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APPEARED IN BULLETIN OF THE

arXiv:math/9304210v1 [math.GT] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 324-328TOPOLOGY OF HOMOLOGY MANIFOLDSJ. Bryant, S. Ferry, W. Mio, and S. WeinbergerAbstract.

We construct examples of nonresolvable generalized n-manifolds, n ≥6,with arbitrary resolution obstruction, homotopy equivalent to any simply connected,closed n-manifold. We further investigate the structure of generalized manifolds andpresent a program for understanding their topology.By a generalized n-manifold we will mean a finite-dimensional absolute neigh-borhood retract X such that X is a homology n-manifold; that is, for all x ∈X,Hi(X, X −{x}) = Hi(Rn, Rn−{0}).

Generalized manifolds arise naturally as fixed-point sets of group actions on manifolds, as limits of sequences of manifolds, andas boundaries of negatively curved groups. See [BM, Bo, B, GPW].

Such spaceshave most of the homological properties of topological manifolds. In particular,generalized manifolds satisfy Poincar´e duality [Bo].Generalized manifolds also share certain geometric and analytic properties withmanifolds.

Modern proofs of the topological invariance of rational Pontrjagin classesshow that Pontrjagin classes can be defined for generalized manifolds and (evenbetter!) that the symbol of the signature operator can be defined for these spaces.See [CSW].

In light of this, the following question seems natural:Question 1. Is every generalized manifold X homotopy equivalent to a topologicalmanifold?By [FP], this is true for compact simply connected homology manifolds in allhigher dimensions.

We shall see below that this is not true in the nonsimply con-nected case. To continue in this vein, we can consider a strong version of Question1 that asserts that, for such an X, a manifold M can be chosen coherently for allof its open subsets.Resolution conjecture (see [C]).

For every generalized n-manifold X there is ann-manifold M and a map f : M →X such that for each open U in X, f −1(U) →Uis a (proper) homotopy equivalence.Quinn [Q2] showed that such a resolution is unique if it exists and reducedthe resolution conjecture for X to the calculation of a locally defined invariantI(X) ∈H0(X; Z). For X (and Y ) connected, this invariant satisfies I(X) ≡1(mod 8), I(X) = 1 iffX has a resolution, and I(X × Y ) = I(X)I(Y ).

He alsoshowed that I(X) is an s-cobordism invariant of generalized manifolds and that an1991 Mathematics Subject Classification. Primary 57N15, 57P10; Secondary 57P05, 57R647.S.

Ferry and S. Weinberger were partially supported by NSF grants. S. Weinberger was partiallysupported by a Presidential Young Investigator Fellowshipc⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2J. BRYANT, S. FERRY, W. MIO, AND S. WEINBERGERaffirmative solution to Question 1 would imply the resolution conjecture as well.We will henceforth assume that X is connected so that the obstruction I(X) is aninteger.Resolutions are useful maps for studying the geometry of generalized manifolds.For n ≥5, Edwards [E] has characterized n-manifolds topologically as being resolv-able generalized manifolds that satisfy the following weak transversality condition:Disjoint disks property (DDP).

X has DDP if for any maps f, g : D2 →Xand ε > 0 there are maps f ′, g′ : D2 →X with d(f, f ′) < ε, d(g, g′) < ε, andf ′(D2)∩g′(D2) = ∅. The resolution conjecture would then imply a characterizationof topological manifolds as generalized manifolds satisfying DDP.Our first theorem says that the resolution conjecture is false.Theorem 1.

Generalized manifolds X with arbitrary index I(X) ≡1 (mod 8)exist within the homotopy type of any simply connected closed n-manifold, n ≥6.Corollary 1. There are generalized manifolds that are not homotopy equivalent tomanifolds.The remainder of this announcement is devoted to more positive statements.Some are theorems and some are conjectures.The first statement requires a little preparation.

Recall that an s-cobordism is amanifold with two boundary components, each of which includes a simple homotopyequivalence. A basic result in high-dimensional topology asserts that s-cobordismsare products.Theorem 2.

