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Open manifolds admit handle filtrations without top-index handles

Statement

Assume the axiom of countable choice. Let $M$ be a nonempty connected open smooth $m$ -manifold without boundary (no compact component, hence noncompact). Then there is a sequence $M_{0} \subseteq M_{1} \subseteq M_{2} \subseteq \dots$ of compact $m$ -submanifolds with boundary with $M_{j} \subseteq int M_{j + 1}$ and $M = ⋃_{j} M_{j}$ such that for every $j$ the band $M_{j + 1} ∖ int M_{j}$ has the following presentation: each connected component $C$ of the band has nonempty outgoing boundary $\partial^{+} C = C \cap \partial M_{j + 1}$ and admits a handle decomposition relative to $\partial^{-} C = C \cap \partial M_{j}$ with no handles of index $m$ . Equivalently, $M_{j + 1}$ is obtained from $M_{j}$ by first extending the boundary through a collar, then adding finitely many disjoint components by $0$ -handles and attaching finitely many further handles of index at most $m - 1$ ; no $m$ -handle is ever needed. Countable choice also selects one finite handle presentation for each band component; the bands are fixed before these independent selections.

Facts & Assumptions

Given: ${A C}_{ω}$ and a connected open smooth $m$ -manifold $M$ without boundary and with no compact component.

[F1]

The cap-free exhaustion: there are compact $m$ -submanifolds with boundary $M_{0} \subseteq M_{1} \subseteq \dots$ with $M_{j} \subseteq int M_{j + 1}$ and $M = ⋃_{j} M_{j}$ , such that no $M_{j}$ has a cap and every connected component of every band $M_{j + 1} ∖ int M_{j}$ has nonempty outgoing boundary (Open manifolds admit exhaustions with no caps).

[L1]

DT-6 dual elimination: a compact connected triad $(W; M_{0}, M_{1})$ with $M_{1} \neq \emptyset$ admits a handle decomposition relative to $M_{0}$ with no handles of index $\dim W$ , and the conventions of the relative handle decomposition and of a smooth cobordism triad allow the empty incoming face and the empty outgoing face (Dual elimination of top-index handles, Handle decomposition relative to the incoming boundary, Smooth cobordism triad for Morse theory); the proposition assumes ${A C}_{ω}$ .

[L2]

Every compact manifold with boundary has a collar, its boundary is a closed embedded submanifold, and a compact locally connected space has finitely many components (Collar neighborhood theorem, The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, Embedded smooth submanifolds with boundary, Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

Proof

technique · direct

1.1F1L2givenconstruct

Fix $j$ and a connected component $C$ of the band $M_{j + 1} ∖ int M_{j}$ . The band is a compact $m$ -manifold with boundary and $C$ is a compact connected component of it; the boundary of $C$ is the disjoint union of the faces $C \cap \partial M_{j}$ and $C \cap \partial M_{j + 1}$ , each a closed embedded submanifold of the corresponding boundary, and [L2] supplies collars of both. Hence $(C; C \cap \partial M_{j}, C \cap \partial M_{j + 1})$ is a compact connected smooth cobordism triad, and its outgoing face $C \cap \partial M_{j + 1}$ is nonempty by the cap-freeness in [F1].

2.1L1step 1.1

Apply [L1] to this triad: because the outgoing face is nonempty, $C$ admits a handle decomposition relative to the incoming face $C \cap \partial M_{j}$ with no handles of index $m = \dim C$ . When the incoming face is empty, the same proposition is applied with the empty incoming face convention, so the presentation begins with $0$ -handles and again uses no $m$ -handle.

3.1F1L1L2step 2.1construct

The band has finitely many connected components by compactness and local connectedness [L2]. Assembling the presentations of its components and the collar $\partial M_{j} \times [0, ε]$ implicit in the relative convention gives a presentation of $M_{j + 1}$ from $M_{j}$ : extend the boundary through the collar, then add each component by $0$ -handles and further handles of index at most $m - 1$ . In particular no $m$ -handle is ever needed.

4.1F1L1L2step 3.1∎

The only in-run suppliers used are [F1] and [L1], both of which assume ${A C}_{ω}$ , and the standard collar and boundary items of [L2]; no handle cancellation, Whitney trick or later page is invoked. The bands have already been fixed; each has finitely many components. Index a band component $C$ by its band number and the least member of a fixed countable coordinate basis contained in ${int}_{M} C$ . Each component has a nonempty ambient interior, so such a member exists. Distinct components of one band have disjoint interiors and therefore cannot receive the same nonempty basis member. This gives an injection into $N \times N$ . Countable choice selects a complete finite handle presentation for each of this at-most-countable family of nonempty witness sets. This proves the claimed filtration and the stated description of the band presentations.

Depends on

Used by

Smale–Hirsch for open source manifolds Theorem

Dependency tree · two levels

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