STEVEN C. FERRY - Wellesleypalmer.wellesley.edu/~sschang/myfiles.d/paperfiles.d/eps-delta.pdfThe...

EPSILON-DELTA SURGERY OVER Z

STEVEN C. FERRY

ABSTRACT. This manuscript fills in the details of the lecture I gave on “squeezing structures” inTrieste in June, 2001. The goal is to develop a controlled surgery theory of the sort discussed/used in[3]. We first prove aπ−π theorem and use it to set up a formal surgery theory. We then examine theimplications of the formal theory in the specific instances where the “alpha approximation theorem”applies. This allows us to deduce stability of the surgery groups and stability of the structure sets.A similar theory has also been developed by Pedersen, Quinn, and Ranicki [13]. This material willappear as part of the writeup of my CBMS lectures.

1. INTRODUCTION

Let p : X → B be a map. We will say that a mapf : Y → X is anε-equivalence overB if thereexist a mapg : X→ Y and homotopiesht : f ◦ g ∼ id, kt : g ◦ f ∼ id so that the tracksp ◦ ht(x)andp ◦ f ◦ kt(y) have diameter less thanε for all x ∈ X andy ∈ Y.

If p : M→ B is a map, anε-structure onM overB will mean an equivalence class of pairs(N, f),wheref : N→M is anε-equivalence overB and pairs(N, f) and(N ′, f ′) are said to beε-relatedif there is an homeomorphismφ : N → N ′ so thatf ′ ◦ φ is ε-homotopic tof overB. Our notionof “equivalence” forε-structures is the equivalence relation generated by this relation. We will use

the symbolS ′ε

(M↓B

)to denote the collection of equivalence classes ofε-structures onM.

The purpose of this paper is to prove anε-δ surgery exact sequence.

Theorem 1.1. If Mn is a compact topological manifold,n ≥ 6, or n ≥ 5 when∂M = ∅, B isa finite polyhedron, andp : M → B is aUV1 map1, then there exist anε0 > 0 and aT > 0

depending only onn andB so that for everyε ≤ ε0 there is a surgery exact sequence

. . .→ Hn+1(B; L)→ Sε(M↓B

)→ [M,G/TOP]→ Hn(B; L)

whereL is the periodicL-spectrum of the trivial group and

Sε(M↓B

)= im

(S ′ε

(M↓B

)→ S ′Tε

(M↓B

)).

Moreover, forε ≤ ε0, Sε(M↓B

)∼= Sε0

(M↓B

).

This research was partially supported by NSF Grants. The author would like to thank the University of Chicago forhospitality while this paper was being written.

1See definition 4.3 below1

2 STEVEN C. FERRY

Our scheme is to prove anε-δ pi-pi theorem and use it to give a formal “chapter 9” development ofsurgery theory. We compute the high-dimensional surgery groups by plugging the Chapman-FerryAlpha-approximation theorem into this theory. This is analogous to Sullivan’s use of the gener-alized Poincare conjecture to compute the homotopy groups of G/TOP. The lower-dimensionalgroups are then computed using a 4-periodic algebraic description of the groups. The stability ofthe surgery groups is then used to deduce the stability of the structure set. The surgery sequenceof theorem 1.1 follows as usual. Here is the statement of the Alpha-approximation theorem:

Theorem 1.2(Alpha-approximation theorem [5]).

(i) LetMn be a closed topological manifold of dimension≥ 5. For everyε > 0 there is aδ > 0 so that iff : N → M is a δ-homotopy equivalence from another manifold of thesame dimension toM, thenf is ε-homotopic to a homeomorphism.

(ii) If Mn is a compact topological manifold, then for everyε > 0 there is aδ > 0 so that iff : N →M is a δ-homotopy equivalence from a compact manifold of the same dimensionto M such thatf|∂N is a homeomorphism from∂N to ∂M, thenf is ε-homotopic to ahomeomorphism.

Remark 1.3.

(i) Since the manifolds in the theorem above are topological, the metric in whichε andδ aremeasured is merely a topological metric. The proof of the alpha-approximation theorem isa handle induction, so the size ofδ is some fraction of the size of the smallest handle ina handle decomposition ofM. As noted by Farrell and Jones, this means that the relationbetweenε andδ is linear in any metric which allows handle decompositions to be subdi-vided linearly. In particular, the relation is linear for PL manifolds with the barycentricmetric. We will always assume that polyhedra in this paper have been given metrics whichare Lipschitz equivalent to barycentric metrics.

(ii) The full statement of the theorem in [5] is valid for noncompact manifolds and uses opencoversα andβ in place ofε andδ, hence the name.

(iii) The alpha-approximation theorem is valid in dimension 4 as an easy consequence of workof Freedman and Quinn. See [9].

(iv) The alpha-approximation theorem is a purely topological theorem. It is false in the PL andsmooth categories.

The derivation of the surgery exact sequence of Theorem 1 will make extensive use of Quinn’sThin h-cobordism theorem, which we state here.

Theorem 1.4(Thin h-cobordism theorem [14]). Let B be a finite polyhedron. Then for everyε > 0 there is aδ > 0 so that ifp : M0 → B is aUV1(δ)-map2 and(W;M0,M1) is a cobordismwith strong deformation retractionsrt and st retractingW toM0 andM1 such that the lengthsof the pathsp(r1(rt(x))) andp(r1(s1(st(x)))) are less thanδ for eachx ∈ W, then there is aproduct structure onW so that the composition ofp with the projection toM0 is ε-homotopic top ◦ r1.

2See definition 4.3 below.

EPSILON-DELTA SURGERY OVERZ 3

2. SETTING THE SCENE

Following [6], [14], [18], we introduce the language of geometric modules.

Definition 2.1.

(i) A geometricZ-module on Eis a free moduleZ[S] on a setS together with a mapf : S→ E.For the most part, the setS will be finite. In any case,S will be locally finite overE. Wewill often suppress the functionf and pretend that the elements ofS are points ofE.

(ii) A geometric morphismh : Z[S] → Z[T ] of geometricZ-modules withf : S → E andg : T → E is a homomorphismZ[S] → Z[T ]. If we write h = (hst) with respect to thebasesS andT , then the radius ofh is sups,t{d(f(s), g(t))|hst 6= 0}. This is less generalthan the definition from [18], but it will suffice for the purposes of this paper.

(iii) A geometric morphismh : Z[S] → Z[T ] of radiusε is anε-isomorphismif there is ageometric morphismk of radiusε, k : Z[T ]→ Z[S], so thath ◦ k = id andk ◦ h = id.

Definition 2.2.

(i) A geometricZ-module chain complexC on E is a sequence of morphisms of geometricZ-modules onE

C : . . . // Cidi // Ci−1

di−1 // . . .

such thatd ◦ d = 0. C will be called anε-chain complex onE if each morphismdi hasradius less thanε.

(ii) A chain map f between geometricZ-module chain complexesC andD is a sequence ofgeometric morphismsfi : Ci → Di so thatdi−1 ◦ fi = fi−1 ◦ di. The mapf hasradius ε ifeachfi has radiusε.

(iii) A chain homotopybetween two geometric chain mapsf andg is a collection{Hi} of geo-metric morphismsHi : Ci → Di+1 so thatdi+1 ◦Hi +Hi−1 ◦ di = fi − gi. Theradius ofH is its radius as a geometric morphism.

(iv) An ε-chain contractions : C∗ → C∗ is anε-chain homotopy between id and0.

Example 2.3. A good example to keep in mind is the case in whichX is a finite polyhedron. Thesimplicialk-chains onX form a geometric moduleCk(X) where eachk-simplexσ is associated toits barycenterσ ∈ X. If the simplices in the subdivision have diameter< ε, then the boundary map∂ : Ck(X)→ Ck−1(X) has radius< ε. Many of the usual constructions of algebraic topology carryover to this situation in the expected manner. In particular, we have the notions of thealgebraicmapping cylinderandalgebraic mapping coneof a morphism of geometric chain complexes.

It is often useful to extend this in the following manner: Letp : K → B be a map from a finitepolyhedron to a compact metric space. The simplicial chains ofK give rise to a chain complex ofgeometric modules overB by associating each simplex with the image of its barycenter inB.

One reason that geometric chain modules are useful is that in certain situations they allow us touse homological data to construct homotopy equivalences without passing to the universal coverand/or dealing with modules over the group ring. See Proposition 4.11 below.

4 STEVEN C. FERRY

Geometric chain modules are a key ingredient in the following controlled version of Browder’sM×R theorem which is a special case of Quinn’s Approximate End Theorem (see p. 283 of [14]).

Theorem 2.4(Controlled M ×R ). Suppose thatn ≥ 6 andX is a finite polyhedron. Then thereexist aδ0 > 0 and ak so that ifδ < δ0 andK → X is aUV1(δ)-map from a finite polyhedron toX andWn → K × R1 is a properδ-equivalence from ann-manifoldW to K × R1 overX × R1,then there is a codimension one submanifoldM ofW which is akδ-strong deformation retract ofW overX. Thek depends on the dimension ofM, but not onδ. The quantityδ0 depends onX.

X × R

K × R

W

X

M

3. EPSILON-DELTA K-THEORY

In this section we will review controlledK-theory as described in [4] and [14].

Definition 3.1. Leth be anε-automorphism of a geometric moduleA over a spaceB. We will saythath is ε-elementary if A can be written as a based direct sumE ⊕ F in such a way thath hasmatrix

(I0AI

).

Definition 3.2. We will identify α : A → A with α ⊕ id : A ⊕ F → A ⊕ F for any geometricZ-moduleF overB. If α andβ areε-automorphisms ofA, we writeα

ε∼ β if α ◦ β−1 is ε-

elementary. The relationε∼ generates an equivalence relation and we denote the set of equivalence

classes ofε-automorphisms by Whε(B). Direct sum makes this set into an additive semigroup.The Whitehead identities(

α 0

0 α−1

)=

(1 1

0 1

) (1 0

−1 1

) (1 1

0 1

) (1 0

α 1

) (1 −α−1

0 1

) (1 0

α 1

)(α 0

0 β

)=

(1 1

0 1

) (1 0

−1 1

) (1 1

0 1

) (1 0

α 1

) (1 −α−1

0 1

) (1 0

α 1

) (1 0

0 αβ

).

show that Whε(B) is an abelian group.

