⊂M from the base point ∗∈M to g(1) ∈M. Any two elements x, y ∈Kn(M) can be represented by such immersions g, h : Sn ↬M with transverse intersections and self-intersections. The intersection of g and h

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is finite. For each intersection point (a, b) ∈D(g, h) let

Γ(A, B) ∈Π1(M) = Π1(X)

be the homotopy class of the loop in M obtained by joining the path in g(Sn) ⊂M from the base point ∗∈M to g(a) to the path in h(Sn) ⊂M from h(b) back to the base point. Choose an orientation for τ∗(M) and transport it to an orientation for τg(a)(M) = τh(b)(M) by the path for g,

→Τg(A)(M)

is orientation-preserving (resp. reversing). The geometric intersection of

With

I(a, b) = ǫ(a, b)γ(a, b) ∈Λ .

Λ(Y, X) = (−1)Nλ(X, Y) ∈Λ

(which also holds from purely homological considerations). (ii) The quadratic function µ : Kn(M) →Q(−1)n(Λ) is defined by geo- metric self-intersection numbers, as follows. Represent x ∈Kn(M) by an immersion g : Sn ↬M as in (i), with transverse self-intersections. The

= {(A, B) ∈Sn × Sn | A̸ = B ∈Sn, G(A) = G(B) ∈M}

is finite, with a free Z2-action (a, b) 7→(b, a). For each (a, b) ∈D2(g) let γ(a, b) be the loop in M obtained by transporting to the base point the image under g of a path in Sn from a to b. The geometric self-intersection

I(A, B) ∈Q(−1)N(Λ) ,

with I(a, b) = ǫ(a, b)γ(a, b) as in (i). Note that µ(x) is independent of the choice of ordering of (a, b) since I(b, a) = (−1)nI(a, b) ∈Λ. (iii) The kernel (−1)n-quadratic form (Kn(M), λ, µ) is such that µ(x) = 0 if (and for n ≥3 only if) x ∈Kn(M) can be killed by surgery on Sn ⊂M 2n, i.e. represented by an embedding Sn × Dn֒ →M with a nullhomotopy in X – the condition µ(x) = 0 allows the double points of a representative framed immersion g : Sn ↬M to be matched in pairs, which for n ≥3 can be cancelled by the Whitney trick. The effect of the surgery is a bordant

(N −1)-Connected Normal Map

(f ′, b′) : M ′2n = cl.(M\Sn × Dn) ∪Dn+1 × Sn−1 →X

0

otherwise . Thus (f ′, b′) is n-connected if and only if x generates a direct summand L = ⟨x⟩⊂Kn(M), in which case L is a sublagrangian of (Kn(M), λ, µ) in

(Kn(M ′), Λ′, Μ′) = (L⊥/L, [Λ], [Μ]) ,

(Kn(M), λ, µ) ∼= (Kn(M ′), λ′, µ′) ⊕H(−1)n(Λ) . (iv) The effect on (f, b) of a surgery on an X-nullhomotopic embedding Sn−1 × Dn+1֒ →M is an n-connected bordant normal map (f ′′, b′′) : M ′′2n = cl.(M\Sn−1×Dn+1)∪Dn×Sn = M#(Sn×Sn) →X

And Kernel (−1)N-Quadratic Form

(Kn(M ′′), λ′′, µ′′) = (Kn(M), λ, µ) ⊕H(−1)n(Λ) . (v) The main result of even-dimensional surgery obstruction theory is that for n ≥3 an n-connected 2n-dimensional normal map (f, b) : M 2n →X is normal bordant to a homotopy equivalence if and only if there exists an isomorphism of (−1)n-quadratic forms over Λ = Z[π1(X)] of the type

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for some k, k′ ≥0. Example 2.16 There is also a relative version of 2.15. An n-connected 2n-dimensional normal map of pairs (f, b) : (M 2n, ∂M) →(X, ∂X) has a kernel (−1)n-quadratic form (Kn(M), λ, µ) over Z[π1(X)] is nonsingular if and only if ∂f : ∂M →∂X is a homotopy equivalence (assuming π1(∂X) ∼= π1(X)).

