We provide a proof of the controlled surgery sequence, including stability,
in the special case that the local fundamental groups are trivial. Stability is
a key ingredient in the construction of exotic homology manifolds by Bryant,
Ferry, Mio and Weinberger, but no proof has been available. The development
given here is based on work of M. Yamasaki.Comment: 5 page

Pedersen, Erik

Quinn, Frank

Ranicki, Andrew

English

arXiv

We provide a proof of the controlled surgery sequence, including stabil-
ity, in the special case that the local fundamental groups are trivial. Stability is a
key ingredient in the construction of exotic homology manifolds by Bryant, Ferry,
Mio and Weinberger, but no proof has been available. The development given here
is based on work of M. Yamasaki

Edinburgh Research Archive

arXiv:math.GT/0111269 v1   27 Nov 2001CONTROLLED SURGERY WITH TRIVIALLOCAL FUNDAMENTAL GROUPSErik Pedersen, Frank Quinn and Andrew RanickiAbstract. We provide a proof of the controlled surgery sequence, including stabil-ity, in the special case that the local fundamental groups are trivial. Stability is akey ingredient in the construction of exotic homology manifolds by Bryant, Ferry,Mio and Weinberger, but no proof has been available. The development given hereis based on work of M. Yamasaki.In this note we provide a proof of the controlled surgery exact sequence usedin the construction of exotic homology manifolds by Bryant, Ferry, Mio and Wein-berger, [BFMW]. A primitive version of controlled surgery was developed by thesecond author in his definition of the invariant that identifies exotic homology man-ifolds, [Q3,Q4]. Surgery with bounded control, including exact sequences, was de-veloped in [FP]. The remarkable limit construction of [BFMW] uses a refinementof the sequences of [FP]. Roughly speaking [FP] describes a limit as ε → 0 while[BFMW] depends crucially on a stability property of the limiting process. Theproof of the refinement was postponed to a planned project that was never com-pleted. The intent was to deduce stability in general from a special case with anindependent solution known to be stable, the “α approximation theorem” of [CF].This is reasonable in principle and may be possible, but it has become clear thatthe authors of [BFMW] have not addressed serious technical issues needed to actu-ally carry it out. As noted in the review [R3] Until [the planned project] or someappropriate substitute becomes available the surgery classification must be regardedas somewhat provisional – although there is little doubt among the experts that it iscorrect . The purpose of this paper is to provide the “appropriate substitute”. Ourproof is direct, and is based on work of Yamasaki [Y].The following is Theorem 2.4 of [BFMW] with minor inaccuracies corrected.1. Theorem. Suppose B is a finite dimensional compact metric ANR and adimension n ≥ 4 is given. There is a stability threshold ε0 > 0 so that for anyε0 > ε > 0 there is δ > 0 with the following property: If f : N → B is (δ, 1)-connected and N is a compact n-manifold then there is a controlled surgery exactsequenceHn+1(B,L) −→ Sε,δ(N, f) −→ [N, ∂N ;G/TOP, ∗] −→ Hn(B;L).1991 Mathematics Subject Classification. 57R67, 57P99.Key words and phrases. Controlled surgery, homology manifolds.Partially supported by the US National Science Foundation, EPSRC Grant GR/M82158/01and the Leverhulme Trust.Typeset by AMS-TEX12 PEDERSEN, QUINN AND RANICKI2. Definitions.(1) Sε,δ(N, f) is the controlled structure set and is the object of interest. Itis defined as the set of equivalence classes of (M,g), where M is a topo-logical manifold and g : M → N is a homeomorphism on boundaries anda δ homotopy equivalence rel boundary. The equivalence relation is givenby: (M,g) ∼ (M ′, g′) if there is a homeomorphism h : M → M ′ whose re-striction to the boundary commutes with g and g′, and which ε homotopycommutes on all of M . It is part of the assertion that this does actuallygive an equivalence relation.(2) Sizes refer to measurements in B. For instance an ε homotopy between mapsM → N is G : M × I → N so that for any x ∈ M , the arc fG({x}× I) liesin the ball of radius ε about fG(x, 0). For basic definitions see Quinn [Q1].