SPINES OF TOPOLOGICAL MANIFOLDS

ERIK KJÆR PEDERSEN

In this paper we prove that a closed 2-connected topological manifold has a PL-spine, i. e. there is a locally tamely embedded complex such that a regular neighborhood of this complex is the manifold with a disc deleted (dimension is assumed to be at least 6). This “spine method” together with the relative edition of regular neighborhoods of complexes in topological manifolds [5] makes it easy to use general position arguments in topological manifolds. This will be used in a forthcoming paper to extend various embedding theorems to the topological category.

The methods we use are PL-approximation theorems due to Cernavskii, Connally, Miller, Rushing ... as quoted in [5, theorem 2] and blocktransersality for PL complexes and PL submanifolds as was first considered by C. Morlet [4] and later extended by D. Stone [6].

Definition 1. A spine of a topological manifold M with ∂M ≠ ∅ is a locally tamely embedded complex K ⊂ M so that K is a strong deformation retract of M and K ⊂ M is a simple homotopy equivalence. In case ∂M = ∅ by the spine of M we mean a spine of M with a disc deleted.

Theorem 2. Let (M, ∂_{−}M, ∂_{+}M) be a triad of topological manifolds dim(M) = m, and assume m ≥ 6 and

π_{j}(M, ∂_{+}M) = 0  for  j < m − r,  r ≤ m − 3.

Further assume there is a PL-complex P locally tamely embedded in the interior of ∂_{+}M, dim(P) = p and m − p ≥ 4. Then there is a complex K of dimension max(p + 1, r, 2), locally tamely embedded in M such that

K ∩ ∂_{+}M = P
K ∩ ∂_{−}M = K′

K′ a subcomplex of K, K′ has a regular neighborhood of the form K′ × I in K and

∂_{−}M ∪ K ⊂ M

is a strong deformation retract and a simple homotopy equivalence.

Theorem 2 has an immediate corollary:

Corollary 3. Let M be a closed topological manifold, dim(M) ≥ 6 and π_{j}(M) = 0 for j ≤ r, r > 2. Then M has a spine of dimension m − r.

Proof. Let \overline{M} = M − (interior of a disc). Put ∂_{−}M = ∅, ∂_{+}M = S^{m−1}, P = ∅, and apply Theorem 2. □

Proof of Theorem 2. First let us consider the case where P = ∅. Put k = max(r, 2). According to Kirby and Siebenmann [3] M has a handlebody decomposition relative to ∂_{−}M with no handles of dimension greater than k: Kirby and Siebenmann prove that (M, ∂_{−}M) has a handle body decomposition, and one can then cancel handles to get a minimal handle decomposition. Because of problems with torsion one needs at least 1- and 2-handles.

We filter M by the handle filtration

∂_{−}M × I = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{s} = M

where M_{i+1} is obtained from M_{i} by adjoining a single handle, no handles of dimension greater than m − 3. The proof will be by downwards induction on the statement:

There is a locally tamely embedded complex

K_{i} ⊂ \overline{M − M_{i}} dim(K_{i}) ≤ k

such that

K′_{i} = K_{i} ∩ ∂_{+}M_{i}

is contained in the interior of ∂_{+}M_{i}, K′_{i} has a neighborhood in K_{i} of the form K′_{i} × [0, 1] and M_{i} ∪ K_{i} is a simple strong deformation retract of M_{i}.

It is easy to start the induction, we let K_{s−1} be the core of the last handle. Then clearly M_{s−1} ∪ K_{s−1} is a simple strong deformation of M = M_{s}, so assume the statement for i + 1. Now

M_{i+1} = M_{i} ∪_{S^{j−1}×D^{m−j}} D^{j} × D^{m−j}

for some j ≤ k. Let

\overline{E} = D^{j} × D^{m−j} ∩ ∂M_{i+1} = D^{j} × S^{m−j−1}

take an outside collar ∂\overline{E} × [0, 2] of ∂\overline{E} in ∂_{+}M_{i+1} and let

E_{1} = \overline{E} ∪ ∂\overline{E} × [0, 1], E_{2} = \overline{E} ∪ \overline{E} × [0, 2].