Remark 2.17 (Realization of even-dimensional surgery obstructions, Wall

[28, 5.8])

(i) Let X2n−1 be a (2n −1)-dimensional manifold, and suppose given an embedding e : Sn−1 × Dn֒ →X, together with a null-homotopy δe of e| : Sn−1֒ →X and a null-homotopy of the map Sn−1 →O comparing the (stable) trivializations of νe| : Sn−1 →BO(n) given by e and δe. Then there is defined an n-connected 2n-dimensional normal map

M 2N = X × [0, 1] ∪E Dn × Dn ,

∂+M = cl.(X\e(Sn−1 × Dn)) ∪Dn × Sn−1 . The kernel (−1)n-quadratic form (Λ, λ, µ) over Λ (2.16) is the (self-)- intersection of the framed immersion Sn−1 × [0, 1] ↬X × [0, 1] defined by the track of a regular homotopy e0 ≃e : Sn−1 × Dn →X from a trivial

Unlinked Embedding

e0 : Sn−1×Dn֒ →S2n−1 = Sn−1×Dn∪Dn×Sn−1֒ →X#S2n−1 = X . Moreover, every form (Λ, λ, µ) arises in this way : starting with e0 construct a regular homotopy e0 ≃e to a (self-)linked embedding e such that the track has (self-)intersection (λ, µ).

(ii) Let (K, λ, µ) be a (−1)n-quadratic form over Z[π], with π a finitely presented group and K = Z[π]k f. g. free. Let n ≥3, so that there exists a (2n−1)-dimensional manifold X2n−1 with π1(X) = π. For any such n ≥3, X there exists an n-connected 2n-dimensional normal map (f, b) : (M 2n; ∂−M, ∂+M) →X2n−1 × ([0, 1]; {0}, {1})

Kn−1(∂+M) = Coker(Λ : K →K∗) ,

Kn(∂+M) = ker(λ : K →K∗) . The map ∂+f : ∂+M →X is a homotopy equivalence if and only if the form (K, λ, µ) is nonsingular. Given (K, λ, µ), X the construction of (f, b)

25

proceeds as in (i). Example 2.18 For π1(X) = {1} the realization of even-dimensional surgery obstructions (2.17) is essentially the same as the Milnor , construction of (n −1)-connected 2n-dimensional manifolds by plumbing together n-plane bundles over Sn. Let G be a finite connected graph with- out loops (= edges joining a vertex to itself), with vertices v1, v2, . . . , vk.

Suppose given an oriented n-plane bundle over Sn at each vertex

Ω1, Ω2, . . . , Ωk ∈Πn(Bso(N)) = Πn−1(So(N)) ,

regarded as a weight. Let (Zk, λ) be the (−1)n-symmetric form over Z defined by the (−1)n-symmetrized adjacency matrix of G and the Euler

If I = J ,

λ : Zk × Zk →Z ; ((x1, x2, . . . , xk), (y1, y2, . . . , yk)) 7→

J=1

λijxiyj . The graph G and the Euler numbers χ(ωi) determine and are determined by the form (Zk, λ).

(i) See Browder [1, Chapter V] for a detailed account of the plumbing con- struction which uses G to glue together the (Dn, Sn−1)-bundles (Dn, Sn−1) →(E(ωi), S(ωi)) →Sn (i = 1, 2, . . . , k) to obtain a connected 2n-dimensional manifold with boundary

E(Ωi))/ ∼

with 1-skeleton homotopy equivalent to G, fundamental group

Π1(P) = Π1(G) = ∗Gz

the free group on g = 1 −χ(G) generators, homology

Otherwise ,

and intersection form (Hn(P), λ). Killing π1(P) by surgeries removing g embeddings S1 ×D2n−1 ⊂P representing the generators, there is obtained

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an (n −1)-connected 2n-dimensional manifold with boundary

0

otherwise .

If G Is A Tree Then G = 0, Π1(P(G, Ω)) = {1}, And

(M(G, ω), ∂M(G, ω)) = (P(G, ω), ∂P(G, ω)) . (ii) By Wall for n ≥3 an integral (−1)n-symmetric matrix (λij)1≤i,j≤k

(A) For 1 ≤I < J ≤K

linking number(xi(Sn−1 × 0) ∩xj(Sn−1 × 0)֒ →S2n−1) = λij ∈Z , (b) for 1 ≤i ≤k xi : Sn−1 × Dn֒ →S2n−1 is isotopic to the embedding

Eωi : Sn−1 × Dn֒ →S2N−1 = Sn−1 × Dn ∪Dn × Sn−1 ;

(s, t) 7→(s, ωi(s)(t)) . Using x to attach k n-handles to D2n there is obtained an oriented (n−1)-

N-Handles Dn × Dn

with boundary an oriented (n−2)-connected (2n−1)-dimensional manifold

K

Dn × Sn−1 .

27

Moreover, every oriented (n −1)-connected 2n-dimensional manifold with non-empty (n−2)-connected boundary is of the form (M(G, ω), ∂M(G, ω)), with (λij, ωi) the complete set of diffeomorphism invariants.

(iii) Stably trivialized n-plane bundles over Sn are classified by Q(−1)n(Z),

Δτsn : Τsn ⊕Ǫ ∼= Ǫn+1

the stable trivialization given by the standard embedding Sn ⊂Sn+1. The

Map

Q(−1)n(Z) = πn+1(BSO, BSO(n)) →πn(BSO(n)) ; 1 7→τSn is an injection for n̸ = 1, 3, 7. With G as above, suppose now that the vertices v1, v2, . . . , vk are weighted by elements µ1, µ2, . . . , µk ∈πn+1(BSO, BSO(n)) = Q(−1)n(Z) .