The most unsatisfactory feature of this definition is the phrase “equivalence relation generated byε∼.” The next lemma makes this relation more palatable.

Lemma 3.3. If α andβ are equivalent inWhε(B), thenα13ε∼ β.

This follows immediately from the next lemma.


Lemma 3.4(Chapman’s swindle). If α,β : A→ A in Whε(B) are automorphisms andeηk. . . eη1

α =

β with radius(αi) < ε for all i, αi = eηi. . . eη1

α, then there existeξj: ⊕2k+1i=1 A → ⊕2k+1i=1 A,

j = 1 . . . 13, with radius(eξj) < ε for all j such that(eξ13

. . . eξ1) (α⊕ id) = β⊕ id.

Proof. We have:

α⊕ (id⊕ id)⊕ (id⊕ id)⊕ . . .⊕ (id⊕ id)ε∼α⊕ (α−1

1 ⊕ α1)⊕ (α−12 ⊕ α2)⊕ · · · ⊕ (α−1

k ⊕ αk)=(α⊕ α−1

1 )⊕ (α1 ⊕ α−12 )⊕ · · · ⊕ (αk−1 ⊕ α−1

k )⊕ αkε∼(αα−1

1 ⊕ id)⊕ (α1α−12 ⊕ id)⊕ · · · ⊕ (αk−1α

−1k ⊕ id)⊕ β

=(e−1η1⊕ id)⊕ (e−1

η2⊕ id)⊕ · · · ⊕ (e−1

ηk⊕ id)⊕ β

ε∼(id⊕ id)⊕ (id⊕ id)⊕ · · · ⊕ (id⊕ id)⊕ β

The first “∼” uses the first 6 term identity disjointlyk times and the next line reassociates paren-theses. The third line uses the second 6 term identity disjointlykmore times and also uses the factthatαk = β. The last line combinesk disjoint elementary operations. �

Theorem 3.5(Controlled vanishing [14]). For any compact metric spaceB andε > 0 there is aδ > 0 so that the mapWhδ(B)→Whε(B) is zero.

Remark 3.6.

(i) The controlled vanishing theorem is due to Quinn. It is closely related to the topologicalinvariance of Whitehead torsion and the alpha approximation theorems of [7], [5].

(ii) The lemma and theorems above have a remarkable consequence: given a compact metricB, for everyε > 0 there is aδ > 0 so that everyδ-automorphism withZ-coefficients canbe written (stably) as a product of at most 13ε-elementary automorphisms.

(iii) The controlled vanishing theorem is also true as stated for noncompact spaces of the formB× R, whereB is compact. Here, we are using a product metric onB× R.

4. REMARKS ON THE PRO CATEGORY

Throughout this paper, we will be working with systems of sets and/or groups. The purpose of thissection is to establish some definitions and notation. The setup we’re describing is pro-theory, butonly for the comparatively uncomplicated case of systems indexed by the natural numbers. For thereader unfamiliar with these things, a good example to keep in mind is the system of homologygroups near infinity in an open manifold. If we have a basis{Ui} of neighborhoods of infinity,it’s actually the system{Hk(Ui)} as a whole that we’re interested in, not the individual groups.A proper map from one manifold to another induces a homomorphism of homology systems atinfinity, but a certain amount of passing to subsequences and reindexing is necessary in order towrite down a pleasant commutative diagram representing the map of homology systems.

Definition 4.1.

6 STEVEN C. FERRY

(i) In this paper, asystemwill be an inverse system of sets and maps indexed by the integers,most often positive, but sometimes including 0. We write such a system as{Ai, αi} whereαi : Ai → Ai−1. The mapsαi are calledbonding maps.

(ii) The relation ofequivalenceon systems is the equivalence relation generated by passingto subsequences. Of course we must also allow passing to “supersequences” in order tomaintain symmetry. When we pass to subsequences we will automatically compose thebonding maps and reindex the remaining spaces.

(iii) Defining a map{Ai, αi} → {A ′i, α

′i} of systems in full generality is messy. The official

definition is limk colimj Maps(Aj, A ′k). This allows allows a bit more flexibility in defining

maps than we’ll need in this paper. Suffice to say that after passing to subsequences andreindexing, maps can be represented by a commuting diagram of level-preserving maps

A1

��

A2α2oo

��

A3α3oo

��

A4α4oo

��

. . .oo

A ′1 A ′

2ooα ′

2oo A ′3

ooα ′

3oo A4α ′

4oo . . .oo

(iv) If we have maps in both directions, it may not be possible to represent both by level-preserving maps in the same diagram and the best we can get by passing to subsequencesand reindexing is a diagram like

A1

β1

��

A2α2oo

β2

��

A3α3oo

β3

��

A4α4oo

β4

��

. . .oo

A ′1 A ′

2ooα ′

2oo

γ2

`ÀAAAAAA

A ′3

ooα ′

3oo

γ3

`ÀAAAAAA

A4α ′

4oo

γ4

`ÀAAAAAAA. . .oo

with commuting squares and parallelograms. Note that if all of the triangles commute, sucha diagram implies equivalence of the systems{Ai, αi} and{A ′

i, α′i}, since it is easy to build

a larger system which contains both as subsystems. The systems would still be equivalentif we only hadαi−1 ◦ γi ◦ βi = γi ◦ α ′

i ◦ βi for eachi, rather than strict commutativityin the diagram above. In general, composing with bonding maps to get commutativity isallowed as long as the commuting subdiagrams are cofinal in the original system. See [10]for more information, including a translation of limk colimj Maps(Aj, A ′

k) into somethingmore readable.

We will dealing with systems{Ai, αi} of “groups” and homomorphisms such that the productab ofelements inAi isn’t well defined but its imageαi(ab) is. The homomorphism property may onlyhold up to composition with bonding maps as well. Such a sequence is equivalent to a sequenceof groups and homomorphisms by passing to subsequences and then passing to the sequence ofimages of the bonding maps.

Definition 4.2.(i) A system isMittag-Lefflerif it is equivalent to a system of epimorphisms.

(ii) A system isstableif it is equivalent to a system of isomorphisms.


After passing to subsequences and reindexing, stability of a system{Ai, αi} leads to a commutingdiagram like the one below, from which one can see that eachαi maps the image ofAi+1 in Aibijectively onto the image ofAi in Ai−1.

A1

β1

��

A2α2oo

β2

��

A3α3oo

β3

��

A4α4oo

β4

��

. . .oo

A Aidoo

γ2

aaBBBBBBBB

Aidoo

γ3

aaBBBBBBBB

Aidoo

γ4

aaBBBBBBBB. . .oo

In this section, we will imitate [8], [4], and Chapter 9 of [17] to give a geometric construction of aformal epsilon-delta surgery theory. We begin with a discussion of controlled Poincare duality.

4.1. Controlled Poincare Duality.

Definition 4.3.

(i) We will say that a mapp : K → B between finite polyhedra isUV1(δ) if for every mapα : P2 → B of a 2-complex intoB with lift α0 : P0 → K defined on a subcomplexP0, thereis a mapα : P → K with α|P0 = α0 so thatp ◦ α is δ−homotopic toα.

P0α0 //

_�

��

K

p

��P2

α //

α>>}

}}

}B

This can be thought of as saying thatp hasδ−simply connected point-inverses. If` is aloop nearp−1(b) for someb, then the image of in B is contractible and the contractioncan beδ−lifted to K, giving a contraction of close top−1(x). The mapp is said to beUV1 if it is UV1(δ) for everyδ > 0. See [12] for details.

(ii) If p : P → B is aUV1(δ) control map, we will say thatf : M→ P is (δ, k)-connected overB if whenever(L, L0) is a CW pair with dim(L) ≤ k andα : L0 → M is a map such thatthere is a mapβ : L→ P with f◦α = β|L0, then there exist a mapγ : L→M with γ|L0 =α and a homotopyht : L→ P relL0 with h0 = f◦γ, h1 = β, and diam(p◦h({x}×I)) < δfor eachx ∈ L.

Mf // P

p

��L0

α>>||||||||

� � // L

γ

OO��β

>>~~~~~~~~B

Definition 4.4. If P is a finite polyhedron,B is compact metric, andp : P → B is an (ε, 1)-connected map, we say thatP is anε-Poincare complex of formal dimensionn overB if thereexist a subdivision ofP so that images of simplices have diameter< ε in B and so that there is acycley in the simplicial chainsCn(P) so thaty ∩ : C#(P) → Cn−#(P) is anε-chain homotopyequivalence.

For simplicity, we will restrict our explicit discussion to the oriented case. The unoriented case canbe handled as usual by using the orientation double cover.

8 STEVEN C. FERRY

Definition 4.5. Let P be aδ-Poincare duality space of formal dimensionn over a metric spaceBand letν be a (TOP, PL or O) bundle overP. A δ-surgery problemor degree one normal mapis atriple (Mn, φ, F) whereφ : M → P is a map from a closed topologicaln-manifoldM to P suchthatφ∗([M]) = [P] andF is a stable trivialization ofτM ⊕ φ∗ν. Two problems(M,φ, F) and(M, φ, F) areequivalentif there exist an(n+ 1)-dimensional manifoldW with ∂W =M

∐M, a

proper mapΦ : W → P extendingφ andφ, and a stable trivialization ofτW ⊕Φ∗ν extendingFandF. Such an equivalence is called a normal bordism. See p. 9 of [17] for further details.

Definition 4.6. If p : K→ L is a simplicial map between finite simplicial complexes, we define thesimplicial mapping cylinder of p to be a subcomplex of the join of the first barycentric subdivisionsK ′ andL ′ of K andL.