(3) [N, ∂N ;G/TOP ] is the set of homotopy classes of maps of N , rel boundary,to the classifying space G/TOP . This is the same term as appears in theuncontrolled surgery sequence.(4) H∗(B;L) is homology with coefficients in the 4-periodic simply-connectedsurgery spectrum L, with π∗(L) = L∗(Z) for ∗ ≥ 0. The spectrum canbe constructed algebraically using (uncontrolled) quadratic Poincaré chaincomplexes over the ring Z ([R2]).3. Outline.The proof divides into two parts. First there is a sequence with more primitiveobstruction terms:Ln+1(B;Z, ε, δ) −→ Sε,δ(N, f) −→ [N, ∂N ;G/TOP, ∗] −→ Ln(B;Z, ε, δ).This is “well known” and quickly assembled from pieces in the literature, thoughnot entirely straightforward. The second part shows that an assembly map definedby Yamasaki,Hn(B;L) −→ Ln(B;Z, ε, δ)is an isomorphism for suitable ε, δ. Theorem 1 follows by using this to replace theobstruction term in the primitive sequence. The proof of assembly isomorphismfollows that of Yamasaki for the L−∞ case, with the substitution of a K-theorysplitting argument for stabilization by ×S1.We remark that the key feature needed in constructing homology manifolds is“stability”: the sequence in Theorem 1 holds for particular ε and δ rather than justfor the ε → 0 limit, even though the homotopy and obstruction terms are explicitlyindependent of ε and δ. Stability in the sequence comes from stability in the assem-bly isomorphism. Yamasaki’s arguments are stable for relatively straightforwardreasons, so the basic source traces back through theK-theory splitting argument tostability of vanishing of controlled Whitehead groups in [Q1, §8]. Finally we notethat for stability the use of ε control, rather than bounded or continuous control,is essential. Bounded versions can be used to identify the inverse limit as ε → 0,but do not give information on how the limit is approached.4. Surgery.We describe how the “primitive” surgery sequence of §3 is a straightforwardcontrolled version of the standard sequence. The only awkward point is that theCONTROLLED SURGERY WITH TRIVIAL LOCAL FUNDAMENTAL GROUPS 3surgery obstruction [N, ∂N ;G/TOP ] → Ln(B;Z, ε, δ) still must be defined usingsurgery rather than by a direct chain-level construction.Ln(B;Z, ε, δ) is defined to be the group ofn-dimensional quadratic Poincaré com-plexes (Z coefficients, [R1]), over B and with radius < δ, modulo bordism through(n + 1)-dimensional quadratic Poincaré pairs with radius < ε. Here we use theroutine version of controlled algebra [Q1] that locates bases for modules at pointsin B, and measures radii of ordinary homomorphisms in terms of distance betweeninvolved basis points. Yamasaki goes further in using “geometric” morphisms thatincorporate paths in a space. This refinement is unnecessary here because the localfundamental groups are constant (in fact trivial).Suppose M → N is a normal map representing an element in [N, ∂N ;G/TOP ].For any δ > 0 we can do surgery below the middle dimension [Q3] to make M → N(δ, j)-connected over B, where n = 2j or 2j + 1. In this case we can give therelative chains a δ controlled quadratic Poincaré structure: nondegenerate formswith a symmetry condition in even dimensions, and “short odd complexes” [R4,§6]in odd dimensions. We define the surgery obstruction function by taking a normalmap to the quadratic Poincaré structure on a highly-connected normal map in thebordism class.The verification that this function is well-defined uses a relative construction. Ifthere is a normal bordism between highly-connected normal maps then do surgeryto make the bordism also δ-connected below the middle dimension. The relativechains of the bordism can then be given the structure of a δ quadratic Poincaréchain bordism between the chains of the two maps.The basis for exactness of the sequence is: if the quadratic structure on thechains of a highly connected normal map is null-bordant through a highly-connectedquadratic chain bordism, then we get the usual algebraic data for doing