aa r X i v : . [ m a t h . G T ] J a n HANDLEBODY NEIGHBOURHOODS AND A CONJECTURE OFADJAMAGBO
DAVID GAULD
Abstract.
Using handlebodies we verify a conjecture of P Adjamagbo that if the frontier of arelatively compact subset V of a manifold is a submanifold then there is an increasing family { V r } of relatively compact open sets indexed by the positive reals so that the frontier of each is asubmanifold, their union is the whole manifold and for each r ≥ r, ∞ )is a neighbourhood basis of the closure of the r th set. Introduction
Throughout this paper we use the euclidean metric on R n . All of our manifolds are assumed tobe metrisable. No extra structure on the manifold is assumed.Pascal Adjamagbo, [1], has proposed the following conjecture :Given a relatively compact non-empty open subset V of a connected topologicalmanifold M m such that the boundary of V is a manifold, there exists an increas-ing family h V r i r ∈ [0 , ∞ ) of relatively compact open subsets of M the boundaries ofwhich are topological manifolds such that M is the union of the elements of thefamily, and that for any r ∈ [0 , ∞ ), the family h V r ′ i r ′ >r is a fundamental system ofneighbourhoods of the closure of V r .Adjamagbo makes no assumptions regarding the tameness of the boundary manifold, so it couldbe wild at every point.In this paper we verify Adjamagbo’s conjecture.An important tool used in our proof is a handlebody, a special structure on (part of) a manifold. Definition 1.
Let M m be a topological manifold with boundary and let k ∈ { , , . . . , m } . Supposethat e : S k − × B m − k → ∂M is an embedding and let M e be the m -manifold obtained from thedisjoint union of M and B k × B m − k by identifying x ∈ S k − × B m − k with e ( x ) ∈ ∂M . Then wesay that M e is obtained from M by adding a k -handle of dimension m to M , with the prefix k suppressed when we do not want to specify it. The image of B k × B m − k in M e is called a ( k )-handle . A handlebody is a manifold obtained inductively beginning at ∅ then successively addinga handle of dimension m to the output of the previous step. If a handlebody has been obtained byadding only finitely many handles then we call it a finite handlebody . In this definition we take B ℓ = { x ∈ R ℓ / | x | ≤ } , the unit ball in R ℓ , so S ℓ − is the boundarysphere of B ℓ . Of course when constructing a handlebody, of necessity the first handle to be addedmust be a 0-handle as ∂ ∅ = ∅ and S k − = ∅ when k >
0. A handlebody is a manifold withboundary.The following two theorems characterise the existence of handlebody structures on topologicalmanifolds.
Date : February 1, 2021. I thank Adjamagbo for drawing my attention to this conjecture.
Theorem 2. [3, Theorem 9.2]
A metrisable manifold fails to have a handlebody decomposition ifand only if it is an unsmoothable 4-manifold.
Theorem 3. [3, Theorem 8.2]
Every connected, non-compact, metrisable 4-manifold is smoothable.
A basic result needed in our proof is the following proposition.
Proposition 4. [4, Proposition 3.17]
Suppose that M is a handlebody and K ⊂ M is compact.Then there is a finite handlebody W ⊂ M which is a neighbourhood of K . We also require the following weak version of the collaring theorem of Brown.
Theorem 5. [2]
Let W be a finite handlebody. Then there is an embedding e : ∂W × [ a, b ] → W ( a < b ) such that e ( x, b ) = x for each x ∈ ∂W . The embedding e is called a collar of the boundary ∂W . Using the notation of Theorem 5,we will call the set e ( c ) a level of the collar and the subset W \ e ( ∂W × ( c, b ]) will be said to be inside the level e ( c ). A set inside a level of a handlebody is a compact subset of W ; moreover theboundary of this set is a manifold of one lower dimension and, when c > a , the set inside the level e ( c ) is homeomorphic to W so is itself a finite handlebody.2. Handlebody Neighbourhoods and Adjamagbo’s Conjecture
In this section we construct a neighbourhood basis of a compactum in a manifold where theneighbourhoods making up the basis are all handlebodies. We then use this construction to proveAdjamagbo’s conjecture.
Theorem 6.