M(p) = {〈τ1, . . . , τk, σk+1, . . . , σk+m〉 | τ1 < . . . < τk, σk+1 < . . . < σk+m, andp(τk) < σk+1}.

Themapping cylinder projection is the simplicial mapM(p)→ P given by

〈τ1, . . . , τk, σk+1, . . . , σk+m〉→ 〈p(τ1), . . . , p(τk), σk+1, . . . , σk+m〉.

We will use the notationMφ // P��B

to denote aδ-surgery problem. WhenB is understood, we will

shorten the notation toφ : M → P or even toφ. We will follow tradition in pretending that ourtopological manifolds are PL in order to simplify details of the proofs. In all cases, the bundleinformation is included as part of the data. Our theorem on surgery below the middle dimensionand its proof are parallel to Theorem 1.2 on p. 11 of [17]. As usual, surgery below the middledimension is unobstructed.

Theorem 4.7. Let (Pn, ∂P) be anε-Poincare duality pair overB, n ≥ 6, or n ≥ 5 if ∂P isempty. Consider anε-surgery problemφ : (M,∂M) → (P, ∂P). Thenφ : (M,∂M) → (P, ∂P)

is equivalent to anε- surgery problemφ : (M,∂M)→ (P, ∂P) such thatφ is (ε,[n2

])-connected

overX andφ| : ∂M→ ∂P is (ε,[n−12

])-connected.

Proof. We start by considering the case in which∂P = ∅. TriangulateM so thatφ is simplicial andthe diametersp ◦ φ(τ), τ ∈M andp(σ), σ ∈ P, are< ε. ReplacingP by the simplicial mappingcylinder ofφ, we can assume thatM ⊂ P. We inductively define a bordismU(i), −1 ≤ i ≤

[n+12

]and mapsΦ(i) : U(i) → M ∪ P(i), so that∂U(i) = M

∐M

(i)and so thatΦ(i) is anε- homotopy

equivalence. We begin by settingU(−1) = M × I, and lettingΦ(−1) → P beφ ◦ proj. LetU(0)

be obtained fromU(−1) by adding a disjoint(n + 1)-ball corresponding to each0-cell of P −M.The mapΦ(0) is constructed by collapsing each new ball to a point and sending the point to thecorresponding0-cell of P − M. Assume thatΦ(i) : U(i) → P has been constructed in such away thatU(i) is an abstract regular neighborhood of a complex consisting ofM together with cellsin dimensions≤ i corresponding to the cells ofP − M in those dimensions. Assume furtherthatΦ(i) is the composition of the regular neighborhood collapse with a map which takes cells tocorresponding cells. Each(i+1)-cell ofP−M induces an attaching mapSi → U(i). If 2i+1 ≤ n,general position allows us to move this map off of the underlying complex and approximate the


attaching map by an embeddingSi → M(i)

. The bundle information tells us how to thicken thisembedding to an embedding of ofSi × Dn−i and attach(i + 1)-handles toU(i), formingU(i+1).We extendΦ(i) toΦ(i+1) in the obvious manner. This process terminates with the construction of

U[n+12 ]. TurningU[n+1

2 ] upside down, we see thatU[n+12 ] is obtained fromM[n+1

2 ] by attaching

handles of index>[n+12

]. Thus, the composite mapM[n+1

2 ] → P is (ε,[n2

])-connected overP.

In case∂P 6= ∅, the argument is similar. We first constructU over the∂ (and, therefore, over acollar neighborhood of the boundary) and then constructU over the interior. �

Remark 4.8.

(i.) The Poincare duality ofP and theUV1(ε)-property of the mapp : P → Bwere never used.This result is true for arbitraryP and arbitrary mapsp : P → B.

(ii.) Notice that direct manipulation of cells and handles has replaced the usual appeals to homo-topy theory and the Hurewicz-Namioka Theorem. This is a general technique for adaptingarguments from ordinary algebraic topology toε-controlled topology.

(iii.) The construction in the proof yields somewhat more – we wind up with(M,∂M) ⊂(P, ∂P). Whenn = 2k+ 1,M andP are equal through the k-skeleton. Whenn = 2k, ∂Mis equal to∂P through the(k− 1)-skeleton andM contains everyk-cell of P − ∂P. SinceM → P is k-connected, everyk-cell in ∂P is homotopic rel boundary to a map intoM.By attaching a(k+ 2)-cell to this homotopy along a face, we can guarantee that for everyk-cell in ∂P−M there is a(k+ 1)-cell in P so that half of the boundary of the(k+ 1)-cellmaps homeomorphically onto thek-cell and the other half maps intoM. Alternatively, thesame effect can be obtained by adding a collar to∂P.

P

M

�P

ekek+1

4.2. Controlled cell-trading. In this section we prove controlled versions of Whitehead’s cell-trading lemma. There are algebraic and geometric versions of this lemma. We will need to use bothin this paper. These operations apply equally well to cells in a finely subdivided CW complex andto handles in a finely subdivided handle decomposition. We will use cell terminology throughout,except for the term “handle addition.”

We need some notation. If we have a sequence

10 STEVEN C. FERRY

Af⊕g// B⊕ C k+l // D

of objects and morphisms, we can represent it pictorially as:

A

f��

g

��>>>

>>>>

B

k��

⊕ C

l��

��

D

Performing an elementary operation – sliding handles corresponding to basis elements inB overhandles inC – corresponding to anε-homomorphismm : B→ C results in the diagram

