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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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.”
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Illustration of the plumbing operation
The previous definition is illustrated as follows byHirzebruch-Neumann-Koh :
Figure: Plumbing of two n-disc bundles
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Murasugi’s notion of primitive s-surface (1963)
Figure: Primitive s-surface of type (n, 1), whose boundary is the(2, n)-torus link
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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”.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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 ].”
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.”
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Patched manifolds
The objects we sum abstractly are :
A
A A
P
M
Figure: A patched manifold (M ,P) with patch (P ,A)
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
An alternative description
A1
A2
A1
A2
P
M2
M1 \ P
Figure: An alternative description of the abstract sum M1
P⊎M2
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Properties of the operation
Theorem
The patch being fixed, the operation of embedded sum ofpatch-cooriented triples is associative, but non-commutative ingeneral.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Cobordisms of manifolds with boundary
∂−W
∂+W
W
Figure: Cobordism of manifolds with boundaryW : ∂−W Z=⇒ ∂+W .
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Mapping torus of an endobordism
glue by a diffeomorphism
W
T (W )
M− M+
M
Figure: Mapping torus of an endobordism
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Splitting
M
W
ΣM(W )M− M+
σM
Figure: Splitting of W along a cooriented properly embeddedhypersurface M
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Adapted angular coordinates
MW
Figure: Angular coordinate of ∂M adapted to M
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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 .
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
Splitting along a Seifert hypersurface
W
Figure: The splitting of W along M .
One gets a cylindrical cobordism.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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).
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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).
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
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.
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums
The end
Happy birthday Victor !
Patrick Popescu-Pampu Generalized plumbings and Murasugi sums