L. C. Siebenmann

Bio: L. C. Siebenmann is an academic researcher from University of Paris. The author has contributed to research in topics: Homotopy group & Contractible space. The author has an hindex of 10, co-authored 10 publications receiving 1557 citations.

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.

Abstract: Since Poincare's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

On the triangulation of manifolds and the Hauptvermutung

Abstract: 1. The first author's solution of the stable homeomorphism conjecture [5] leads naturally to a new method for deciding whether or not every topological manifold of high dimension supports a piecewise linear manifold structure (triangulation problem) that is essentially unique (Hauptvermutung) cf. Sullivan [14]. At this time a single obstacle remains—namely to decide whether the homotopy group 7T3(TOP/PL) is 0 or Z2. The positive results we obtain in spite of this obstacle are, in brief, these four: any (metrizable) topological manifold M of dimension ^ 6 is triangulable, i.e. homeomorphic to a piecewise linear ( = PL) manifold, provided H*(M; Z 2 ) = 0 ; a homeomorphism h: MI—IMI of PL manifolds of dimension ^ 6 is isotopic to a PL homeomorphism provided H(M; Z2) = 0 ; any compact topological manifold has the homotopy type of a finite complex (with no proviso) ; any (topological) homeomorphism of compact PL manifolds is a simple homotopy equivalence (again with no proviso). R. Lashof and M. Rothenberg have proved some of the results of this paper, [9] and [ l0] . Our work is independent of [ l0 ] ; on the other hand, Lashofs paper [9] was helpful to us in that it showed the relevance of Lees' immersion theorem [ l l ] to our work and reinforced our suspicions that the Classification theorem below was correct. We have divided our main result into a Classification theorem and a Structure theorem.

The obstruction to finding a boundary for an open manifold : of dimension greater than five

Abstract: For dimensions greater than five the main theorem gives necessary and sufficient conditions that a smooth open manifold W be the interior of a smooth compact manifold with boundary. The basic necessary condition is that each end ! of W be tame. Tameness consists of two parts (a) and (b): (a) The system of fundamental groups of connected open neighborhoods of ! is stable. This means that (with any base points and connecting paths) there exists a cofinal sequence G1 G2 f1 !! . . . f2 !! so that isomorphisms are induced Image(f1) Image(f2) ∼= !! . . . ∼= !! . (b) There exist arbitrarily small open neighborhoods of ! that are dominated each by a finite complex. Tameness for ! clearly depends only on the topology of W . It is shown that if W is connected and of dimension ! 5, its ends are all tame if and only if W ×S is the interior of a smooth compact manifold. However examples of smooth open manifolds W are constructed in each dimension ! 5 so that W itself is not the interior of a smooth compact manifold although W ×S is. When (a) holds for ! , the projective class group K̃0(π1(!)) = lim ←− j Gj is well defined up to canonical isomorphism. When ! is tame an invariant σ(!) ∈ K̃0(π1!) is defined using the smoothness structure as well as the topology of W . It is closely related to Wall’s obstruction to finiteness for CW complexes (Annals of Math. 81 (1965) pp. 56-69). Copyright declaration is printed here 2 Laurence C. Siebenmann Main Theorem. A smooth open manifold W , n > 5, is the interior of a smooth compact manifold if and only if W has finitely many connected components, and each end ! of W is tame with invariant σ(!) = 0. (This generalizes a theorem of Browder, Levine, and Livesay, A.M.S. Notices 12, Jan. 1965, 619-205). For the study of σ(!), a sum theorem and a product theorem are established for C.T.C. Wall’s related obstruction. Analysis of the different ways to fit a boundary onto W shows that there exist smooth contractible open subsets W of R , n odd, n > 5, and diffeomorphisms of W onto itself that are smoothly pseudo-isotopic but not smoothly isotopic. The main theorem can be relativized. A useful consequence is Proposition. Suppose W is a smooth open manifold of dimension > 5 and N is a smoothly and properly imbedded submanifold of codimension k %= 2. Suppose that W and N separately admit completions. If k = 1 suppose N is 1-connected at each end. Then there exists a compact manifold pair (W, N) such that W = IntW , N = IntN . If W is a smooth open manifold homeomorphic to M × (0, 1) where M is a closed connected topological (n − 1)-manifold, then W has two ends !− and !+ , both tame. With π1(!−) and π1(!+) identified with π1(W ) there is a duality σ(!+) = (−1)σ(!−) where the bar denotes a certain involution of the projective class group K̃0(π1W ) analogous to one defined by J.W. Milnor for Whitehead groups. Here are two corollaries. If M is a stably smoothable closed topological manifold, the obstruction σ(M) to M having the homotopy type of a finite complex has the symmetry σ(M) = (−1)σ(M). If ! is a tame end of an open topological manifold W and !1 , !2 are the corresponding smooth ends for two smoothings of W , then the difference σ(!1)−σ(!2) = σ0 satisfies σ0 = (−1)σ0 . Warning: In case every compact topological manifold has the homotopy type of a finite complex all three duality statements above are 0 = 0. It is widely believed that all the handlebody techniques used in this thesis have counterparts for piecewise-linear manifolds. Granting this, all the above results can be restated for piecewise-linear manifolds with one slight exception. For the proposition on pairs (W, N) one must insist that N be locally unknotted in W in case it has codimension one.