A

f��

g−mf

��>>>

>>>>

B

k+lm��

⊕ C

l��

��

D

We will write this operation schematically as

A

f��

g

@@@

@@@@

B

k��

⊕m // C

l~~~~~~

~~~

D

and call itadding theB-cells toC via m. When the sequenceA → B ⊕ C → D is a part of acellular chain complex, this operation is realized geometrically by handle-addition by taking eachgeneratorx in B and sliding it acrossm(x). Changing the attaching maps of the cells this wayhas the effect described above on the cellular chains. Ifm is anε−morphism, then the new chaincomplex isε−isomorphic to the old one.

The next construction iscancellation of cells. If a portion of a chain complex looks like

. . .→ A→ B⊕ C→ D⊕ C′ → . . .

and the compositeC→ B⊕ C→ D⊕ C′ → C′


is an isomorphism sending generators to generators, then the chain complex is chain homotopyequivalent to. . . A → B → D . . . with predictable control. This has a geometric counterpartin cancellation ofn- and (n + 1)-cells. Note that it is not sufficient that the mapC → C′ bean isomorphism. It must send generators to generators. The complementary process of changing

. . . // An∂ // An−1

// . . . to . . . // An ⊕D∂⊕1 // An−1 ⊕D // . . . is calledintro-

ducing cancellingn− 1 andn cells. Neither of these processes affects size estimates.

Here is our algebraic cell-trading lemma. It involves introducing cells, adding cells, and cancellingcells, the final result being thatn-cells are “traded for”(n+ 2)-cells.

Lemma 4.9. Suppose given anε-chain complex decomposed as modules asB#⊕A# for which theboundary map has the form

B#

��

⊕ A#

��zzvvvvvvvvv

B#−1 ⊕ A#−1

If there is anε-chain homotopys, with (s|B#) = 0, from the identity to a morphism which is0 onA# for #< `, thenB# ⊕A# is chain-homotopy equivalent toB# ⊕A′

# whereA′# = 0 for #< ` and

A′# = A# for #≥ `+ 2. The new chain complex is a3`ε-chain complex with a3`ε-contraction.

Proof. First introduce cancelling 1- and 2-cells corresponding toA0 to obtain

B2

��

⊕ A2

��||yyyy

yyyy

⊕ A0

��B1

��

⊕ A1

��||yyyy

yyyy

⊕ A0

B0 ⊕ A0.

Now add the newA0-cells in dimension 1 toA1 via s to obtain

B2

��

⊕ A2

��zzvvvvv

vvvv

v⊕ A0

��B1

��

⊕ A1

��zzvvvvv

vvvv

vA0

⊕soo

B0 ⊕ A0.

12 STEVEN C. FERRY

The lower map fromA0 toA0 is the identity, so the lower copies ofA0 may be canceled to obtain

B2

��

⊕ A2

��zzvvvvv

vvvv

v⊕ A0

{{vvvvvvvvv

B1

��

⊕ A1

zzvvvvvvvvv

B0.

Repeat this process, and defineA′# so thatB# ⊕A′

# is the resulting chain complex. �

Lemma 4.10. Let ` > 0 be given and letB be a polyhedron with the barycentric metric. Thenthere existδ0 = δ0(`) andk = k(`) > 0 so that ifδ < δ0 and

i. (X, Y) is aδ-CW pair overB.ii. p : (X, Y)→ B is a map so thatp andp|Y areUV1(δ)-maps.

iii. The cellular chain complexC#(X) is decomposed as (based) modulesC(Y)# ⊕ C#(X− Y)for which the boundary map has the form

C(Y)#

��

⊕ C#(X− Y)

��vvnnnnnnnnnnnn

C(Y)#−1 ⊕ C#−1(X− Y)

iv. There is anδ-chain homotopys with s|C(Y)# = 0, from the identity to a morphism whichis 0 onC#(X− Y) for #< `.

thenX may be changed by a simple homotopy equivalence of sizekδ to a complexX′, so that thecellular chainsC#(X

′) have the formC(Y)# ⊕ C#(X′ − Y) whereC#(X

′ − Y) = 0 for # < ` andC#(X

′ − Y) = C#(X− Y) for #≥ `+ 2.

Proof. Using theUV1(δ) condition we can trade away 0- and 1-cells ofX − Y. Now perform thesame operations as in the algebraic cell-trading lemma, but do them geometrically, using handleadditions and cell cancellations, rather than algebraically. The constantδ0 is included to guaranteethat all intersections take place in simply connected regions of the space, so that geometric inter-sections can be manipulated to agree with algebraic intersection numbers without taking the globalfundamental group into account. There are discussions of controlled cell-trading in section 6 of[14] and on page 84 of [4]. �

Controlled cell-trading is a very useful tool in epsilon-delta topology. Here’s a controlled White-head theorem whose proof relies on cell-trading. LetB be a finite polyhedron endowed with thebarycentric metric.Proposition 4.11(Controlled Hurewicz–Whitehead). Let an integern > 0 be given. There exista k > 0 and aδ0 > 0 depending onn so that if

i. δ < δ0


ii. (Y, X) is ann−dimensional polyhedral pair with cells of sizeδ overB andp : Y → B is aUV1(δ)-map such thatp|X is alsoUV1(δ).

iii. C∗(X)→ C∗(Y) is aδ-chain homotopy equivalence.

ThenX→ Y is akδ-homotopy equivalence.

Proof. By cell-trading, there is ak1 depending onn so that(Y, X) is k1(n)δ-homotopy equivalentrelX to a CW pair(Y ′, X) so that all cells ofY ′−X are of dimension greater thann. Letφ : Y → Y ′

andψ : Y ′ → Y be thek1(n)δ-homotopy equivalence and homotopy inverse.

By cellular approximation (or general position), we can takeφ to be a map fromY into X ⊂ Y ′.This approximation loses as much as3dimYδmore in control, since cell trading may make the cellslarger. The controlled homotopy fromψ ◦ φ to the identity gives a controlled strong deformationretraction fromY toX, establishing the desired controlled homotopy equivalence. �

The epsilon-deltaπ-π theoremDefinition 4.12. If f# : A# → B# is a chain homomorphism, we defineK(f)# to be the algebraicmapping cone off# : B# → A#. We defineK(f)# to be the dual ofK(f)#. Unraveling this, we seethatK(f)k = Ak ⊕ Bk+1, K(f)k = Ak ⊕ Bk+1, and that up to signs in the boundary map,K(f)# isthe algebraic mapping cone off# with dimensions shifted by one.

Now, suppose that we are given a degree one mapφ : M → X from a manifold to aδ-Poincarespace overB, a finite polyhedron. As usual, we equipB with the barycentric metric. We need toshow that the complexK#(φ) described above haskδ-Poincare duality for somek = k(n). Wehave a commuting diagram

(∗) Cn−k(M)

[M]∩��

Cn−k(X)

[X]∩��

φ#oo

Ck(M)φ# // Ck(X)

which shows thatφ# is µ-chain homotopy split by theumkehr map

φ! = ([M] ∩ ) ◦ φ# ◦ ([X] ∩ )−1

for µ some predictable small multiple ofδ. This splitting shows thatC#(M) is µ-chain homotopyequivalent toC#(X) ⊕ C#(φ

!) for µ a somewhat larger multiple ofδ, whereC#(φ!) denotes the

algebraic mapping cone ofφ!. This last assertion relies on the work of Ranicki, [15], Proposition2.2, which gives explicit formulas for this chain homotopy equivalence in terms of the (controlled)chain equivalences[M] ∩ , [X] ∩ , their chain homotopy inverses, and the various chain homo-topies.3 Ranicki’s argument also shows that

(∗∗) ([M] ∩ ) : Cn−#(φ!)→ C#(φ!)

3The interested reader might also want to look at [16], where Ranicki constructs explicit splittings from chainhomotopy splittings.

14 STEVEN C. FERRY

is a controlled chain-homotopy equivalence. The diagram(∗) above shows thatC#(φ!) is µ-chain

homotopy equivalent to the algebraic mapping coneCn−#(φ#) which, in turn isµ-chain homotopyequivalent to the mapping coneC#(φ

#) by (∗∗). The upshot is that the geometric kernel complexK#(φ) haskδ-Poincare duality for somek = k(n).

The next lemma is standard, as in Browder-Levine-Livesay [2] or Siebenmann’s thesis.

Lemma 4.13. If M is a topological manifold andH ⊂ M is a handle attached to∂M, then theeffect of excising the interior ofH from(M,∂M) is to kill im(H∗(H,H∩∂M)→ H∗(M,∂M)) andto create homology in the next dimension corresponding toker(H∗(H,H∩∂M)→ H∗(M,∂M)).

Theorem 4.14(Simply connected controlledπ-π theorem). If B is a finite polyhedron with thebarycentric metric, then there existk > 0 andε0 > 0 so that if(Pn, ∂P), n ≥ 6, is anε-Poincareduality space overB, ε ≤ ε0, and

(M,∂M)φ // (P, ∂P)

p

��B

is an ε-surgery problem with bundle information assumed as part of the notation so that bothp : P → B andp| : ∂P → B areUV1(ε), then we may do surgery to obtain a normal bordismfrom (M,∂M) → (P, ∂P) to (M′, ∂M′) → (P, ∂P), where the second map is akε-homotopyequivalence of pairs. Here,k andε0 depend onn.

Proof. The argument is a translation intoε-terms of the boundedπ-π Theorem of [8]. We firstfocus on the casen = 2`. By Theorem 4.7 we may do surgery below the middle dimension.

We obtain a surgery problemM ′ φ ′// P so thatφ ′ is an inclusion which is the identity through

dimension .

This means that cancelling cells in the algebraic kernelK#(P, ∂P;M′, ∂M ′) yields a complex which

is 0 through dimension − 1. Abusing the notation, we will assume that the chain complexK#(P, ∂P;M

′, ∂M′) is 0 for # ≤ ` − 1. The generators ofK`−1(P,M) correspond to -cells in∂P −M. Cancelling these against the(` + 1)-cells described in Remark 4.8ii and leaving out theprimes for notational convenience, we have

K#(P, ∂P;M,∂M) = 0 #≤ `− 1

K#(P,M) = 0 #≤ `− 1.


SinceKn−#(P, ∂P;M,∂M) is ε− chain homotopic toK#(P,M), there is an algebraicε-homotopyσ onK#(P, ∂P;M,∂M) satisfyingσδ+δσ = 1 for #≥ `+1. Taking duals, there is an algebraic ho-motopy s onK#(P, ∂P;M,∂M) such thats∂+∂s = 1 for #≥ `+1. SinceK# = K#(P, ∂P;M,∂M)is finite-dimensional, the “cell trading” procedure may be applied upside down, so that theK# ischanged to

0 // K′`+2∂ // K′`+1

∂ // K` // 0

together with a homotopy s so thats∂+ ∂s = 1 except at degree. Again, we leave out the primesfor notational convenience. Corresponding to each generator ofK`+1 (and at a point near wherethe generator sits in the control space) we introduce a pair of cancelling(` − 1)- and`-handlesand excise the interior of the(` − 1)-handle from(M,∂M), modifying the map so that the newboundary maps to∂P. The chain complex for this modifiedM is:

0 // K`+3 // K`+2 // K`+1 // 0

⊕K`+2