Let M m be a connected manifold and V ⊂ M a non-empty, relatively compact, opensubset of M such that the frontier of V in M is an ( m − -manifold. Suppose further in the casewhere M is closed that V = M . Then for each real number r ∈ (0 , there is a finite handlebody W r such that for each r ∈ [0 , , the collection { Int( W r ′ ) / r ′ > r } is a neighbourhood basis of W r ,where W is the closure of V .Proof. Suppose given M m and V ⊂ M as in Theorem 6. In the case where M is closed pick apoint p ∈ M \ V . It follows that M (in the case where M is open) or M \ { p } (in the case where M is closed) may be embedded properly in some euclidean space, see [4, Theorem 2.1(1 ⇔ d be the metric on M or M \ { p } as appropriate inherited from the euclidean metricunder some fixed proper embedding.For each natural number n let U n = (cid:26) x ∈ M / d ( x, V ) < n (cid:27) . Then U n +1 ⊂ U n for each n and the collection { U n / n = 1 , , . . . } is a neighbourhood basis for V .For each n use Proposition 4 to find a finite handlebody X n ⊂ U n containing U n +1 in its interior.By Theorem 5 we may find a collar e n : ∂X n × h n +2 , n i → X n such that e n (cid:0) x, n (cid:1) = x for each x ∈ ∂X n . Further, compactness of the disjoint sets U n +1 and ∂X n allows us to assume that theimage of e n is disjoint from U n +1 .For each r ∈ (0 ,
1) choose W r as follows. There is a unique natural number n such that n +1 ≤ r < n . Let W r be the set inside the level e n ( r ) of the handlebody X n . As noted above, W r is ahandlebody. Moreover, for each r ∈ [0 , W r ′ ) / r ′ > r } is a neighbourhoodbasis of W r . (cid:3) We are now ready to prove Adjamagbo’s conjecture.
ANDLEBODY NEIGHBOURHOODS AND A CONJECTURE OF ADJAMAGBO 3
Corollary 7.
Let M m be a connected topological manifold and V ⊂ M be a relatively compactnon-empty open subset V of M such that the boundary of V is a manifold. Then there existsa family { V r / r ∈ [0 , ∞ ) } of relatively compact open subsets of M the boundaries of which aretopological manifolds such that M is the union of the elements of the family, and that for any r ∈ [0 , ∞ ) the family { V r ′ / r ′ > r } is a neighbourhood basis of the closure of V r .Proof. The case where V = M is trivial so we assume that V = M .Applying Theorem 6 to the case V = M , for each r ∈ (0 ,
1) set V r = Int( W r ). Then each V r isopen and relatively compact with boundary a topological manifold, and for each r < { V r ′ / r ′ > r } is a neighbourhood basis of the closure of V r . The proof is complete in the casewhere M is closed if we set V r = M for each r ≥ M is open. In this case follow the procedure in the proof of Theorem 6 but now replacethe sets U n by sets U ′ n = (cid:8) x ∈ M / d ( x, V ) < n (cid:9) and the handlebodies X n by finite handlebodies X ′ n whose boundaries lie in U ′ n +1 \ U ′ n . For each natural number n and each r ∈ [ n, n + 1) we thenconstruct the open sets V r to be the interiors of sets inside appropriate levels of the handlebody X ′ n . (cid:3) Remark 8. If V is connected in Corollary 7 then we may also assume that each of the neighbour-hoods V r is also connected. To ensure this, when we construct the handlebodies X n , we discard anysupernumerary components of the handlebodies. Remark 9.
The assumption in Corollary 7 that M is connected can be weakened to the assumptionthat M has countably many components. We may construct sets such as the sets V r in the proofof Corollary 7 within each component of M , beginning with an arbitrary point of the component inplace of V when a component contains no point of V , and then taking V r to be the union of theresulting sets. Questions
In our construction of the neighbourhoods V r in the proof of Corollary 7 the neighbourhoodsgrow gradually for a while then jump: for example V \ ∪ r< V r is non-empty. So the followingquestion might be addressed. Question 10.
Given a relatively compact non-empty open subset V of a topological manifold M such that ∂V is a manifold, is it possible to find a family { V r / r ∈ [0 , ∞ ) } of relatively compact opensubsets of M the boundaries of which are topological manifolds such that M = V ∪ ( ∪ r ∈ [0 , ∞ ) ∂V r ) ,and that for any r ∈ [0 , ∞ ) the family { V r ′ / r ′ > r } is a fundamental system of neighbourhoods ofthe closure of V r ? (Here ∂V denotes the boundary of V .) We might also explore Adjamagbo’s conjecture in other categories. As an example:
Question 11.
Given a relatively compact non-empty open subset V of a connected smooth man-ifold M m such that the boundary of V is a smooth submanifold, there exists an increasing family h V r i r ∈ [0 , ∞ ) of relatively compact open subsets of M the boundaries of which are smooth submanifoldssuch that M is the union of the elements of the family, and that for any r ∈ [0 , ∞ ) , the family h V r ′ i r ′ >r is a fundamental system of neighbourhoods of the closure of V r . Morse functions may well play a role in answering the latter question.
References [1] Pascal Adjamagbo, private communication, 3 January, 2021.[2] Morton Brown,
Locally flat imbeddings of topological manifolds , Ann. Math. 75(1962), 331–341.
DAVID GAULD [3] Michael H Freedman and Frank Quinn: Topology of 4-manifolds. Princeton University Press, Princeton (1990).[4] David Gauld,
Non-metrisable manifolds , Springer, Singapore, 2014, https://link.springer.com/book/10.1007%2F978-981-287-257-9 . The University of Auckland
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