TOPOLOGICAL CONCORDANCES

ERIK KJÆR PEDERSEN

1. Introduction

In this announcement we outline a proof of Hudson’s theorem, “Concordance implies isotopy in codimension 3” [3], for topological manifolds. This theorem is also proved by Armstrong [1] in case the ambient manifold is PL, using methods similar to those employed by Hudson. Our approach is to generalize the methods of Morlet [5] to the topological case, using PL approximation theorems locally. We have also generalized Morlet’s results on the higher homotopy groups of concordance spaces to the topological case [6]. Detailed proofs will appear elsewhere [7]. I wish to thank professor Lashof at the University of Chicago for many useful discussions and suggestions.

2. Statement of results

Let M and V be topological manifolds. By an embedding ϕ : M → V we understand a locally flat embedding such that ϕ^{−1}(∂V) = ∂M and ϕ meets the boundary regularly. We prove the following theorems:

Theorem 1. Let V^{n} be a topological manifold and M^{q} a compact topological handlebody. Let

ϕ : (M × I, M × 0, M × 1) → (V × I, V × 0, V × 1)

be an embedding. Denote ϕ|M × 0 by g and assume

ϕ|∂M × I = (g|∂M) × 1)I

(∂M may be empty) and n − q ≥ 3, n ≥ 5. If n = 5 we further assume that ∂V is a stable manifold. Then there is an ambient isotopy H^{t} of V × I, fixing ∂V × I ∪ V × 0 such that H^{1} ∘ ϕ = g × 1_{I}.

Theorem 2. Let V^{n} be a topological manifold and (K, K′) a finite polyhedral pair, dim(K − K′) = q (K′ may be empty). Let

ϕ : (K × I, K × 0, K × 1) → (V × I, V × 0, V × 1)

be a locally nice embedding such that ϕ^{−1}(∂V ×I) = K′ ×I. Denote ϕ|V ×0 by g and assume ϕ is equal to g × 1_{I} in a neighborhood of K′ × I. Also assume n − q ≥ 3 and n ≥ 5. If n = 5 we assume ∂V is a stable manifold. Then there is an ambient isotopy H^{t} of V × I fixing ∂V × I ∪ V × 0 such that H^{1} ∘ ϕ = g × 1_{I}.

3. Outline of proof

First we prove Theorem 1 for M a disc. So consider a concordance ϕ : D^{q} × I → V × I and denote ϕ(D^{q} × I) by L_{1} and g(D^{q}) × I by L_{0}.

(1) We find PL product neighborhood U of L_{0}. Using PL approximation theorems we can then isotop L_{1} and shrink U such that U ∩L_{1} ⊂ U is a PL embedding. We can further isotop L_{1} such that L_{1} intersects L_{2} blocktransversally, and both these isotopies can be made such that the restriction to ϕ(∂D^{q} × I) = g(∂D^{q}) × I is a product isotopy.

(2) Having moved L_{1} to intersect L_{0} block-transversally, denote X = L_{0} ∩ L_{1}, X^{0} = X ∩ V × 0 and X^{1} = X ∩ V × 1. We define τ(X) ≥ α if X is obtained from X^{0} × I by adding handles not meeting ∂X^{0} of dimension j, α ≤ j ≤ dim(X) − α. Assume τ(X) ≥ α − 1 and let μ_{s} : D^{α−1} → X be the cores of the α − 1 handles relative to X^{0} and X^{1}. μ_{s} can be bounded by embedded discs in L_{0} as well as in L_{1}, and the union of the bounding discs can be bounded in V × I − L_{0} ∪ L_{1} by an embedded disc F_{2}(D^{α+1}. We prove this using homotopy-theoretic considerations and Less’ immersion theorem [4]. Now we slide L_{1} across F_{s}(D^{α+1}) (analogously to the standard Whitney trick for deletion of double points) so as to remove a regular neighborhood of μ_{s}(D^{α−1}). We thus obtain that τ(resulting X) ≥ α.

(3) We inductively obtain that τ(X) > \frac{1}{2} dim(X), so X is a product. This product structure extends to L_{0}∪L_{1} and further to V ×I (using the topological s-cobordism theorem). Using this product structure we move L_{1} back so that its image finally coincides with L_{0}, and we can then use an Alexander isotopy to straighten L_{1}.

(4) The isotopy we obtained above does not fix ∂D^{q} × I ∪ D^{q} × 0, but it may be modified to do so by using a collar of the boundary. We finally obtain an ambient isotopy using the isotopy extension theorem [2]

The general case follows from that of a disc by induction on skeleta. First we straighten the 0-skeleton. The we straighten a neighborhood of the 0-skeleton using local PL approximation and Hudson’s theorem. Cutting out a regular neighborhood of the 0-skeleton makes the 1skeleton into a disjoint union of 1-discs which can be straightened from what we already have proved and, we proceed inductively.

References

1. M. A. Armstrong, Collars and concordances of topological manifolds, Comment. Math. Helv. 45 (1970), 119–128.

2. R. D. Edwards and R. C. Kirby, Deformations of spaces of embeddings, Ann. of Math. (2) 93 (1971), 63–88.

3. J. F. P. Hudson, Piecewise Linear Topology, W. A. Benjamin,Inc., New York, 1969.

4. J. A. Lees, Immersions and surgeries on topological manifolds, Bull. Amer. Math. Soc. (N.S.) 475 (1969), 529–534.

5. C. Morlet, Mimeographed notes, Cours Pecot, 1969.

6. E. K. Pedersen, Topological concordances, Ph.D. thesis, University of Chicago, 1974.

7. , Topological concordances, Invent. Math. 38 (1977), 255–267.

TOPOLOGICAL CONCORDANCES 3

Department of Mathematics, University of Chicago, Chicago, Illinois 60637 Department of Mathematics, Aarhus University, Aarhus, Denmark