All generators ofK` ⊕ K`+1 are represented by discs. We may represent any linear combination ofthese discs by an embedded disc, and these embedded discs may be assumed to be disjoint by theusual piping argument. See p. [17], p. 39. TheUV1 condition on the interior is used here. We dosurgery on the following elements: For each generatorx of K`, we do surgery on(x−∂sx, sx) andfor each generatory of K`+2, we do surgery on(0, ∂y). We can think of the process as introducingpairs of cancelling- and(`+1)-handles, performing handle additions with the`-handles, and thenexcising the -handles from(M,∂M). The resulting chain complex is:

0 // K`+2∂ // K`+1 // K`

⊕ ⊕

K`

1−∂s

<<xxxxxxxxxxxx s // K`+1 // 0

⊕

K`+2

∂

<<xxxxxxxxxxx

16 STEVEN C. FERRY

which is easily seen to be contractible, the contraction being

0 K`+2oo K`+1soo K`

1−∂s

||xxxxxxxxxxxxoo

⊕ ⊕

K` K`+1∂oo

s

||xxxxxxxxxxx0oo

⊕

K`+2

Dualizing, we see that after surgery,K#(P, ∂P;M,∂M) is ε-chain contractible. Poincare dualityshows thatK#(P,M) isk ′ε-chain contractible. Together, these imply thek ′′ε- chain contractibilityof K#(∂P, ∂M). Using the controlled Hurewicz-Whitehead theorem now shows that∂P → ∂M

andP → M arekε-homotopy equivalences for somek depending onn. This application of thecontrolled Hurewicz-Whitehead theorem, proposition 4.11, uses bothUV1 conditions. An easyargument composing deformations in the mapping cylinder of(M,∂M)→ (P, ∂P) completes theproof that(M,∂M)→ (P, ∂P) is a controlled homotopy equivalence.

To obtain theπ-π-theorem in the odd dimensional case we resort to a trick.

(1) Cross withS1 to get back to an even dimension and do the surgery.(2) Go to the cyclic cover and split using the controlledM×R theorem to obtain a controlled

homotopy equivalence of the ends.

This completes the proof. �

4.3. Manifold 1-skeleton. Our construction of epsilon-delta surgery groups is modeled on Ch. IXin Wall [17]. An essential ingredient there is Wall’s lemma 2.8, which says that Poincare dualityspaces have manifold 1-skeleta.

In what follows,B will be a finite polyhedron endowed with the barycentric metric.

Proposition 4.15.Suppose thatB is a finite polyhedron. Given an integern ≥ 4 andδ > 0, thereis a closedn-manifoldM with aUV1(δ)-map ontoB.

Proof. Suppose thatp2 : Mn1 → B isUV1(µ1) for someµ1 < δ. Forn > 4, we will show how to

find a mapp2 : Mn−12 → B which isUV1(µ2) for someµ2 < δ.

TriangulateM1 by a triangulation with simplices whose diameters, measured inB, are less than(δ−µ1)2

. Let M2 be the boundary of a regular neighborhood of the two-skeleton ofM1 in thistriangulation and letp2 be the restriction ofp1 to M2. Given a 2-dimensional relative liftingproblem, we first find aµ1-lift into M1 and then move the image of the polyhedron, call itK, offof the two skeleton ofM1 by general position, and then retract the image ofK intoM2. This givesus aµ2 = (δ+µ1)

2−lifting into M2.


If we start by takingM1 to be the boundary of a regular neighborhood ofB in a high-dimensionaleuclidean space, this process can be continued to give us manifolds with the desired property inany dimension≥ 4. �

Proposition 4.16.Givenn, there is an integerk = k(n) so that an unrestrictedδ-Poincare dualityspaceP → B of dimensionn ≥ 5, is kδ-homotopy equivalent overB to akδ-Poincare space withmanifold 1-skeleton, i.e., there exist a manifold with boundary(W,∂W)→ B and akδ-homotopyequivalenceX ∼ W ∪∂W Y whereY is obtained from∂W by attaching cells of dimension 2 andhigher which have diameters≤ kδ, and(Y, ∂W) is akδ-Poincare pair.

Proof. The proof is a line-by-line translation of Theorem 8.2 of [8] from bounded topology intoepsilon-delta topology. �

We will also need the following:

Lemma 4.17. Let Xp // B be aδ-Poincare duality complex4, and let W

f // X be a degree

1 normal map of a manifoldW toX. Then there is a normal bordism ofWf // X to a manifold

(which we will again callW) so thatf is a homeomorphism over the 1-skeleton ofX found inProposition 4.16, i.e., so thatf|f−1 (regular neighborhood of 1-skeleton) is a homeomorphism.

Proof. We start withF = f ◦ proj : W × I → X and makeF|W × 1 transverse to the barycenterof each n-cell of X. Sincef is degree 1, the inverse image of this point counted with signs mustbe 1. On pairs of points of opposite sign we attach a 1-handle toW × 1, and extend F over theresulting bordism, sending the core of the 1-handle to the point and the normal bundle of the coreto normal bundle of the point. The restriction of F to the new boundary has 2 fewer double points.Continuing this process,F|(new boundary) becomes a homeomorphism over the 0-handles ofour 1-skeleton.

Now consider the 1-handles. AssumingF| transverse to the core of a 1-handle, the inverse imagemust be a union of finitely manyS1’s and one interval. After a homotopy the interval may beassumed to map homeomorphically onto its image, and we only need to eliminate theS1’s. ButeachS1 maps to the interior of an interval so this map is homotopy trivial. Attaching a 2-handleto eachS1 in the induced framing, extending the map to the coreD2 by the nullhomotopy, andextending to the normal direction by the framing removesS1 from the inverse image of the 1-handle. Doing this to all of the 1-handles completes the process. �

4.4. Geometricδ-surgery groups. Our construction follows [17], p. 86 and [8].

Definition 4.18. Let B be a finite polyhedron endowed with the barycentric metric. We define ann-dimensional unrestrictedδ-object to be.

4This lemma does not require theUV1(δ) condition on the control mapp.

18 STEVEN C. FERRY

(i) A δ-Poincare pair(Y, X) of formal dimensionn with control mapp : Y → B. For thisdefinition, we omit our usual requirement thatp be aUV1(δ)−map. We will refer to sucha pair as anunrestricted Poincare pair.

(ii) A degree one normal mapφ : (W,∂W) → (Y, X) with φ|∂W a δ-homotopy equivalenceoverB.

(iii) A stable framingF of τW ⊕ φ∗(τ).(iv) For technical reasons, there should be a bound on the actual dimension ofY as a function

of n and dimB. Something like dimY ≤ (Avogadro’s number)(dimB+n+1) should leaveroom for most sensible constructions. The purpose of this is to limit the number of layersof cell trading in some of the arguments later on. It is not important what the limit is, but itis necessary that there should be some limit.

Bordism of these objects is defined in the usual fashion. At this point we begin to model ourdefinitions on Chapman’s formal construction ofε-controlled Whitehead groups. See p. 13 of [4].

We denote the collection of unrestrictedδ-objects byURO1n,B,δ(e). There is a natural addition onthis set which is given by disjoint union of manifolds and Poincare spaces over the common controlspaceB. An identity is given by the empty set. This construction is the reason that we cannot yet

require our control maps to beUV1(δ). For elementsφ1, φ2 ∈ URO1n,B,δ(e), we defineφ1δ∼ 0

if φ1 is bordant through unrestricted objects to aδ-homotopy equivalence and we defineφ1δ∼ φ2

if φ1 + (−φ2)δ∼ 0. Note that, as in [17] and [8], this notion of bordism allows the unrestricted

Poincare pair(Y, X) to vary along with(W,∂W).

Clearly,δ∼ is a reflexive and symmetric relation, so it generates an equivalence relation∼ on

URO1n,B,δ(e). DefineL1n,B,δ(e) to be the set of all equivalence classes of this relation. Disjointunion makesL1n,B,δ(e) into a commutative monoid with identity. Elements inL1n,B,δ(e) have in-verses, since crossing with[0, 1] provides a bordism from the sum of an element with its oppositelyoriented self to the empty set. We see, therefore, thatL1n,B,δ(e) is an abelian group.

Proposition 4.19.There is an integerk = k(n) such that if(P1, P0) and(P2, P0) are unrestrictedδ-Poincare pairs over a finite polyhedronB with the barycentric metric, then(P1 ∪P0

P2) is anunrestrictedkδ-Poincare space overB.

Proof. First, we note that, up toδ-chain equivalence, we can choose any chain-level representativewe like for the orientation class. Ifσ2 andσ1 are representatives of the orientation class by singularsimplices with diameter< δ, chooseτ with ∂τ = σ2 − σ1 so thatτ has simplices of diameter< δ.Capping withτ provides a chain homotopy from capping withσ2 to capping withσ1 as in theunestimated case. The small size of the simplices ofτ gives us the needed size estimate.

We can therefore assume that we have a chain representativeσ for the fundamental class of(P1∪P0

P2) which is the sum of fundamental classes onP1 andP2 and whose common boundary gives afundamental class onP0. Our goal is to show that capping withσ gives akδ-chain homotopy


equivalence for some predictablek. This is equivalent to showing that the mapping cone of(σ∩ )is δ-chain homotopy contractible.

C1

2C

C0 × I

�

Corollary 4.20. If φ1, φ2, φ3 ∈ L1n,B,δ(e), thenφ1δ∼ φ2 andφ2

δ∼ φ3 implyφ1

kδ∼ φ3 for some

predictable integerk.

Addendum 4.21. If the control maps onP0, P1, andP2 in proposition 4.19 above happen to beUV1(δ) for someδ, then the control map on(P1 ∪P0

P2) isUV1 − kδ for some predictablek.

Proof. Given an initial diagram

K0α0 //

_�

��

(P1 ∪P0P2)

p

��K

α // B

with K a 2-dimensional polyhedron, we can assume thatP0 has a bicollared neighborhood in(P1 ∪P0

P2). After making the initial liftα0 transverse toP0, we retriangulate and define a PLmapβ : (P1 ∪P0

P2) → [1, 2] so thatβ(P1) ⊂ [1, 3/2], β(P2) ⊂ [3/2, 2] andβ is transverse to{3/2}. We now solve the lifting problem in stages by first liftingβ−1(3/2) to P0 rel α−1

0 (P0) andthen liftingβ−1([1, 3/2]) andβ−1([3/2, 2]) toP1 rel α−1

0 (P1) andP2 rel α−10 (P2), respectively. �

Proposition 4.22.There is an integerk so that ifφ1, φ2 ∈ L1n,B,δ(e), thenφ1 ∼ φ2 ⇒ φ1kδ∼ φ2.

Proof. This is the Chapman swindle of lemma 3.4 again. �

Remark 4.23.

(i) It follows that if φ = 0 in L1n,B,δ(e), thenφkδ∼ 0. This estimate will be useful in setting up