= Ker(Πn(Bso(N)) →Πn(Bso))

and let (Zk, λ, µ) be the (−1)n-quadratic form over Z with λ as before and

Such That

λii = χ(ωi) = (1 + (−1)n)µi ∈Z .

The (N −1)-Connected 2N-Dimensional Manifold

M(G, µ1, µ2, . . . , µn) = M(G, ω1, ω2, . . . , ωn) is stably parallelizable, with an n-connected normal map (M(G, µ1, µ2, . . . , µn), ∂M(G, µ1, µ2, . . . , µn)) →(D2n, S2n−1) with kernel form (Zk, λ, µ).

(iv) For n ≥3 the realization of a (−1)n-quadratic form (Zk, λ, µ) over Z (2.17) is an n-connected 2n-dimensional normal map

(F, B) : (M 2N; S2N−1, ∂+M)

= (cl.(M(G, µ1, . . . , µk)\D2n); S2n−1, ∂M(G, µ1, . . . , µk)) →S2n−1 × ([0, 1]; {0}, {1}) . with kernel form (Zk, λ, µ). If (Zk, λ, µ) is nonsingular then

Andrew Ranicki

is a homotopy equivalence, and Σ2n−1 is a homotopy sphere with a poten- tially exotic differentiable structure (Milnor , Kervaire and Milnor ) – see 2.20, 3.6 and 3.7 below.

Example 2.19 (i) Consider the special case k = 1 of 2.18 (i). Here G =

Ω ∈Πn(Bso(N)) = Πn−1(So(N))

classifies an n-plane bundle over Sn. The plumbed (n −1)-connected 2n- dimensional manifold with boundary is the (Dn, Sn−1)-bundle over Sn

= D2N ∪Eω Dn × Dn

obtained from D2n by attaching an n-handle along the embedding

Eω : Sn−1 × Dn֒ →S2N−1 = Sn−1 × Dn ∪Dn × Sn−1 ;

(x, y) 7→(x, ω(x)(y)) . (ii) Consider the special case k = 1 of 2.18 (iii), the realization of a (−1)n- quadratic form (Z, λ, µ) over Z, with G = {v1} as in (i). An element

If N ≡1(Mod 2)

classifies an n-plane bundle ω : Sn →BSO(n) with a stable trivialization

And

(M(G, µ), ∂M(G, µ)) = (E(ω), S(ω)) . For n̸ = 1, 3, 7 δω is determined by ω. For even n µ ∈Q+1(Z) = Z and

/ Bso(N)

is the unique stably trivial n-plane bundle over Sn with Euler number χ(ω) = 2µ ∈Z .

Ǫn

if µ = 0 .

29

and δω is the (stable) trivialization of ω with mod 2 Hopf invariant µ. The plumbed (n −1)-connected 2n-dimensional manifold (M(G, µ), ∂M(G, µ)) = (M(G, ω), ∂M(G, ω)) = (E(ω), S(ω)) (as in (i)) is stably parallelizable. The trace of the surgery on the normal

(F−, B−) = Id. : ∂−Mω = S2N−1 →S2N−1

killing eω : Sn−1 × Dn֒ →S2n−1 is an n-connected 2n-dimensional normal

Kn(Mω) = Z

and kernel form (Z, λ, µ). If µ = 0 ∈Q(−1)n(Z) then

Ω = Τsn : Sn →Bso(N) ,

∂+Mω = S(τSn) = O(n + 1)/O(n −1) . (iii) Consider the special case k = 2 of 2.18 (i), with G = I the graph with

1 Edge And 2 Vertices

...................................................

I

For any weights ω1, ω2 ∈πn(BSO(n)) there is obtained an (n −1)-

M(I, Ω1, Ω2) = D2N ∪Eω1 ∪Eω2 (Dn × Dn ∪Dn × Dn)

by plumbing as in Milnor , , with intersection form the (−1)n-

Λ : Z ⊕Z × Z ⊕Z →Z ;

((x1, x2), (y1, y2)) 7→χ(ω1)x1y1 + χ(ω2)x2y2 + x1y2 + (−1)nx2y1 . (iv) Consider the special case k = 2 of 2.18 (iii), with G = I as in (iii). For

Ωi = [Μi] ∈Im(Q(−1)N(Z) →Πn(Bso(N)))

the (−1)n-quadratic form (Z ⊕Z, λ, µ) over Z defined by µ : Z ⊕Z →Q(−1)n(Z) ; (x1, x2) 7→µ1(x1)2 + µ2(x2)2 + x1x2