Let S(M) denote the set of s-cobordism classes of generalized man-ifolds mapping to M by homotopy equivalences which are homeomorphisms on theboundary. If dim M ≥6, then S(M) = S(M × D4).This periodicity theorem was proven by Siebenmann [KS] for manifold structuresets, but there was a Z obstruction to its universal validity (see [N]).

As suggestedby Cappell, the theorem is valid in the class of generalized manifolds. The followingtheorem will not be precisely stated here but explains the functorial significance ofthese spaces:Theorem 3.

The algebraic surgery exact sequence (see [R]) is valid for high-dimensional generalized manifolds up to s-cobordism.Thus, one knows that if X is simply connected, S(X) contains generalized man-ifolds of every index and there is a one-to-one correspondence between s-cobordismclasses of generalized manifolds homotopy equivalent to X with different indices.Simply connected surgery theory describes the structure of the set of such general-ized manifolds with I(X) = 1.On the other hand, for “rigid” manifolds X like tori or nonpositively curvedmanifolds, this theorem asserts that all generalized manifolds homotopy equivalentto X are s-cobordant to the standard model. (This can be proven more directly byusing [FJ] and Quinn’s resolution theorem [Q1, Q2]).Our main conjectures are the following:Homogeneity conjecture.

If X is a connected generalized n-manifold with DDP,n ≥5, then given p, q ∈X, there is a homeomorphism h : X →X with h(p) = q.

TOPOLOGY OF HOMOLOGY MANIFOLDS3S-cobordism conjecture. S-cobordisms of DDP generalized manifolds are prod-ucts.A corollary of these conjectures is that n-dimensional generalized manifolds of agiven index that satisfy the DDP are “noncartesian manifolds” modeled on uniquelocal models.

The following is very attractive and seems to be a step en route tothe s-cobordism conjecture.Revised resolution conjecture. Every generalized manifold X has a resolutionby a DDP generalized manifold.

If X satisfies DDP, then any such resolution of Xis a uniform limit of homeomorphisms.With these ideas in place, we suggest a modification to another standard con-jecture:Rigidity conjecture. If X is an aspherical Poincar´e complex, then X is homotopyequivalent to a unique generalized manifold satisfying DDP.If the polyhedron X is nonpositively curved in the sense of [Gr], then one canshow, by using [FW], that the resolution obstruction is a well-defined invariantof π1(X).

Farrell has informed the authors that this also follows from the workof Hu. The Borel rigidity conjecture (which for nonpositively curved π1(X) alsohas been claimed by Hu) for manifolds with boundary and the authors’ realizationtheorem imply that a generalized manifold homotopy equivalent to X exists.

Thes-cobordism conjecture would then provide the uniqueness. The authors’ methodsshow that metric analogues of the usual rigidity conjecture are false in situationswhere this version is correct (up to s-cobordism).Construction of a counterexample among groups of nonpositive curvature couldbe very useful in terms of producing natural examples of the anticipated localmodels: these would arise as the spaces at infinity in natural compactifications ofthe group (see [Gr]).Finally, we close with the following problem:Dimension conjecture.

The local index described above is realized in dimensionfour, but no three-dimensional examples of this sort exist.Remarks on the proof. The spaces are built using controlled surgery as developedin [Q1], [Q2], and [FP].

In surgery one tries to take manifold approximations toa space and make them homotopy equivalent to the target. Our construction is a“controlled” or local version of this.

The first step in the construction is to constructa model of a Poincar´e space X by gluing together two manifolds with boundary byusing a homotopy equivalence between their boundaries.Successive approximations are refinements of this first step.We construct amodel for X with better local Poincar´e duality by gluing together two manifoldpieces with a homotopy equivalence which is controlled over the first stage. Thecommon boundary is constructed in such a way that its image in the first stage isnearly dense.

