Generalized plumbings and Murasugi sums Patrick Popescu-Pampu - PowerPoint PPT Presentation

Generalized plumbings and Murasugi sums Patrick Popescu-Pampu Universit´ e de Lille 1, France Liverpool 2 April 2016 Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

This is joint work with Burak OZBAGCI (Ko¸ c 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 related operations : • following Mumford , a cut-and-paste operation used to describe the boundary of a tubular neighborhood of a union of submanifolds of a smooth manifold, intersecting generically ; • following Milnor , a purely pasting operation used to describe the tubular neighborhoods themselves. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

The sources John Milnor , Differentiable manifolds which are homotopy spheres. Mimeographed notes (1959). Published for the first time in Collected papers of John Milnor III. Differential topology. American Math. Soc. 2007, 65-88. David Mumford , The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes ´ Etudes Sci. Publ. Math. No. 9 (1961), 5-22. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

The definition of plumbing According to Hirzebruch-Neumann-Koh (“ Differentiable manifolds and quadratic forms ”, 1971) : Definition “Let ξ 1 = ( E 1 , p 1 , S n 1 ) and ξ 2 = ( E 2 , p 2 , S n 2 ) be two oriented n -disc bundles over S n . Let D n i ⊂ S n i be embedded n -discs in the base spaces and let : i × D n → E i | D n f i : D n i be trivializations of the restricted bundles E i | D n i for i = 1 , 2. To plumb ξ 1 and ξ 2 we take the disjoint union of E 1 and E 2 and identify the points f 1 ( x , y ) and f 2 ( y , x ) for each ( x , y ) ∈ D n × D n .” Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Illustration of the plumbing operation The previous definition is illustrated as follows by Hirzebruch-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 ) consisting of : a codimension 2 submanifold K ⊂ W , called the binding , 1 with a trivialized normal bundle ; a fibration θ : W \ K → S 1 which, in a tubular neighborhood 2 D 2 × K of K is the normal angular coordinate (that is, the composition of the first projection D 2 × K → D 2 with the angular coordinate D 2 \ { 0 } → S 1 ). Before the appearance of the name “open book” (Winkelnkemper 1973), pages of open books in 3-dimensional manifolds were also named “fibre surfaces”. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Stallings’ generalization (1978) “Consider two oriented fibre surfaces T 1 and T 2 . On T i let D i be 2 -cells, and let h : D 1 → D 2 be an orientation-preserving homeomorphism such that the union of T 1 and T 2 identifying D 1 with D 2 by h is a 2 -manifold T 3 . That is to say : h ( D 1 ∩ Bd T 1 ) ∪ ( D 2 ∩ Bd T 2 ) = Bd D 2 . (1) [Here Bd X denotes the boundary of X].” Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Stallings’ theorem on fiber surfaces Theorem If T 1 and T 2 are fibre surfaces, so is T 3 . Corollary The oriented link ˆ β obtained by closing a homogeneous braid β is fibered. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Homogeneous braids are fibered Figure : On the left : the link ˆ β which is the closure of the homogeneous braid β = σ − 1 1 σ 2 σ − 1 1 σ 2 . On the right : the top two disks with twisted bands connecting them form a primitive s-surface of type (2 , − 1), while the lower two disks with twisted bands connecting them form a primitive s-surface of type (2 , 1). By gluing these primitive s-surfaces in the obvious way, we get a Seifert surface for ˆ β . Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Gabai’s credo (1983) Gabai (1983) coined the name “ Murasugi sum ” for a slightly restricted operation. He proved different instances of : “ The Murasugi sum is a natural geometric operation. ” Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Lines’ extension to higher dimensions (1985) Definition Let K 1 and K 2 be two simple knots in S 2 k +1 bounding ( k − 1)-connected Seifert surfaces F 1 and F 2 respectively. Suppose that S 2 k +1 is the union of two balls B 1 and B 2 with a common boundary which is a (2 k )-sphere S . Let ψ : D k × D k → S be an embedding such that : F 1 ⊂ B 1 , F 2 ⊂ B 2 ; 1 F 1 ∩ S = F 2 ∩ S = F 1 ∩ F 2 = ψ ( D k × D k ) ; 2 ψ ( ∂ D k × D k ) = ∂ F 1 ∩ ψ ( D k × D k ) and 3 ψ ( ∂ D k × ∂ D k ) = ∂ F 2 ∩ ψ ( D k × D k ). Then the submanifold F := F 1 ∪ F 2 ⊂ S 2 k +1 , after smoothing the corners, is said to be obtained by plumbing together the Seifert surfaces F 1 and F 2 . Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Our motivations Our work is motivated by the search of the most general operation of Murasugi-type sum (that is, embedded Milnor-style plumbing) for which one has an analog of Stallings’ theorem . We figured out that we do not need to restrict in any way the full-dimensional submanifolds which are to be identified in the plumbing operation. That is why we define a general operation of “summing” of manifolds, which reduces to the classical plumbing operation when the identified submanifolds have product structures D n × D n . Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Patched manifolds The objects we sum abstractly are : M A A A P 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 : M 2 P � A 1 P P A 2 A 2 A 1 M 1 P = P � Figure : The abstract sum M 1 M 2 of M 1 and M 2 along P Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

An alternative description M 2 A 1 P A 2 A 2 A 1 M 1 \ P P � Figure : An alternative description of the abstract sum M 1 M 2 Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Embedded summing Our generalization of Murasugi sum is : positive thick patch M 2 P = � M 1 negative thick patch P � Figure : Embedded sum ( W 1 , M 1 ) ( W 2 , M 2 ) 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 of patch-cooriented triples is associative, but non-commutative in general. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Our generalization of Stallings’ theorem Theorem Let ( W i , M i , P ) i =1 , 2 be two summable patched pages of open books on the closed manifolds W i . Then the hypersurface P � associated to the sum ( W 1 , M 1 ) ( W 2 , M 2 ) 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 in dimension 3, we prove also : Theorem Let ( W i , M i , P ) i =1 , 2 be two regular pages of Morse open books on the closed manifolds W i . Then the hypersurface associated to the P � sum ( W 1 , M 1 ) ( W 2 , M 2 ) is again a regular page of a Morse open book, whose multigerm of singularities is isomorphic to the disjoint union of the multigerms of singularities of the initial Morse open 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 the manifolds : the only important issues are about coorientations , which makes the setting rather non-standard when compared with the 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 boundary W : ∂ − W � = ⇒ ∂ + W . Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Mapping torus of an endobordism M + M − W glue by a diffeomorphism M T ( W ) Figure : Mapping torus of an endobordism Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Splitting M W σ M M − M + Σ M ( W ) Figure : Splitting of W along a cooriented properly embedded hypersurface M Patrick Popescu-Pampu Generalized plumbings and Murasugi sums

Seifert hypersurfaces Definition Let W be a manifold with boundary. A compact hypersurface with boundary M ֒ → W is a Seifert hypersurface if : the boundary of each connected component of M is non-empty ; M ֒ → int( W ) ; M is cooriented. Patrick Popescu-Pampu Generalized plumbings and Murasugi sums