Andrew Ranicki

is the kernel form of an n-connected 2n-dimensional normal map (f, b) : M(I, µ1, µ2) = M(I, ω1, ω2) →D2n . If µ1 = µ2 = 0 then (Z⊕Z, λ, µ) = H(−1)n(Z) is hyperbolic (−1)n-quadratic

Form Over Z, With

λ : Z ⊕Z × Z ⊕Z →Z ; ((x1, x2), (y1, y2)) 7→x1y2 + (−1)nx2y1 , µ : Z ⊕Z →Q(−1)n(Z) = Z/{1 + (−1)n−1} ; (x1, x2) 7→x1x2 ,

And The Plumbed Manifold Is A Punctured Torus

(M(I, 0, 0)2n, ∂M(I, 0, 0)) = (cl.(Sn × Sn\D2n), S2n−1) . The hyperbolic form is the kernel of the n-connected 2n-dimensional normal

Map

(f, b) : (M; ∂−M, ∂+M) = (cl.(M(I, 0, 0)\D2n); S2n−1, S2n−1)

→S2N−1 × ([0, 1]; {0}, {1})

defined by the trace of surgeries on the linked spheres

Sn−1 ∪Sn−1֒ →S2N−1 = Sn−1 × Dn ∪Dn × Sn−1

with no self-linking.

M(I, 0, 0) = D2N ∪Dn × Dn ∪Dn × Dn ,

using the standard framings of Sn−1 ⊂S2n−1. If n is odd, say n = 2k + 1, and µ0 = µ1 = 1 ∈Q−1(Z) the form in 2.18 (i) is just the Arf (−1)-

Quadratic Form Over Z (Z ⊕Z, Λ, Μ′) With

µ′ : Z ⊕Z →Q(−1)(Z) = Z2 ; (x, y) 7−→x2 + xy + y2 .

The Plumbed Manifold

M(I, 1, 1)4k+2 = D4k+2 ∪D2k+1 × D2k+1 ∪D2k+1 × D2k+1 has the same attaching maps for the cores of the (2k + 1)-handles as M(I, 0, 0), but now using the framings of S2k ⊂S4k+1 classified by

Τs2K+1 ∈Π2K+1(Bso(2K + 1)) = Π2K(So(2K + 1))

(which is zero if and only if k = 0, 1, 3). The Arf form is the kernel of the (2k + 1)-connected (4k + 2)-dimensional normal map (f ′, b′) : (M ′; ∂−M ′, ∂+M ′) = (cl.(M(I, 1, 1)\D4k+2); S4k+1, Σ4k+1)

→S4K+1 × ([0, 1]; {0}, {1})

defined by the trace of surgeries on the linked spheres

S2K ∪S2K֒ →S4K+1 = S2K × D2K+1 ∪D2K+1 × S2K ,

with self-linking given by the non-standard framing. (See 3.7 below for a brief account of the exotic sphere Σ4k+1).

31

Example 2.20 The sphere bundles S(ω) of certain oriented 4-plane bundles ω over S4 (the special case n = 4 of 2.19 (i)) give explicit exotic 7-spheres. An oriented 4-plane bundle ω : S4 →BSO(4) is determined by the Euler

∼=

→Z ⊕Z ; ω 7−→((2χ(ω) + p1(ω))/4, (2χ(ω) −p1(ω))/4) .

P1(Ω) = 2ℓ∈H4(S4) = Z

for some odd integer ℓ. The 7-dimensional differentiable manifold Σ7

ℓ=

S(ω) is homeomorphic to S7 (by Smale’s generalized Poincar´e conjecture,

ℓd8

is a closed 8-dimensional differentiable manifold with

P1(M) = P1(Ω) = 2ℓ∈H4(M) = Z ,

σ(M) = 1 ∈Z .

= (7P2(M) −P1(M)2)/45

= (7p2(M) −4ℓ2)/45 = 1 ∈Z .

ℓis An Exotic 7-Sphere (Mil-

nor , Milnor and Stasheff[14, p.247]). Definition 2.21 An isomorphism of ǫ-symmetric forms

Is A Λ-Module Isomorphism F : K ∼= K′ Such That

λ′(f(x), f(y)) = λ(x, y) ∈Λ . An isomorphism of ǫ-quadratic forms f : (K, λ, µ) ∼= (K′, λ′, µ′) is an iso- morphism of the underlying ǫ-symmetric forms f : (K, λ) ∼= (K′, λ′) such

That

µ′(f(x)) = µ(x) ∈Qǫ(Λ) .