ourε− δ pro-surgery theory.(ii) For anyδ ≤ ε there is a stabilization homomorphismL1n,B,δ(e)→ L1n,B,ε(e)

Still following Wall, we proceed to define restricted objects.

Definition 4.24. We define ann-dimensional restrictedδ-objectto be:

(i) A δ-Poincare pair(Y, X) with UV1(δ) control mapp : Y → B. This means that bothp andp|X areUV1(δ) wheneverX is nonempty.

20 STEVEN C. FERRY

(ii) A degree one normal mapφ : (W,∂W) → (Y, X) with φ| : ∂W → X a δ-homotopyequivalence overB.

(iii) A stable framingF of τW ⊕ φ∗(τ).We defineL2n,B,δ(e) to be the set of restricted bordism classes of restricted objects. Note thatL2n,B,δ(e) is only a set, since disjoint union destroys theUV1(δ) property.Proposition 4.25. Letφ : (W,∂W) → (Y, X) be a restricted object. Then there is a predictablek so that ifφ is equivalent to aδ-equivalence inL2n,B,δ(e), thenφ is normally bordant to akδ-equivalence.

Proof. The point here is that if we can do controlled surgery changing the target, then we can docontrolled surgery without changing the target. The case in whichX = ∅ is a direct application ofthe epsilon-deltaπ − π theorem. The case in whichφ|∂W is aδ-homotopy equivalence followsby extending our proof of theπ− π theorem to cover that case. �

Proposition 4.26. Let {δi} be a monotone decreasing sequence of positive numbers whose limitis 0. Then the systems{L2n,B,δi

(e), i∗} and {L1n,B,δi(e), i∗} are equivalent. Here, the mapsi∗ are

induced by inclusion.

Proof. We have a diagram:

L1n,B,δ1(e) L1n,B,δ2

(e)oo L1n,B,δ3(e)oo . . .oo

L2n,B,δ1(e)

OO

L2n,B,δ2(e)oo

OO

L2n,B,δ3(e)oo

OO

. . .oo

We need to show that after passing to a suitable subsequence, which we will also label{δi}, thatwe can fill in the dotted arrows below so that theγi’s are eventually 1-1 and onto.

L1n,B,δ1(e) L1n,B,δ2

(e)oo

γ2xxqq

qq

qL1n,B,δ3

(e)oo

γ3xxqq

qq

q. . .oo

L2n,B,δ1(e)

OO

L2n,B,δ2(e)oo

OO

L2n,B,δ3(e)oo

OO

. . .oo

“Eventually,” in this case, will mean that compositions agree after composing with one more mapto the left.

We begin the construction ofγ3 by showing that every unrestrictedδ3-object is unrestricted bordantto a restrictedδ2-object. The construction is the same as in [8]. Letf : W → P be a degree onenormal map withP δ3-Poincare with control mapp : P → B which has no good properties.We may assume thatP has a fine manifold 1-skeleton and thatf is a homeomorphism over aneighborhood of this 1-skeleton. Letq : Mn → B be aUV1(δ3)-map as constructed in Proposition4.15. Take the disjoint union off with id : M→M. Now do 0- and 1- surgeries onP

∐M→ B

using the manifold part ofP to cobordP∐M → B to aUV1(δ3) control map. This is the same

construction as in the first stages of surgery below the middle dimension. Mimic the constructionback inW to get a normal cobordism toW ′ → P ′ with P ′ a kδ3-Poincare space withUV1(δ3)


control map. Here,k is predictable as in the statement of Theorem 4.16. Thus, as long asδ2 ≥ kδ3,we can define our map for elements in each equivalence class. A relative version of this argumentshows that the mapγ3 will be well-defined if we push forward intoL2n,B,δ1

(e), so long asδ1 ≥ kδ2.By assuming thatkδi+1 < δi for eachi and then passing to subsequences using every other term,we can assume that the mapsγi are well-defined.

It is now easy to check that the required commutativities hold. If we start in, say,L2n,B,δ3(e),

include intoL1n,B,δ1(e), map viaγ3 to a restricted object which is unrestrictedlyδ2-equivalent to

our original object, then the same construction shows that our original object is restrictedlyγ1-equivalent to the original. The other compositions,L1n,B,δ3

(e) → L2n,B,δ2(e) → L1n,B,δ2

(e), say,commute on the nose. �

Corollary 4.27. This allows us to put a group structure on the system{L2n,B,δi(e), i∗}. After rein-

dexing, the group operation becomes well-defined on the image inL2n,B,δi−1(e).

This group operation is, of course, disjoint union followed by surgery to make the result into arestricted object. Next, we show that our sequence of surgery groups measures our ability to solvecontrolled surgery problems.

Notation 4.28. At this point, we will drop the superscript 1 from our notation. From here on,Ln,B,δ(e) will be used in place ofL1n,B,δ(e).

Proposition 4.29. If φ : M→ P is a restrictedδ-surgery problem, then there is a predictablek sothat ifφ is equivalent to0 in Ln,B,δ(e), thenφ is normally bordant to akδ-equivalence.

Proof. The point here is that inLn,B,δ(e) we are allowed to vary both the source and the target,while in constructing a normal bordism we are only allowed to vary the source. The proof is toinvoke theπ− π theorem rel the end of the bordism where the map is aδ-equivalence. �

Theorem 4.30(Wall Realization). Letn ≥ 6 be given. Then there existk andδ0 such that givena UV1(δ)-mapp : Vn−1 → B andα ∈ Ln,B,δ(e), δ < δ0, we can represent the image ofα inLn,B,kδ(e) by a map withV × I as target.

Proof. The proof is the same as the proof of Theorem 9.5 of [8], which is, in turn, taken fromQuinn’s thesis, but we repeat it here because of its importance in what follows. IfX 6= ∅, letφ : (W,∂W) → (Y, X) be a degree one normal map and letp : Y → B beUV1(δ) with p|X alsoUV1(δ) if X is nonempty. By adjoining the mapping cylinder ofφ|∂W to X, we can assume thatX = ∂W andp|∂W is the identity. ConstructQn with ∂Q = V ∪ ∂W ∪ P as follows: Begin with∂W× [0, 1

2]∪V× [1

2, 1]. Take a fine 1-skeleton inB and lift it into∂W× 1

2andV× 1

2. Take regular

neighborhoods of these skeleta and identify the neighborhood in∂W × 12

with the neighborhoodin V × 1

2. The resulting manifold isQ. It has three boundary components and a map toB which is

UV1(3δ).

AttachQ to W andY along the copies of∂W. The result is a surgery problem equivalent tothe original. Now doπ − π surgery on(W ∪ Q, V), fixing P to get akδ-equivalence for somepredictablek. The trace of this surgery is a bordism from the original surgery problem to a trivial

22 STEVEN C. FERRY

problem together with a problem with targetV × I, so the original problem is equivalent to aproblem with targetV × I in Ln,B,kδ(e).

If X = ∅, choose a manifoldZ with nonempty boundary and aUV1-map toB. Take a union of theproblem(Z, ∂Z) → (Z, ∂Z) with the original surgery problem along a regular neighborhood of a1-complex as above to construct an equivalent problem with nonempty boundary. Then proceed asbefore. �

For manifold targets, at least, we can always replaceUV1(δ)−maps byUV1-maps. Here is atheorem from [11] which is a modified version of a theorem of Bestvina [1].

Theorem 4.31.

(i) Let a finite polyhedronB andn ≥ 5 be given. Then givenε > 0 there is aδ > 0 so that ifp ′ : Nn → B is aUV1(δ)-map from a compact manifold toB, thenp ′ is ε-homotopic to aUV1-mapp : N→ B.

(ii) LetN be a compact manifold and suppose that aUV1-mapq : N→ B onto a polyhedronB is given. Then for eachε > 0 there is aδ = δ(ε, q) > 0 such that for each mapf : M → N of any compact PL manifoldM with dimM ≥ 5 which is1-connected withδ-control with respect toq, there is aUV1-mapg : M → N which isε-close tof asmeasured inB.

Using these results, we can represent images of elements ofLn,B,δ(e) in Ln,B,ε(e) by surgeryproblems whose targets are manifolds withUV1-maps toB. Thus, theδ has been removed fromPoincare duality space. The only role theδ still plays is in determining our criterion for success –

whether a givenφ isδ∼ 0.

Remark 4.32. Theorem 4.31 asserts the existence of a variety of interesting space-filling curves.Let f : S5 → S5 be a degree two map and letB be a point. Sincef is 2-connected, the theoremasserts the existence of aUV1-map homotopic tof. Thus, there is a degree 2 map fromS5 to itselfwhich has connected and even simply-connected point-inverses.

4.5. Stability of Ln,B,δ(e)-systems for n large.

Definition 4.33. A system consisting of groupsA1 A2oo A3oo . . .oo is stableif it isequivalent to a sequence of groups and isomorphisms.

Our goal in this section is to prove the stability of the sequence

Lq,B,δ1(e) Lq,B,δ2

(e)oo Lq,B,δ3(e)oo . . .oo

whenever{δi} is a sequence of positive real numbers converging monotonically to zero andq ≥2dimB + 2. In section 5 we will show that these groups are 4-periodic forn > 4, which willestablish stability forq > 4.


LetN = N(B) be a regular neighborhood ofB in Rq, q ≥ 2dimB + 2, and letN be the groupof normal bordism classes of degree one normal maps(M,∂M) → (N,∂N) which restrict to ahomeomorphism on the boundary.

Taking rel boundary surgery obstructions gives us a commuting diagram:

N

σ��

Nidoo

σ��

N

σ��

idoo . . .oo



We will show that the vertical maps are eventual isomorphisms. Consider an elementα of Ln,B,δ2(e).

By a rel boundary adaptation of Wall realization, this element is realized by a degree one normalmapφ : (W,∂W,N ′,N ′′)→ (N ′× I, ∂(N ′× I),N ′× 0,N ′× 1) whereN ′ is a regular neighbor-hood ofB in Rq−1 andφ restricts to homeomorphisms overN ′×0 ∪ ∂N ′×I and akδ−equivalenceoverB onN ′′.

The restriction ofφ to N ′′ is a homeomorphism on the boundary. In order to show thatφ is inthe image ofN, we apply the alpha approximation theorem of the introduction together with avariation on the classical Alexander trick.

Here is the Alexander trick. Letp be the mapping cylinder projectionp : N ′′ → B. The homotopyequivalenceφ is controlled over this map. Now, repeat the boundary homeomorphism over aboundary collar inN ′′, rescaling so that the image of this homeomorphism contains all but a verysmall regular neighborhood ofB in N ′ × 1. After this modification,φ becomes a controlledhomotopy equivalence overN ′ × 1, not just overB, and we can use the alpha approximationtheorem to find a small homotopy fromφ to a homeomorphism. This means that ifkδ2 is chosento be so small that alpha approximation works onN ′ to produce a homeomorphismδ1−homotopicto the originalφ, then the image ofα in Lq,B,δ1

