generalized plumbings and murasugi sumspjgiblin/vvg60/slides/... · the sources john milnor,...

Generalized plumbings and Murasugi sums

Patrick Popescu-Pampu

Universite de Lille 1, France

Liverpool2 April 2016

Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

This is joint work with Burak OZBAGCI

(Koc University, Istanbul, Turkey)

It appeared in :

Arnold Mathematical Journal 2 (2016), 69-119.


Plumbing according to Mumford and Milnor

The term “plumbing” is a name for two different but relatedoperations :

• following Mumford, a cut-and-paste operation used todescribe the boundary of a tubular neighborhood of a union ofsubmanifolds of a smooth manifold, intersecting generically ;

• following Milnor, a purely pasting operation used to describethe tubular neighborhoods themselves.


The sources

John Milnor, Differentiable manifolds which are homotopyspheres. Mimeographed notes (1959). Published for the first timein Collected papers of John Milnor III. Differential topology.American Math. Soc. 2007, 65-88.

David Mumford, The topology of normal singularities of analgebraic surface and a criterion for simplicity. Inst. Hautes EtudesSci. Publ. Math. No. 9 (1961), 5-22.


The definition of plumbing

According to Hirzebruch-Neumann-Koh (“Differentiablemanifolds and quadratic forms”, 1971) :

Definition

“Let ξ1 = (E1, p1,Sn

1) and ξ2 = (E2, p2,Sn

2) be two oriented n-discbundles over Sn. Let Dn

i⊂ S

n

ibe embedded n-discs in the base

spaces and let :fi : D

n

i × Dn → Ei |Dn

i

be trivializations of the restricted bundles Ei |Dn

ifor i = 1, 2. To

plumb ξ1 and ξ2 we take the disjoint union of E1 and E2 andidentify the points f1(x , y) and f2(y , x) for each (x , y) ∈ Dn×Dn.”


Illustration of the plumbing operation

The previous definition is illustrated as follows byHirzebruch-Neumann-Koh :

Figure: Plumbing of two n-disc bundles


Murasugi’s notion of primitive s-surface (1963)

Figure: Primitive s-surface of type (n, 1), whose boundary is the(2, n)-torus link


Murasugi’s construction

Figure: Disks in primitive s-surfaces of type (2, 1) and of type (2,−1)are identified to give a Seifert surface for a figure-eight knot.


Open books

Definition

An open book in a closed manifold W is a pair (K , θ) consistingof :

1 a codimension 2 submanifold K ⊂ W , called the binding,with a trivialized normal bundle ;

2 a fibration θ : W \ K → S1 which, in a tubular neighborhood

D2 × K of K is the normal angular coordinate (that is, the

composition of the first projection D2 × K → D

2 with theangular coordinate D

2 \ {0} → S1).

Before the appearance of the name “open book” (Winkelnkemper1973), pages of open books in 3-dimensional manifolds were alsonamed “fibre surfaces”.


Stallings’ generalization (1978)

“Consider two oriented fibre surfaces T1 and T2. On Ti

let Di be 2-cells, and let h : D1 → D2 be anorientation-preserving homeomorphism such that theunion of T1 and T2 identifying D1 with D2 by h is a2-manifold T3. That is to say :

h(D1 ∩ Bd T1) ∪ (D2 ∩ Bd T2) = Bd D2. (1)

[Here Bd X denotes the boundary of X ].”


Stallings’ theorem on fiber surfaces

Theorem

If T1 and T2 are fibre surfaces, so is T3.

Corollary

The oriented link β obtained by closing a homogeneous braid β isfibered.


Homogeneous braids are fibered

Figure: On the left : the link β which is the closure of the homogeneousbraid β = σ−1

1 σ2σ−11 σ2. On the right : the top two disks with twisted

bands connecting them form a primitive s-surface of type (2,−1), whilethe lower two disks with twisted bands connecting them form a primitives-surface of type (2, 1). By gluing these primitive s-surfaces in theobvious way, we get a Seifert surface for β.


Gabai’s credo (1983)

Gabai (1983) coined the name “Murasugi sum” for a slightlyrestricted operation. He proved different instances of :

“The Murasugi sum is a natural geometricoperation.”


Lines’ extension to higher dimensions (1985)

Definition

Let K1 and K2 be two simple knots in S2k+1 bounding

(k − 1)-connected Seifert surfaces F1 and F2 respectively. Supposethat S2k+1 is the union of two balls B1 and B2 with a commonboundary which is a (2k)-sphere S . Let ψ : Dk × D

k → S be anembedding such that :

1 F1 ⊂ B1, F2 ⊂ B2 ;

2 F1 ∩ S = F2 ∩ S = F1 ∩ F2 = ψ(Dk × Dk) ;

3 ψ(∂Dk × Dk) = ∂F1 ∩ ψ(D

k × Dk) and

ψ(∂Dk × ∂Dk) = ∂F2 ∩ ψ(Dk × D

k).

Then the submanifold F := F1 ∪ F2 ⊂ S2k+1, after smoothing the

corners, is said to be obtained by plumbing together the Seifertsurfaces F1 and F2.


Our motivations

Our work is motivated by the search of the most generaloperation of Murasugi-type sum (that is, embeddedMilnor-style plumbing) for which one has an analog ofStallings’ theorem.