Andrew Ranicki

Proposition 2.22 If there exists a central element s ∈Λ such that

There Is An Identification Of Categories

{ǫ-quadratic forms over Λ} = {ǫ-symmetric forms over Λ} . Proof : The ǫ-symmetrization map 1+Tǫ : Qǫ(K) →Qǫ(K) is an isomor-

Phism For Any Λ-Module K, With Inverse

Qǫ(K) →Qǫ(K) ; λ 7−→((x, y) 7−→sλ(x, y)) . For any ǫ-quadratic form (K, λ, µ) the ǫ-quadratic function µ is determined

By The Ǫ-Symmetric Pairing Λ, With

µ(x) = sλ(x, x) ∈Qǫ(Λ) . Example 2.23 If 2 ∈Λ is invertible then 2.22 applies with s = 1/2 ∈Λ.

For Any Ǫ-Symmetric Form (K, Λ) And X ∈K

λ(x, x) ∈Qǫ(Λ) . Definition 2.24 An ǫ-symmetric form (K, λ) is even if for all x ∈K λ(x, x) ∈im(1 + Tǫ : Qǫ(Λ) →Qǫ(Λ)) .

Proposition 2.25 Let ǫ = 1 or −1. If the ǫ-symmetrization map

1 + Tǫ : Qǫ(Λ) →Qǫ(Λ)

is an injection there is an identification of categories {ǫ-quadratic forms over Λ} = {even ǫ-symmetric forms over Λ} . Proof : Given an even ǫ-symmetric form (K, λ) over Λ there is a unique

(1 + Tǫ)(Μ(X)) = Λ(X, X) ∈Qǫ(Λ) ,

which then automatically satisfies the conditions of 2.13 for (K, λ, µ) to be an ǫ-quadratic form.

1 + T = 2 : Q+1(Z) = Z →Q+1(Z) = Z

is an injection, so that quadratic forms over Z coincide with the even sym- metric forms.

33

is not an injection, so that (−1)-quadratic forms over Z have a richer struc- ture than even (−1)-symmetric forms.

Λ : K × K →Z ; ((A, B), (C, D)) 7−→Ad −Bc

admits two distinct (−1)-quadratic refinements (K, λ, µ), (K, λ, µ′), with

Μ : K →Q−1(Z) = Z/2 ; (X, Y) 7−→Xy ,

µ′ : K →Q−1(Z) = Z/2 ; (x, y) 7−→x2 + xy + y2 . See §3 below for the definition of the Arf invariant, which distinguishes the hyperbolic (−1)-quadratic form (K, λ, µ) = H−1(Z) from the Arf form (K, λ, µ′) (which already appeared in 2.19 (iv)).

§3. The Even-Dimensional L-Groups

The even-dimensional surgery obstruction groups L2n(Λ) will now be defined, using the following preliminary result. Lemma 3.1 For any nonsingular ǫ-quadratic form (K, λ, µ) there is defined

(K, Λ, Μ) ⊕(K, −Λ, −Μ) ∼= Hǫ(K) ,

with Hǫ(K) the hyperbolic ǫ-quadratic form (2.14). Proof : Let L be a f. g. projective Λ-module such that K ⊕L is f. g. free,

Λij = (Λ ⊕0)(Xj, Xi) ∈Λ (1 ≤I < J ≤K)

and choose representatives µi ∈Λ of µ(xi) ∈Qǫ(Λ) (1 ≤i ≤k). Define

1≤I

bjλijai) .

Λ = Ψ + Ǫψ∗: K →K∗,

µ(x) = ψ(x, x) ∈Qǫ(Λ) (x ∈K) . As (K, λ, µ) is nonsingular ψ + ǫψ∗: K →K∗is an isomorphism. The

Is Such That

(ψ + ǫψ∗)−1 = eψ + ǫ eψ∗: K∗→K .

: K ⊕K∗→K ⊕K .

Definition 3.2 The 2n-dimensional L-group L2n(Λ) is the group of equiv- alence classes of nonsingular (−1)n-quadratic forms (K, λ, µ) on stably f. g. free Λ-modules, subject to the equivalence relation

(K, Λ, Μ) ∼(K′, Λ′, Μ′)

if there exists an isomorphism of (−1)n-quadratic forms

→(K′, Λ′, Μ′) ⊕H(−1)N(Λk′)

for some f. g. free Λ-modules Λk, Λk′.

Addition And Inverses In L2N(Λ) Are Given By

(K1, λ1, µ1) + (K2, λ2, µ2) = (K1 ⊕K2, λ1 ⊕λ2, µ1 ⊕µ2) , −(K, λ, µ) = (K, −λ, −µ) ∈L2n(Λ) . The groups L2n(Λ) only depend on the residue n(mod 2), so that only two L-groups have actually been defined, L0(Λ) and L2(Λ). Note that 3.2 uses Lemma 3.1 to justify (K, λ, µ) ⊕(K, −λ, −µ) ∼0.