(e) is in the image ofN. This proves eventual

24 STEVEN C. FERRY

surjectivity and we pass to a subsequence so that “surjectivity” takes place in the(i − 1)st placefor eachi.

Proving “injectivity” amounts to constructing dotted arrows in the diagram below so that the dia-gram eventually commutes.

N

σ��

Nidoo

σ��

N

σ��

idoo . . .oo



The vertical maps are homomorphisms, at least in the sense of systems, see [17], p. 111, so itsuffices to consider an elementφ ∈ N whose image in, say,Lq,B,δ3

(e) is trivial. This means thatφ : M→ N with φ a homeomorphism over∂N and thatφ is bordant as a restrictedδ3−object toaδ3−equivalence.

The restriction ofΦ to ∂P −◦M gives aδ3−equivalence to∂(N × I) −

◦N× 1 which is a home-

omorphism on the boundary. This is the darker part of the boundary in the rather stylized pictureabove. Reparameterizing, this gives aδ3−equivalence toN. Using the same Alexander trick,this isε−homotopic rel boundary to a homeomorphism, whereε is related toδ3 as in the alphaapproximation theorem. This shows that the sequence{Lq,B,δi

(e)} is equivalent to the sequenceN← N← . . ..

The groupN is isomorphic to[N,∂N; G/TOP], which is in turn isomorphic toHq(N,G/TOP),Hq(B,G/TOP), andHq(B,L(e)). Thus, we have shown:Proposition 4.34.GivenB andq ≥ 2dimB+2, we can choose a sequenceδi of positive real num-bers monotonically approaching zero so that the image ofLq,B,δi

(e) in Lq,B,δi−1(e) is isomorphic

toHq(B,L(e)) for all i.

Remark 4.35.(i) The knowledgeable reader will have recognized the proof of the surgery exact sequence

embedded in the argument above. The point of our Alexander trick is to show that the rel

boundary structure setSδ

(N��B

)is equivalent to a trivial system and then to use this fact to

prove that the normal maps are isomorphic to the surgery groups as systems. Rather thanset up a structure set which is going to be zero in this specific instance anyway, we have


chosen to delay setting up the official surgery sequence and make our argument on the levelof individual representatives of elements of the putative structure set.

(ii) Our plan is to show that this stability holds forq ≥ 5 by showing that the sequence{Lq,B,δi

(e)} is isomorphic as a system to an algebraically defined system of groups forwhich periodicity is clear. As a side benefit, this will prove stability of the algebraicallydefined groups in all dimensions.

Next, we want to show how to use stability of the controlledL−groups to prove a similar stabilityresult for manifold structures.

Proposition 4.36. Suppose that the systems{Lq,B,δi(e)} are stable forq = n, n + 1. If Mn is a

closedn−manifold andp : M→ B is aUV1−map, then for everyε > 0 there is aδ > 0 so thatfor anyµ > 0, if φ : N →M is a δ−homotopy equivalence overB then there is anε−homotopyoverB fromφ to aµ−homotopy equivalenceψ : N→M.

In other words, if we are willing to allow a homotopy of fixed size, then if we start with a suffi-ciently well-controlled homotopy equivalence we can improve the control of that homotopy equiv-alence by an arbitrary amount. This is the “squeezing theorem” for structures.

Proof. As above, the proof consists of working our way through the not-yet-existent surgery exactsequence. We start with the systems{Lq,B,δi

(e)} reindexed so that forq = n, n + 1, we havecommutative diagrams:

N(M)

σ

��

N(M)idoo

σ

��

N(M)

σ

��

idoo . . .oo


(e)γ2

wwnnnnnnnnnnnnoo Lq,B,δ3

(e)γ3

wwnnnnnnnnnnnnoo . . .oo

Hq(B; L(e))

OO

Hq(B; L(e))oo

OO

Hq(B; L(e))oo

OO

. . .oo

Assume, further, that theδi’s have been chosen so that forq = n, n+ 1

(i) Degree one normal mapsφ : N→M with vanishing surgery obstruction inLq,B,δi(e) are

normally bordant toδi−1−equivalences.(ii) Eachα ∈ Lq,B,δi

(e) can be Wall-realized by a normal cobordism to anδi−1−equivalenceoverB.

(iii) Eachδi−thin h-cobordism has aδi−1−product structure overB.(iv) δ0 = ε andδ4 = µ.

We takeδ = δ3. Suppose that we are given aδ3−controlled homotopy equivalenceφ : N → M.The surgery obstruction of[φ] vanishes inLn,B,δ3

(e), so an easy diagram chase shows that thesurgery obstruction of[φ] vanishes inLn,B,δi

(e) for all i ≥ 3, so[φ] is normally bordant to someδi−equivalence for eachi. Choose aδ5−equivalenceφ5 : N5 → M normally bordant toφ. Thenormal bordismΦ5 : W5 → M × I has a surgery obstructionα in Ln,B,δ3

(e). Wall realize an

26 STEVEN C. FERRY

elementα ∈ Ln,B,δ3(e) whose image inLn,B,δ2

(e) is the same as the image of−α starting withφ5 : N5 →M to get a normal bordismW fromφ5 to aδ4−homotopy equivalenceφ ′ : N ′ →M.

The obstruction to surgeringW5 ∪ W to a controlledh−cobordism dies inLn,B,δ2(e), so the

surgered cobordism is aδ1-h-cobordism and has aδ0−product structure. This provides the desiredε−homotopy fromφ to a δ5−equivalenceφ ′ ◦ h, whereh : N → N ′ is the homeomorphismcoming from the product structure. �

We can now define ourδ−controlled structure sets.

Definition 4.37. Let M be a closed manifold and letp : M → B be aUV1−map. We define

Sδ

(M��B

)to consist of equivalence classes ofδ−homotopy equivalencesφ : N → M overB

modulo the relation thatφ : N→M andφ ′ : N ′ →M are equivalent if there is a homeomorphismh : N→ N ′ so thatφ is δ−homotopic toφ ′ ◦ h overB.

Given stability of the system of surgery groups, proposition 4.36 shows that the system

Sδi

(M��B

)is stable as a sequence of sets, that is, it is equivalent to a sequence of bijections. The propositionshows immediately that the sequence is equivalent to a sequence of surjections and a relative ver-sion of the proposition shows that the sequence is equivalent to a system of injections. Thus, wewill be finished as soon as we can show that our geometrically defined surgery groups are periodic.For the full result, we need an algebraic description of the surgery groups. In any case, though, wehave the following:

Corollary 4.38. Let B be a finite polyhedron and letn ≥ 2dimB + 2. If Mn is a closedn−manifold andp : M → B is a UV1−map, then for everyε > 0 there is aδ > 0 so thatfor anyµ > 0, if φ : N →M is a δ−homotopy equivalence overB then there is anε−homotopyoverB fromφ to aµ−homotopy equivalenceψ : N→M.

4.6. Algebraic δ-surgery groups. In this section, we will define our algebraicδ−surgery groupsover a polyhedronB. We will define these groups to be systems ofδ−Witt groups overB andB×Rin even dimensions. The even-dimensional groups overB×R will be our odd-dimensional groups.


Thus, in this section we will be looking at locally finite geometric modules over finite polyhedraand very special noncompact polyhedra. Restricting ourselves to locally finite polyhedra of thisspecial form allows us to phrase our work in terms of epsilons and deltas, rather than working withcollections of open covers.

Definition 4.39. Let η = ±1. Let Iη = {0} for η = 1 and2Z for η = −1. By a special geometric (Z−)quadraticη−form overB, we will mean a triple(A, λ, µ) whereA is a geometricZ−moduleoverB, λ : A×A→ Z is Z−bilinear,µ : A→ Z/Iη

(i) λ(x, y) = ηλ(y, x) x, y ∈ A(ii) λ(x, x) = µ(x) + ηµ(x) x ∈ A

(iii) µ(x+ y) − µ(x) − µ(y) = λ(x, y) mod Iη x, y ∈ A(iv) µ(xa) = a2µ(x) x ∈ A, a ∈ Z(v) Aϕ : A→ A∗ defined byAλ(x)(y) = λ(x, y) for x, y ∈ A is an isomorphism.5

The radius of this form is< ε if λ(x, y) = 0 whend(x, y) ≥ ε.

Definition 4.40. If A is a geometric module overB, thenonsingularη−hyperbolic quadratic formonA ⊕ A is the formηH(A) which has matrix

(0η10

)corresponding to each basis element ofA

with µ(1, 0) = µ(0, 1) = 0.

Definition 4.41. Two special geometric quadraticη−forms(A, λ, µ) and(A ′, λ ′, µ ′) of radius≤ δareδ−isomorphic overB if there is aδ−isomorphismh : A→ A ′ overB so thatλ ′(h(x), h(y)) =λ(x, y) for all x, y ∈ A andµ ′(h(x)) = µ(x) for all x ∈ A.

Definition 4.42. If (A, λ, µ) and (A ′, λ ′, µ ′) are geometric quadraticη−forms of radius≤ δ,

we will write (A, λ, µ)δ∼ (A ′, λ ′, µ ′) if there are geometric modulesF andG overB such that

(A ′, λ ′, µ ′)⊕ ηH(F) is δ−isomorphic to(A, λ, µ)⊕ ηH(G).

Following Chapman’s lead again, we defineLη,B,δ(Z) to be the collection of geometric special

quadraticη−forms of radius≤ δ, modulo the equivalence relation generated byδ∼.

We digress to discuss the motivation for these definitions. LetM = M2` be a closed manifold withaUV1−mapp : M→ B to a finite polyhedronB and consider a(δ, `− 1)−connected degree onenormal mapφ : N→M. The kernel complex is akδ−Poincare duality chain complex, as always,for some predictablek, with homology concentrated in dimension`. Trading cells from the bottomand then flipping the complex over (algebraically) and trading down from the top shows that thekernel complex iskδ−chain homotopy equivalent to a complex of geometric modules with cellsin only two dimensions, − 1 and` or, if we choose, and` + 1. This means that we have thediagram below, whereϕ andψ are controlled chain-homotopy inverses.

5The usual simplicity condition is not needed because of the vanishing of controlledZ-Whitehead groups. We’regetting away with a certain amount in this section because the controlled Whitehead group vanishes and we don’t haveto prove simplicity at each stage of our argument.

28 STEVEN C. FERRY

0 C`−1

ψ

��

oo C`

ψ=id��

∂oo 0oo 0oo

0 0oo C`

ϕ=id

OO

oo C ′`+1

ϕ

OO

∂oo 0oo

Since the compositionϕ ◦ψ is controlled chain homotopic to the identity, we have a diagram:

0 C`−1

0��

oo

s

!!DDD

DDDD

DC`

id��

∂oo 0oo

0 C`−1oo C`oo ∂oo 0oo

with ∂◦s = id. This controlled splitting exhibitsC` as being controlled isomorphic toC`−1⊕C ′`+1.