To produce the required controlled homotopy equivalence, controlledsurgery theory comes into the picture. The new generalized manifolds “predicted”by the surgery theory result from the difference between controlled and uncontrolledsurgery obstructions on the putative X.Each stage in the construction is a space with better Poincar´e duality measuredover the previous stage of the construction.

As the control on the gluing homotopy

4J. BRYANT, S. FERRY, W. MIO, AND S. WEINBERGERequivalences improves, the resulting spaces have better local Poincar´e duality.

Thisforces the limit of the approximations to be a homology manifold because X is ahomology manifold if and only if X satisfies Poincar´e duality locally measured overitself, that is, if and only if the constant sheaf is Verdier dual to itself in the derivedcategory of X.Homotopy equivalences with good metric properties are precisely the outputof the surgery theory of [FP]. To achieve the desired goal, the codimension-onesubmanifold we glue along will become denser and denser in X.

This is inevitable,since the resolution obstruction can be measured on any open subset of a homologymanifold X. The nonresolvable generalized manifold is obtained as an inverse limitof these approximations.

In the limit, all of the approximate self-dualities becomea genuine local self-duality.The surgery theory for generalized manifolds follows from more controlled andrelative versions of the basic construction.References[B]G. Bredon, Introduction to compact transformation groups, Academic Press, New York,1972.[BM]M. Bestvina and G. Mess, The boundary of negatively curved groups, J. Amer.

Math. Soc.4 (1991), 469–481.[Bo]A.

Borel, Seminar on transformation groups, Ann. of Math.

Stud., no. 46, PrincetonUniv.

Press, Princeton, NJ, 1960.[C]J. Cannon, The characterization of topological manifolds of dimension 5, Proc.

Internat.Congr. Math., Helsinki, 1980.

[CSW] S. Cappell, J. Shaneson, and S. Weinberger, Topological characteristic classes for groupactions on Witt spaces, C. R. Acad. Sci.

Paris 313 (1991), 293–295.[E]R. D. Edwards, The topology of manifolds and cell-like maps, Proc.

Internat. Congr.Math., Helsinki, 1980.[FJ]F.

T. Farrell and L. Jones, Rigidity and other topological aspects of compact nonpositivelycurved manifolds, Bull. Amer.

Math. Soc.

(N.S.) 22 (1990), 59–64.[FP]S.

Ferry and E. Pedersen, Epsilon surgery theory, preprint.[FW]S. Ferry and S. Weinberger, The Novikov conjecture for compactifiable groups, in prepa-ration.

[GPW] K. Grove, P. Petersen, and J. Wu, Geometric finiteness theorems in controlled topology,Invent. Math.

99 (1990), 205–213.[Gr]M. Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.

), Math. Sci.Res.

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8, Springer-Verlag, New York, 1987, pp. 75–263.[KS]R.

Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smooth-ings, and triangulations, Princeton Univ. Press, Princeton, NJ, 1977.[N]A.

Nicas, Induction theorems for groups of manifold structure sets, Mem. Amer.

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267, Amer. Math.

Soc., Providence, RI, 1982.[Q1]F. Quinn, Resolutions of homology manifolds and the topological characterization of man-ifolds, Invent.

Math. 72 (1987), 267–284.

[Q2], An obstruction to the resolution of homology manifolds, Michigan Math. J.

301(1987), 285–292.[R]A. A. Ranicki, The total surgery obstruction, Lecture Notes in Math., vol 763, Springer-Verlag, Berlin and New York, 1979, pp.

275–316.(J. Bryant and W. Mio) Department of Mathematics, Florida State University, Tal-lahassee, Florida 32306E-mail address: bryant@math.fsu.eduE-mail address:mio@math.fsu.edu(S. Ferry) Department of Mathematics, State University of New York at Bingham-ton, Binghamton, New York 13901

TOPOLOGY OF HOMOLOGY MANIFOLDS5E-mail address: steve@math.binghamton.edu(S. Weinberger) Department of Mathematics, University of Chicago, Chicago, Illi-nois 60637

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