Remark 3.3 The surgery obstruction of an n-connected 2n-dimensional

Σ∗(F, B) = (Kn(M), Λ, Μ) ∈L2N(Z[Π1(X)])

such that σ∗(f, b) = 0 if (and for n ≥3 only if) (f, b) is normal bordant to a homotopy equivalence.

G Be The Orientable 2-Manifold (= Surface) Of

genus g, with degree 1 map f : M →S2.

A Choice Of Framing Of The

stable normal bundle of an embedding M֒ →R3 determines a 1-connected 2-dimensional normal map (f, b) : M →S2.

For A Standard Choice Of

framing (i.e. one which extends to a 3-manifold N with ∂N = M) the kernel form and the surgery obstruction are given by

35

and (f, b) is normal bordant to a homotopy equivalence, i.e. M is framed null-cobordant. Example 3.5 The even-dimensional L-groups of the ring Λ = R of real

0

if n is odd . Since 1/2 ∈R there is no difference between symmetric and quadratic forms over R.

The signature (alias index) of a nonsingular symmetric form (K, λ) over

R Is Defined By

Here, the symmetric form λ ∈Q+1(K) is identified with the symmetric k × k matrix (λ(xi, xj)1≤i,j≤k) ∈Mk,k(R) determined by any choice of ba- sis x1, x2, . . . , xk for K. By Sylvester’s law of inertia the rank k and the signature σ(K, λ) define a complete set of invariants for the isomorphism classification of nonsingular symmetric forms (K, λ) over R, meaning that two forms are isomorphic if and only if they have the same rank and sig- nature. A nonsingular quadratic form (K, ψ) over R is isomorphic to a hyperbolic form if and only if it has signature 0. Two such forms (K, λ),

If And Only If They Have The Same Signature

σ(K, λ) = σ(K′, λ′) ∈Z . Moreover, every integer is the signature of a form, since 1 ∈Z is the signature of the nonsingular symmetric form (R,1) with

1 : R →R∗; X 7−→(Y 7−→Xy)

and for any nonsingular symmetric forms (K, λ), (K′, λ′) over R

Σ((K, Λ) ⊕(K′, Λ′)) = Σ(K, Λ) + Σ(K′, Λ′) ,

σ(K, −λ) = −σ(K, λ) ∈Z . The isomorphism of 3.5 in the case n ≡0(mod 2) is defined by

∼=

→Z ; (K, λ) 7−→σ(K, λ) . L2(R) = 0 because every nonsingular (−1)-symmetric (alias symplectic) form over R admits is isomorphic to a hyperbolic form.

Andrew Ranicki

It is not possible to obtain a complete isomorphism classification of non- singular symmetric and quadratic forms over Z – see Chapter II of Milnor and Husemoller for the state of the art in 1973. Fortunately, it is much easier to decide if two forms become isomorphic after adding hyperbolics then whether they are actually isomorphic. Define the signature of a non- singular symmetric form (K, λ) over Z to be the signature of the induced

Nonsingular Symmetric Form Over R

σ(K, λ) = σ(R ⊗K, 1 ⊗λ) ∈Z . It is a non-trivial theorem that two nonsingular even symmetric forms

If And Only If They Have The Same Signature

σ(K, λ) = σ(K′, λ′) ∈Z . Moreover, not every integer arises as the signature of an even symmetric form, only those divisible by 8. The Dynkin diagram of the exceptional Lie

Group E8 Is A Tree

............................................................................................................................................................................................................................................................................................................. ...................................................

V8

Weighing each vertex by 1 ∈Q+1(Z) = Z gives (by the method recalled in 2.18) a nonsingular quadratic form (Z8, λE8, µE8) with signature

: Z8 →(Z8)∗

and µE8 is determined by λE8.

37

Example 3.6 (i) The signature divided by 8 defines an isomorphism

Σ : L4K(Z) →Z ; (K, Λ, Μ) 7→Σ(K, Λ)/8

so that (Z8, λE8, µE8) ∈L4k(Z) represents a generator. (ii) See Kervaire and Milnor and Levine for the surgery classification of exotic spheres in dimensions n ≥7, including the expression of the h- cobordism group Θn of n-dimensional exotic spheres as

And The Exact Sequence

. . . →πn+1(G/O) →Ln+1(Z) →Θn →πn(G/O) →. . . . (iii) In the original case n = 7 (Milnor ) there is defined an isomorphism

→Z28 ; Σ7 7→Σ(W)/8

for any framed 8-dimensional manifold W with ∂W = Σ7. The realization (2.17) of (Z8, λE8, µE8) as the kernel form of a 4-connected 8-dimensional

Normal Bordism

(f, b) : (M 8, S7, ∂+M) = (cl.(M(E8, 1, . . . , 1)\D8); S7, ∂M(E8, 1, . . . , 1))