Deformation of homeomorphisms on stratified sets

Abstract: T H E O R E M 0. The topological group H(X) of homeomorphisms of a finite simplicial complex X onto itself is locally contractible. This result does not extend to ENR's (euclidean neighborhood retracts). To see this let a space X be obtained from S 3 = R 3 u o o by crushing to a point each of a sequence of mutually disjoint wild non-cellular arcs in R 3 :,41, a 2 , A 3 . . . . such that each A,, n~> 1, is a copy of the same wild arc A in the unit ball in R 3 translated by the vector (4n, 0, 0). This X is an ENR; indeed X x R is homeomorphic to S a x R = R 4 0 by a result of Andrews and Curtis [4]. Clearly this compactum admits self-homeomorphisms h : X ~ X arbitrarily near the identity which nontrivially permute the images o fA 1, A2, A3, .... But no such h is isotopic to the identity because these are isolated points at which Y fails to be a manifold. (See also the fish skeleton of w The treatment of non-manifolds rests roughly speaking on a method for deforming homeomorphisms on R" x cX, cY being the open cone on X, once one is given such a method on R "+1 xX-. Then the proof proceeds by induction on the depth of X. Here Xis regarded as a stratified set, and depth is the greatest difference of dimensions of nonempty strata of X. Stratified sets are vital to the proof because their open subsets are themselves stratified sets, and often of a lesser depth. Thus it will only clarify matters to deal from the outset with suitable stratified sets. I take this opportunity to introduce classes of pleasant stratified sets that may come to be the topological analogues of polyhedra in the piecewise-linear realm or of Thom's stratified sets in the differentiable realm. This technique of proof almost automatically provides strong relative and respectful deformation theorems (w 4.3, w 5.10), which a counterexample (w 2.3.1) suggests are

Infinitesimal computations in topology

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Surgery on compact manifolds

Abstract: Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important special case The even-dimensional case The odd-dimensional case The bounded odd-dimensional case The bounded even-dimensional case Completion of the proof Patterns of application: Manifold structures on Poincare complexes Applications to submanifolds Submanifolds: Other techniques Separating submanifolds Two-sided submanifolds One-sided submanifolds Calculations and applications: Calculations: Surgery obstruction groups Calculations: The surgery obstructions Applications: Free actions on spheres General remarks An extension of the Atiyah-Singer $G$-signature theorem Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces Applications: Free uniform actions on euclidean space Fake tori Polycyclic groups Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work Function space methods Topological manifolds Poincare embeddings Homotopy and simple homotopy Further calculations Sullivan's results Reformulations of the algebra Rational surgery References Index.

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.

Abstract: Since Poincare's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

Intersection homology II

Abstract: In [19, 20] we introduced topological invariants IH~,(X) called intersection homology groups for the study of singular spaces X. These groups depend on the choice of a perversity p: a perversity is a function from {2, 3, ...} to the non-negative integers such that both /~(c) and c 2 / ~ ( c ) are positive and increasing functions of c (2.1). The group IHr is defined for spaces X called pseudomanifolds: a pseudomanifold of dimension n is a space that admits a stratification X ~ X n ~ X n _ z ~ X n _ 3 ~ . . . ~ x 1 ~ X 0