middle-dimensional surgery to get an equivalence. More specifically if ε > 0 then thereis δ > 0 so that if the quadratic chains of a highly δ-connected normal map ishighly δ-connected algebraically nullbordant, then the normal map is bordant (bysurgery) to an ε equivalence. The ε and δ here come from the controlled Hurewiczand Whitehead theorems [Q1, §5] rather than subtle stability issues.This is the point at which dimension issues arise. Standard surgery requiresdimension ≥ 5, and gives surgery sequences for smooth and PL manifolds whenthe structure set and homotopy terms are changed appropriately. The topologicalversion holds in dimension 4: since the local fundamental groups are trivial [Q2]gives the controlled embeddings of 2-spheres in 4-manifolds needed for the surgery.In fact there is a weak 3-dimensional version in which objects in the structure setare δ homology equivalences and equivalences are ε homology s-cobordisms, but wehave not tried to include this in the main statement.The algebra and topology are brought together by algebraic surgery on controlledquadratic complexes. The results needed are exactly analogous to the topologicalcase, and easily obtained by adding control to the arguments of [R1]: given nand ε > 0 there is δ so that an n-dimensional δ quadratic Poincaré complex overB is ε bordant to one that is connected up to the middle dimension. Similarlyif two highly-connected quadratic complexes are δ-bordant then there is a highlyconnected ε-bordism.5. Assembly.The assembly map Hn(B;L) −→ Ln(B;Z, ε, δ) is defined by Yamasaki [Y, §3]4 PEDERSEN, QUINN AND RANICKIusing “cycles”. The characterization of assembly maps in Weiss andWilliams [WW]shows this agrees with other definitions. We review [Y] to explain how it reducesour case to a K-theory splitting problem.An element of Hn(B;L) is represented by (1) a triangulated codimension 0 sub-manifold V of some sphere SN ; (2) a quadratic Poincaré pair over each simplexof V so that the pairs over ∂σ fit together to give the boundary of the pair overσ, and so that they all glue together to give a Poincaré complex of dimension n;and (3) a map V → B. The complex obtained by glueing them all together is theassembly. To get control on the assembled complex one subdivides V so finely thatimages of simplices have diameter < δ in B. Define a function from the basis of theassembled complex into B as follows: each basis element in the assembled complexcomes from one of the fragments, lying over some simplex. Take the basis elementto an arbitrary point in the image of the simplex. Since the structure maps in theassembled complex keep this basis element inside its fragment, therefore over pointsin a small image, the assembled complex has radius < δ.This description of the assembly makes it clear that to show it is an isomorphismwe need to start with a controlled complex and split it up as a union of small pieces.Full splitting follows inductively from being able to split once, so we are reducedto the following analog of [Y, Lemma 2.5]:6. Lemma: splitting Poincaré complexes. Suppose B is a metric space, W ⊂B, ε > 0, and a dimension n are given. Then there is δ > 0 so that an n-dimensional quadratic Poincaré complex D over B with radius < δ is ε equivalentto a union of ε Poincaré pairs D′ ∪C D′′ with D′ located over B − W , D′′ locatedover W ε, and the common boundary C located over W ε ∩ (B − W ).Yamasaki comes close to producing such a decomposition. He produces pairsC → D′, C → D′′ with D′, D′′ over B−W , W ε respectively, as desired, but C maybe nontrivial over all of B. In a nutshell, D′′ is the maximal based subcomplex ofD lying over B − W , D′ is the quotient D/D′′ with basis the complement of thebasis of D′′ in the basis of D. Since these are n-dimensional complexes of radius< δ, D′ lies over W nδ. Finally C is Ranicki’s “algebraic boundary” defined asthe mapping cone of a duality homomorphism. This homomorphism uses all of Dso has no restrictions on its location. However the duality homomorphism is a δchain equivalence outside W nδ ∩ (W−nδ), so the mapping cone C is δ contractiblethere. Over