Σ7 = ∂+M = ∂M(E8, 1, . . . , 1)

generating Θ7 : the framed 8-dimensional manifold W = M(E8, 1, . . . , 1) obtained by the E8-plumbing of 8 copies of τS4 (2.18) has σ(W) = 8. The 7-

ℓdefined For Any Odd Integer ℓin 2.20 Is The

boundary of a framed 8-dimensional manifold Wℓwith σ(Wℓ) = 8(ℓ2 −1). For any nonsingular (−1)-quadratic form (K, λ, µ) over Z there exists a

0

otherwise . The Arf invariant of (K, λ, µ) is defined using any such basis to be

I=1

µ(xi)µ(xi+m) ∈Z2 . Example 3.7 (i) The Arf invariant defines an isomorphism

C : L4K+2(Z) →Z2 ; (K, Λ, Μ) 7→C(K, Λ, Μ)

The nonsingular (−1)-quadratic form (Z ⊕Z, λ, µ) over Z defined by

Andrew Ranicki

has Arf invariant c(Z ⊕Z, λ, µ) = 1, and so generates L4k+2(Z). (ii) The realization (2.19 (iv)) of the Arf form (Z ⊕Z, λ, µ) as the kernel form of a 5-connected 10-dimensional normal bordism (f, b) : (M 10, S9, ∂+M) = (cl.(M(I, 1, 1)\D10); S9, ∂M(I, 1, 1))

→S9 × ([0, 1]; {0}, {1})

is obtained by plumbing together 2 copies of τS5 (2.18) where I is the tree with 1 edge and 2 vertices, and Σ9 = ∂+M = ∂M(I, 1, 1) is the exotic 9- sphere generating Θ9 = Z2. Coning offthe boundary components gives the closed 10-dimensional PL manifold cS9 ∪M ∪cΣ9 without differentiable structure of Kervaire .

Which Can Be Regarded As A Sesquilinear Pairing

ψ : K × K →Λ ; (x, y) 7→ψ(x, y) . Split forms are more convenient to deal with than ǫ-quadratic forms in describing the algebraic effects of even-dimensional surgery (in §5 below), and are closer to the geometric applications such as knot theory.

The main result of §4 is that the ǫ-quadratic structures (λ, µ) on a f.g. projective Λ-module K correspond to the elements of the ǫ-quadratic group

Of 2.6

Qǫ(K) = coker(1 −Tǫ : S(K) →S(K)) . The pair of functions (λ, µ) used to define an ǫ-quadratic form (K, λ, µ) can thus be replaced by an equivalence class of Λ-module morphisms ψ :

Μ(X) = Ψ(X, X) ∈Qǫ(Λ)

i.e. by an equivalence class of split forms. Definition 4.1 (i) A split form (K, ψ) over Λ is a f. g. projective Λ-module K together with an element ψ ∈S(K).

(ii) A morphism (resp. isomorphism) of split forms over Λ

F : (K, Ψ) →(K′, Ψ′)

is a Λ-module morphism (resp. isomorphism) f : K →K′ such that f ∗ψ′f = ψ : K →K∗.

39

(iii) An ǫ-quadratic morphism (resp. isomorphism) of split forms over Λ

(F, Χ) : (K, Ψ) →(K′, Ψ′)

is a Λ-module morphism (resp. isomorphism) f : K →K′ together with

An Element Χ ∈Q−Ǫ(K) Such That

f ∗ψ′f −ψ = χ −ǫχ∗: K →K∗. (iv) A split form (K, ψ) is ǫ-nonsingular if ψ+ǫψ∗: K →K∗is a Λ-module isomorphism.

Proposition 4.2 (i) A split form (K, ψ) determines an ǫ-quadratic form

(K, Λ, Μ) By

λ = ψ + ǫψ∗: K →K∗; x 7−→(y 7−→ψ(x, y) + ǫψ(y, x)) , µ : K →Qǫ(Λ) ; x 7−→ψ(x, x) . (ii) Every ǫ-quadratic form (K, λ, µ) is determined by a split form (K, ψ),

Which Is Unique Up To

ψ ∼ψ′ if ψ′ −ψ = χ −ǫχ∗for some χ : K →K∗. (iii) The isomorphism classes of (nonsingular) ǫ-quadratic forms (K, λ, µ) over Λ are in one-one correspondence with the ǫ-quadratic isomorphism classes of (ǫ-nonsingular) split forms (K, ψ) over Λ.