\overline{E} has a PL structure being a codimension 0 submanifold of the boundary of D^{j} × D^{m−j} and we can extend this PL structure to E_{1} and E_{2} using the collar. K′_{i+1} is of codimension more than 3 in ∂_{+}M_{i+1}, so by [1], see e. g. [5, Theorem 2], since dim ∂_{+}M_{i+1} ≥ 5, there is an ambient ε-isotopy of E_{2} fixing ∂E_{2} that moves K′_{i+1} to be PL embedded in E_{2}, except in a neighborhood of ∂E_{2} which can be assumed small. So we may assume, since this can be taken to be the restriction of an ambient isotopy of M, that K′_{i+1} ∩ E_{1} ⊂ E_{1} is PL. Using [4] we can isotop K′_{i+1} further by a small ambient isotopy so that K′_{i+1} intersects ∂\overline{E} = S^{j−1} × S^{m−j−1} blocktransversally. Assume this done and denote

Z = K′_{i+1} ∩ ∂\overline{E}

Since the normal blockbundle of ∂\overline{E} in E_{1} is a trivial one dimensional bundle we obtain that ∂\overline{E} has a neighborhood in E of the form ∂ → E × (−1, 1) and

∂\overline{E} × (−1, 1) ∩ K′_{i+1} = Z × (−1, 1)

since it is the restriction of the trivial blockbundle to Z, by blocktransversality. Z is a PL subcomplex of S^{j−1} ×S^{m−j−1}, which is the boundary of S^{j−1} ×D^{m−j} of dimension m−1−r, so of codimension at least 3. By [2, Theorem 5.2] there is a subcomplex Z′ of S^{j−1} × D^{m−j} of dimension min(dim(Z) + 1, j) so that

Z′ ∩ S^{j−1} × S^{m−j−1} = Z

and S^{j−1} × D^{m−j} simplicially collapses to Z′ (S^{j−1} × D^{m−j} is the mapping cylinder of the projection S^{j−1} × S^{m−j−1} → S^{j−1}, so take Z′ to be the mapping cylinder of the restriction to Z). Using [2, lemma 2.20] this implies that

S^{j−1} × D^{m−j} × I

simplicially collapses to

S^{j−1} × D^{m−j} × 0 ∪ Z′ × I ∪ S^{j−1} × D^{m−j} × I

so taking S^{j−1} × D^{m−j} × I to be a collar of S^{j−1} × D^{m−j} in D^{j} × D^{m−j} we see that if we define D to be

D = \overline{D^{j} × D^{m−j} − S^{j−1} × D^{m−j} × I}

there is a simple strong deformation retract of M_{i+1} ∪ K_{i+1} to M_{i} ∪ Z′ × I ∪ K_{i+1} ∪ D. However D is a disc, and Z′ × 1 ∪ K_{i+1} ∩ ∂D is of codimension bigger than 3 in ∂D, so we may as before assume it is PL-embedded and D now simplicially collapses to the cone of Z′ × 1 ∪ K_{i+1} ∩ ∂D thus finishing the induction step. It is clear by construction that K′_{i} has a product neighborhood in K_{i}.

In case P ≠ ∅ the proof is the same except we have to go through the motions of the induction step in the initial step of the induction too. □

References

1. R. Connelly, Unknotting close embeddings of polyhedra, Topology of Manifolds, (Univ. of Georgia, 1969), Markham Publishing Company, Chicago, 1970, pp. 384–389.

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

3. R. Kirby and L. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Mathematics Studies, vol. 88, Princeton Univ. Press, 1977.

4. C. Morlet, Topologie des variétés semi-linéaires, Ann. Sci. École Norm. Sup. (4) 1 (1968), 313–394.

5. E. K. Pedersen, Topological concordances, Invent. Math. 38 (1977), 255–267.

6. D. A. Stone, Stratified polyhedra, Lecture Notes in Mathematics, vol. 252, Springer, 1972.

Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Ill. 60637, USA

4 ERIK KJÆR PEDERSEN

Present address: Matematisk Institut, Odense Universitet, Niels Bohrs Allé, 5000 Odense, Danmark