the region where it is contractible Yamasaki uses the usual foldingargument to get C concentrated in two adjacent degrees, in which case the boundaryhomomorphism is an nδ isomorphism.Recall that we would be finished if we could show that given ε there is δ >0 so that C is ε equivalent to a complex located over W ε ∩ (B − W−ε). Theproblem is therefore reduced to a problem about δ isomorphisms. Namely we havea homomorphism d : Cj → Cj−1 that is a δ isomorphism off Wδ ∩ (W−δ) and wewant to split off and discard a contractible summand containing the part outsideW ε ∩ (B − W−ε).It is at this point that our argument diverges from Yamasaki’s. Splitting anisomorphism is a controlled K-theory problem and generally not possible. Yamasakistabilizes by multiplying by S1. This canonically kills K-theory so splitting becomespossible. However it also changes the surgery problem, so he defines L−∞ byfactoring out these changes and concludes that the assembly is an isomorphism inthis context. In our special case (Z coefficients) the controlled K-theory vanishesCONTROLLED SURGERY WITH TRIVIAL LOCAL FUNDAMENTAL GROUPS 5and splitting is possible without stabilization.7. Lemma: splitting isomorphisms. Suppose B ⊃ V and ε > 0 are given.Then there is δ > 0 so that if d : A → A′ is a δ homomorphism of Z-modules overB and is a δ isomorphism over B−V then there are ε automorphisms H, H ′ of A,A′ respectively so that (1) H, H ′ are the identity over V ε, and (2) H ′dH is inducedby a bijection of bases over B − V 2ε.This follows easily from [Q1, Theorem 8.4].We apply this to the homomorphism d : Cj → Cj−1 in the boundary complexconsidered above. Conclusion (1) enables us to extend H, H ′ by the identity onthe rest of C to get an ε equivalence of quadratic Poincaré complexes. Conclu-sion (2) shows that the new complex is the sum of a trivial complex (d = 1) andone lying over B − V 2ε. Deleting the trivial summand gives a splitting satisfyingthe conclusions of Lemma 6. This proves Lemma 6, which shows that the assem-bly Hn(B;L) −→ Ln(B;Z, ε, δ) is an isomorphism and so completes the proof ofTheorem 1.References[BFMW] J. Bryant, S. Ferry, W. Mio and S. Weinberger,Topology of homology manifolds, Annalsof Maths. 143 (1996), 435–467.[CF] T.A.Chapman and S.Ferry, Approximating homotopyequivalences by homeomorphisms,Amer. J. Math. 101 (1979), 583–607.[FP] S.Ferry and E.K.Pedersen, Epsilon surgery theory, Novikov conjectures, index theoremsand rigidity, Vol. 1, LMS Lecture Notes, vol. 226, Cambridge, 1995, pp. 167–226.[Q1] F.Quinn, Ends of maps I., Annals of Maths. 110 (1979), 275–331.[Q2] , Ends of maps III: dimensions 4 and 5, J. Diff. Geo. 17 (1982), 503–521.[Q3] , Resolutions of homology manifolds, and the topological characterization ofmanifolds, Invent. Math. 72 (1983), 267–284.[Q4] , An obstruction to the resolution of homology manifolds, Michigan Math. J. 34(1987), 284–291.[R1] A.Ranicki, The algebraic theory of surgery, Proc. Lond. Math. Soc. 40 (1980), 87–283.[R2] , Algebraic L-theory and topological manifolds, Tracts in Mathematics, vol. 102,Cambridge, 1992.[R3] , Featured Review of [BFMW], Mathematical Reviews 97 (1997), MR97b:57017.[R4] , An introduction to algebraic surgery, Surveys on Surgery Theory: Volume 2,Annals of Mathematics Study, vol. 149, Princeton, 2001, pp. 81–163.[WW] M.Weiss and B.Williams, Assembly, Novikov conjectures, index theorems and rigidity,Vol. 2, LMS Lecture Notes, vol. 227, Cambridge, 1995, pp. 332–352.[Y] M.Yamasaki, L-groups of crystallographic groups, Invent. Math. 88 (1987), 571–602.Dept. of Mathematical SciencesBinghamton UniversityBinghamton, New York 13901, USAE-mail address : erik@math.binghamton.eduDept. of MathematicsVirginia TechBlacksburg VA 24061-0123, USAE-mail address : quinn@math.vt.eduDept. of Mathematics and StatisticsUniversity of EdinburghEdinburgh EH9 3JZ, Scotland, UKE-mail address : aar@maths.ed.ac.uk

CONTROLLED SURGERY WITH TRIVIAL  LOCAL FUNDAMENTAL GROUPS

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Controlled surgery with trivial local fundamental groups

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