Proof : (i) By construction. (ii) There is no loss of generality in taking K to be f.g. free, K = Λk. An ǫ-quadratic form (Λk, λ, µ) over Λ is determined by a k × k-matrix

Λij = Ǫλji ∈Λ

and a collection of elements µ = {µi ∈Qǫ(Λ) | 1 ≤i ≤k} such that µi + ǫµi = λii ∈Qǫ(Λ) . Choosing any representatives µi ∈Λ of µi ∈Qǫ(Λk) there is defined a split form (Λk, ψ) with ψ = {ψij ∈Λ | 1 ≤i, j ≤k} the k × k matrix defined by

0

otherwise . (iii) An ǫ-quadratic (iso)morphism (f, χ) : (K, ψ) →(K′, ψ′) of split forms determines an (iso)morphism f : (K, λ, µ) →(K′, λ′, µ′) of ǫ-quadratic forms. Conversely, an ǫ-quadratic form (K, λ, µ) determines an ǫ-quadratic isomorphism class of split forms (K, ψ) as in 3.1, and every (iso)morphism of ǫ-quadratic forms lifts to an ǫ-quadratic (iso)morphism of split forms.

Thus Qǫ(K) is both the group of isomorphism classes of ǫ-quadratic forms and the group of ǫ-quadratic isomorphism classes of split forms on a f. g. projective Λ-module K.

Andrew Ranicki

The following algebraic result will be used in 4.6 below to obtain a ho- mological split form ψ on the kernel Z[π1(X)]-module Kn(M) of an n- connected 2n-dimensional normal map (f, b) : M →X with some extra structure, which determines the kernel (−1)n-quadratic form (Kn(M), λ, µ) as in 4.2 (i).

Lemma 4.3 Let (K, λ, µ) be an ǫ-quadratic form over Λ.

: (K, 0, 0) →(K, Λ, Μ) ⊕(K, −Λ, −Μ)

defines a morphism of ǫ-quadratic forms then (K, λs) is a split form which determines the ǫ-quadratic form (K, λ, µ). (ii) If (K, λ, µ) is nonsingular and (K, ψ) is a split form which determines

: (K, 0, 0) →(K, Λ, Μ) ⊕(K, −Λ, −Μ)

defines a morphism of ǫ-quadratic forms.

Proof :

(i) By 4.2 there exist a split form (K, ψ) which determines (K, λ, µ) and an ǫ-quadratic morphism of split forms

: (K, 0) →(K, ψ) ⊕(K, −ψ) .

With

χ′ = χ −s∗ψ : K →K∗. (ii) From the definitions.

= Ker(( (−1)N−1Ψ∗

ψ ) : K ⊕K →K∗) . Example 4.4 A (2n −1)-knot is an embedding of a homotopy (2n −1)-

Sphere In A Standard (2N + 1)-Sphere

ℓ: Σ2n−1֒ →S2n+1 . For n = 1 this is just a classical knot ℓ: Σ1 = S1֒ →S3; for n ≥3 Σ2n−1 is homeomorphic to S2n−1, by the generalized Poincar´e conjecture, but may have an exotic differentiable structure. Split forms (K, ψ) first appeared as the Seifert forms over Z of (2n−1)-knots, originally for n = 1. See Ranicki [21, 7.8], for a surgery treatment of high-dimensional knot theory. In particular, a Seifert form is an integral refinement of an even-dimensional surgery kernel form, as follows.

(I) A (2N −1)-Knot ℓ: Σ2N−1֒ →S2N+1 Is Simple If

πr(S2n+1\ℓ(Σ2n−1)) = πr(S1) (1 ≤r ≤n −1) . (Every 1-knot is simple). A simple (2n −1)-knot ℓhas a simple Seifert surface, that is an (n −1)-connected framed codimension 1 submanifold

M 2N ⊂S2N+1 With Boundary

∂M = ℓ(Σ2n−1) ⊂S2n+1 .

The Kernel Of The N-Connected Normal Map

(f, b) = inclusion : (M, ∂M) →(X, ∂X) = (D2n+2, ℓ(Σ2n−1)) is a nonsingular (−1)n-quadratic form (Hn(M), λ, µ) over Z. The Seifert form of ℓwith respect to M is the refinement of (Hn(M), λ, µ) to a (−1)n- nonsingular split form (Hn(M), ψ) over Z which is defined using Alexander

Ψ = I∗: Hn(M) →Hn(S2N+1\M) ∼= Hn(M) ∼= Hn(M)∗

with i : M →S2n+1\M the map pushing M offitself along a normal direction in S2n+1.

I′

∗= (−1)n+1ψ∗: Hn(M) →Hn(S2n+1\M) ∼= Hn(M) ∼= Hn(M)∗

= ([M] ∩−)−1 : Hn(M) →Hn(M) ∼= Hn(M)∗

the Poincar´e duality isomorphism. If x1, x2, . . . , xk ∈Hn(M) is a basis then (ψ(xj, xj′)) is a Seifert matrix for the (2n−1)-knot ℓ. For any embeddings

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