We figured out that we do not need to restrict in any way thefull-dimensional submanifolds which are to be identified in theplumbing operation. That is why we define a general operation of“summing” of manifolds, which reduces to the classical plumbingoperation when the identified submanifolds have product structuresDn × D

n.


Patched manifolds

The objects we sum abstractly are :

A

A A

P

M

Figure: A patched manifold (M ,P) with patch (P ,A)


Abstract summing

Our generalization of plumbing is :

M1

M2

A1

A1 P P

P

A2 A2

P⊎

=

Figure: The abstract sum M1

P⊎M2 of M1 and M2 along P


An alternative description

A1

A2

A1

A2

P

M2

M1 \ P

Figure: An alternative description of the abstract sum M1

P⊎M2


Embedded summing

Our generalization of Murasugi sum is :

M1

M2P⊎ =

positivethickpatch

negativethickpatch

Figure: Embedded sum (W1,M1)

P⊎(W2,M2) of two

patch-cooriented triples


Properties of the operation

Theorem

The patch being fixed, the operation of embedded sum ofpatch-cooriented triples is associative, but non-commutative ingeneral.


Our generalization of Stallings’ theorem

Theorem

Let (Wi ,Mi ,P)i=1,2 be two summable patched pages of openbooks on the closed manifolds Wi . Then the hypersurface

associated to the sum (W1,M1)P⊎(W2,M2) is again a page of an

open book.


Extension to Morse open books

Generalizing work of Weber, Pajitnov and Rudolph (2002) done indimension 3, we prove also :

Theorem

Let (Wi ,Mi ,P)i=1,2 be two regular pages of Morse open books onthe closed manifolds Wi . Then the hypersurface associated to the

sum (W1,M1)

P⊎(W2,M2) is again a regular page of a Morse

open book, whose multigerm of singularities is isomorphic to thedisjoint union of the multigerms of singularities of the initial Morseopen books.


The importance of coorientations

Let us see the principle of the proof.

We work without any assumptions about orientability of themanifolds : the only important issues are about coorientations,which makes the setting rather non-standard when compared withthe usual literature in differential topology.

∂−W

∂+W

W

Figure: A classical cobordism.


Cobordisms of manifolds with boundary

∂−W

∂+W

W

Figure: Cobordism of manifolds with boundaryW : ∂−W Z=⇒ ∂+W .


Mapping torus of an endobordism

glue by a diffeomorphism

W

T (W )

M− M+

M

Figure: Mapping torus of an endobordism


Splitting

M

W

ΣM(W )M− M+

σM

Figure: Splitting of W along a cooriented properly embeddedhypersurface M


Seifert hypersurfaces

Definition

Let W be a manifold with boundary. A compact hypersurface withboundary M → W is a Seifert hypersurface if :

the boundary of each connected component of M isnon-empty ;

M → int(W ) ;

M is cooriented.


Adapted angular coordinates

MW

Figure: Angular coordinate of ∂M adapted to M


The radial blow-up

M ′W

Figure: The radial blow-up of W along the boundary of the Seiferthypersurface M , and the strict transform M ′ of M .


Splitting along a Seifert hypersurface

W

Figure: The splitting of W along M .

One gets a cylindrical cobordism.


Cylindrical cobordisms and Seifert hypersurfaces

Lemma

The operations of taking the circle-collapsed mapping torus of acylindrical cobordism and of splitting along a Seifert hypersurfaceare inverse to each other.


Stiffened cylindrical cobordisms

In fact, splitting along a Seifert hypersurface produces a stiffenedcylindrical cobordism :

W

baseM

height

core C

I

Figure: A stiffened cylindrical cobordism W with directing segment I


Summable cylindrical cobordisms

The main idea of the proof may be seen on the figure :

P

W1

baseM1

baseM2 W2

heightI

coreC2

coreC1

P

Figure: Two summable stiffened cylindrical cobordisms


Two equivalent definitions

We show that the two summing operations give the same result :

Theorem

Let (W1,M1,P) and (W2,M2,P) be two summable patchedSeifert hypersurfaces. Then their embedded sum :

M1

P⊎M2 → (W1,M1)

P⊎(W2,M2)

is diffeomorphic, up to isotopy, to the Seifert hypersurfaceassociated to the sum of cylindrical cobordisms obtained bysplitting along the starting Seifert hypersurfaces :

ΣM1(W1)

P⊎ΣM2

(W2).


Open questions I

Question : We call an open book indecomposable if it cannot bewritten in a non-trivial way as a sum of open books. Find sufficientcriteria of indecomposability.

Question : Find sufficient criteria on germs of holomorphicfunctions f : (X , 0) → (C, 0) with isolated singularity to defineindecomposable open books.

Question : Find natural situations leading to triples (Xi , fi )1≤1≤3

of isolated singularities and holomorphic functions with isolatedsingularities on them, such that the Milnor open book of (X3, f3) isa sum of the Milnor open books of (X1, f1) and (X2, f2).


Open questions II

Question : Consider an open book and a contact structuresupported by this open book on a closed manifold. Describe anadapted position of a patch inside a page, relative to the contactstructure, allowing to extend the operation of sum of open booksto a sum of open books which support contact structures.


The end

Happy birthday Victor !


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