Since the controlled Whitehead group vanishes, we can manipulate the cells ofC` to make a setof generators inC` map to the corresponding generators ofC`−1, after which we can cancel cells.Using Chapman’s swindle again, we can insure that this maneuver loses only a predictable amountof control. The upshot is that the kernel complex iskδ−chain homotopic to a complex with a singlegeometric module concentrated in dimension`. Controlled Poincare duality gives this complex thestructure of a geometric quadratic(−1)`−form.

Now, suppose that our normal mapφ : N → M is normally cobordant to aδ−homotopy equiva-lence. We can controlled surger the normal bordism rel ends to make the map from the bordism toM× I into an`−connected map. By handle trading, first up fromM and then down from the otherend, it follows thatφ is normally bordant to a controlled homotopy equivalence via a bordism withsmall cells in dimensionsand`+ 1 and no cells outside those dimensions.

We now look at the effect on the surgery kernel of passing through these layers of cells. Startingfrom the left in the diagram below, the first set of cells is trivially attached, so the algebraic effecton the surgery kernel is to add a geometric hyperbolic form to the algebraic kernel fromM. Thisis the surgery kernel at the level ofM ′′ below. On the other hand, we can begin at the right end ofthe bordism, where the surgery kernel is trivial, and see that adding the`+ 1-cells, which become`−cells when viewed from that side, exhibitsM ′′’s surgery kernel as a hyperbolic form. Combined,this shows that in order forφ to be normally bordant to a controlled homotopy equivalence, the

original surgery kernel must be controlled stably hyperbolic, i.e., it must bekδ∼ 0 in the sense of

definition 4.42. Moreover, if the surgery kernel of a degree one normal map is stably controlledhyperbolic, we can proceed exactly as in [8], which is modelled on chapter 5 of [17] to surger toa kδ− equivalence for some predictablek. Controlled stable equivalence is therefore the relationon geometric special quadratic forms that we need to understand in order to understand controlledeven-dimensional surgery.


Finally, we note that the algebraic proof of Wall realization given in Theorem 5.8 of [17] workswithout modification to show that every geometric special quadratic(−1)`−form can be realizedon a manifold with boundary. The statement is identical to the statement of Theorem 4.30 withalgebraic Wall groups in place of geometric ones.6

We now return to algebra. The next proposition is needed so that we can use Chapman’s swindleto set up our algebraic Wall groups.Proposition 4.43. If (A, λ, µ) is a geometric special quadraticη−form of radius≤ δ, then

(A, λ, µ)⊕ (A,−λ,−µ)kδ∼ ηH(A) for some predictablek.

Proof. We revert to old (bad) habits and give a geometric argument. Given a geometric specialquadratic(−1)`−form, we Wall realize it on a manifold(W2`;N,N ′) with boundary. Considerthe2`+ 1-dimensional surgery problem obtained by crossing our problem withJ = [0, 1].

If (A, λ, µ) is the surgery kernel ofW → N× I, thenW× J gives a normal bordism from the darkregion in the figure, which isW ∪N ′ × J ∪ (−W), toN × J, where we have a homeomorphism.The surgery kernel for the dark region is(A, λ, µ) ⊕ (A,−λ,−µ). This shows that a controlledsurgery problem with kernel(A, λ, µ) ⊕ (A,−λ,−µ) can be solved and that the sum(A, λ, µ) ⊕(A,−λ,−µ) is therefore stably hyperbolic. �

Remark 4.44. It is straightforward to give an algebraic proof of proposition 4.43 over rings where2 is invertible. The case above appears to require more work. Ranicki has proven very generalresults along these lines.

We can now set up our system{Lη,B,δi(Z)} of even-dimensional algebraic surgery groups. We can

add elements ofLη,B,δi(Z) immediately this time, because direct sum does not increase the radius.

Each element ofLη,B,δi(Z) has a negative inLη,B,δi

(Z), but the sum is onlykδ∼ 0, so elements don’t

have inverses until we increaseδ by a factor ofk. Chapman’s swindle then shows that the image ofLη,B,δi

(Z) in Lη,B,δi−1(Z) is an abelian group, provided thatδi−1 is bigger thanδi by a predictable

factor for eachi.6The groups haven’t quite been set up yet, but it doesn’t hurt to point out that we can realize all of the forms.

30 STEVEN C. FERRY

Proposition 4.45.Letφ : (W2`, ∂W)→ (M2`, ∂M) be a degree one normal map withp : M→ B

and p|∂M bothUV1−maps. Here,B is either a finite polyhedron or a finite polyhedron×R.Let η = (−1)`. Suppose, in addition, thatφ|∂W is a δ−equivalence. Then there is a surgeryobstructionσ(φ) ∈ Lη,B,δ(Z). There are predictable numbersk1 andk2 so that:

(i) σ(φ) is well-defined inLη,B,k1δ(Z). In particular, ifφ can be surgered to aδ−equivalence,thenσ(φ) = 0 in Lη,B,k1δ(Z).

(ii) If σ(φ) = 0 in Lη,B,k1δ(Z), thenφ can be surgered to ak2δ− equivalence.(iii) Every element ofLη,B,k1δ(Z) is realized on a manifold with two boundary components such

that the restriction ofφ to the first boundary component is the identity and the restrictionofφ to the second boundary component is ak1δ−equivalence.

Remark 4.46. The algebraic part of this theory is pretty independent of the control spaceB, thatis, specific properties ofB play no role in the development of the theory. The base space doesenter the picture when we start pushing geometric cells around to mimic the algebraic moves. Atthat point, we need to be able to find disks in the manifold which have small projections intoB sothat we can make algebraic and geometric intersection numbers agree. This is easy to arrange ifthe map isUV1 and the spaceB is uniformly locally simply connected. Both finite polyhedra andfinite polyhedra timesR satisfy this condition, so for these control spaces there are no difficultiesin maneuvering the geometry to match the algebra.

5. PERIODICITY

We have defined even-dimensional surgery groups and shown that they do indeed determine whensurgery to a controlled homotopy equivalence is possible. Here is a proposition that shows wecan do odd-dimensional surgery by crossing withR and splitting back. Note that this is the basicphilosophy of [8], as expressed in the introduction – the major difference being that here we use aproduct metric rather than a conelike metric.

Proposition 5.1. Letp : M→ B beUV1 and letφ : N→M be a degree one normal map.

(i) There is ak so that ifφ× id : N×R→M×R is normally cobordant to aδ−equivalencecontrolled overB×R, thenφ is normally cobordant to akδ−equivalence controlled overB.

(ii) If φ is normally bordant to aδ−equivalence, thenφ× id : N× R→M× R

Proof. Part (i) is a direct application of the controlledM× R theorem stated in section 2 plus thethinh-cobordism theorem. To see part (ii), suppose thatφ is normally bordant to aδ−equivalenceφ ′ : N ′ →M by a bordism(P;N, N ′). Attaching the bordism×[0, 1] toN× R × I overM× 0gives a normal bordism fromφ × id to φ ′ : W → M × I which is split overM × 0. W is thetop boundary on the right in the picture below. Pushing the collar onN ′ out to cover all ofM×Rgives a normal bordism fromφ× id toφ ′ × id and completes the argument. �


Corollary 5.2 (of the proof). There is a 1-1 correspondence between normal bordism classes ofof degree one normal mapsW → M × R and normal bordism classes of degree one normalmapsN → M. This means that the groupsL(−1)`+1,B×R,δ(Z) give the obstructions for(2` +1)− δ−controlled dimensional surgery in the same “pro-” sense as the even-dimensional groupsLη,B,δδ

(Z) which we have already discussed.Notation 5.3. By Ln,B,δ(Z), we will meanL(−1)`,B,δ(Z) for n = 2` even andL(−1)`+1,B×R,δ(Z) forn = 2`+ 1 odd.

Proof. Given a degree one normal mapψ : W → M × R, taking the transverse inverse imageof M × 0 givesφ : N → M. Conversely, the spreading procedure used above shows that everysurgery problemψ : W → M × R is bordant to one of the formφ × id : N × R → M × R andthat two such are normally bordant if and only if the restrictions to the transverse inverse imagesofM× 0 are normally bordant. �

Proposition 5.4. There is a natural equivalence of systems{Ln,B,δ(e)} → {Lη,B,δkδ(Z)} for all

n ≥ 6.Proof. Take an element in{Ln,B,δ(e)} and Wall realize it. Send it to it’s algebraic surgery obstruc-tion in {Lη,B,δkδ

(Z)}. This is a well-defined map on the level of systems. There is a similar map inthe reverse direction and the compositions are clearly pro-equivalent to the identity. �

Since the algebraic groups are clearly 4-periodic, this establishes stability, periodicity, and theexistence of an epsilon-delta surgery exact sequence in dimensions≥ 6.

32 STEVEN C. FERRY

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3. J. Bryant, S. Ferry, W. Mio, and S. Weinberger,Topology of homology manifolds, Ann. of Math. (2)143 (1996),no. 3, 435–467.

4. T. A. Chapman,Controlled simple homotopy theory and applications, Springer-Verlag, Berlin, 1983.5. T. A. Chapman and Steve Ferry,Approximating homotopy equivalences by homeomorphisms, Amer. J. Math.101

(1979), no. 3, 583–607.6. E. H. Connell and John Hollingsworth,Geometric groups and Whitehead torsion, Trans. Amer. Math. Soc.140

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12. R. C. Lacher,Cell-like mappings and their generalizations, Bull. Amer. Math. Soc.83 (1977), no. 4, 495–552.13. Erik Pedersen, Frank Quinn, and Andrew Ranicki,Controlled surgery with trivial local fundamental group,

preprint.14. Frank Quinn,Ends of maps. I, Ann. of Math. (2)110(1979), no. 2, 275–331.15. Andrew Ranicki,The algebraic theory of surgery. II. Applications to topology, Proc. London Math. Soc. (3)40

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