Cone complexes and PL transversality

McCrory, Clint

doi:10.1090/s0002-9947-1975-0400243-7

Cited by 26 publications

(23 citation statements)

References 21 publications

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“…Now the two preceding propositions coupled with general theory ( [6,14]) immediately imply the following. Recall that a finite regular cell complex is a Hausdorff space X with a finite collection of closed cells c i ⊂ X whose interiors c Before continuing on, let us also record the local structure of arboreal singularities.…”

Section: From the Above Equations We Conclude That L T (P ) Is In Thmentioning

confidence: 80%

Arboreal singularities

Nadler

2017

Geom. Topol.

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Abstract. We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.

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Section: From the Above Equations We Conclude That L T (P ) Is In Thmentioning

confidence: 80%

Arboreal singularities

Nadler

2017

Geom. Topol.

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“…More material can be found in Rourke and Sanderson [37]. The notion of intrinsic stratification is taken from Akin [1], Armstrong [4], McCrory [32] and Stone [38] and described in Section 2.3. Stein factorization (which we take from Costantino and Thurston [11]) is introduced in Section 2.4.…”

Section: Piecewise-linear Topologymentioning

confidence: 99%

“…We recall the notions of intrinsic dimension and strata of polyhedra; see Akin [1], Armstrong [4], McCrory [32] and Stone [38].…”

Section: Intrinsic Stratamentioning

confidence: 99%

Complexity of PL manifolds

Martelli

2010

Algebr. Geom. Topol.

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We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant).Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group). 57Q99; 57M99 IntroductionThe complexity c.M / of a compact 3-manifold M (possibly with boundary) was defined in a nice paper of Matveev [29] as the minimum number of vertices of an almost simple spine of M . In that paper he proved the following properties:Finiteness There are finitely many closed irreducible (or cusped hyperbolic) 3-manifolds of bounded complexity.Monotonicity If M S is obtained by cutting M along an incompressible surface S , then c.M S / 6 c.M /.Thanks to the combinatorial nature of spines, it is not hard to classify all manifolds having increasing complexity 0; 1; 2; : : :. Tables have been produced Table 1 below). Some of these classifications were actually done using the dual viewpoint of singular triangulations, which turns out to be equivalent to Matveev's for the most interesting 3-manifolds.We extend here Matveev's complexity from dimension 3 to arbitrary dimension. To do this, we need to choose an appropriate notion of spine. In another paper [27] written in 1973, Matveev defined and studied simple spines of manifolds in arbitrary dimension. A simple spine of a compact manifold is a (locally flat) codimension-1 subpolyhedron with generic singularities, onto which the manifold collapses. If the manifold is closed there cannot be any collapse at all and we therefore need to priorly remove one ball.Simple spines are actually not flexible enough for defining a good complexity. In dimension 3, as an example, any simple spine for S 3 (or, equivalently, D 3 ) is a complicated and unnatural object, such as Bing's house or the abalone. Every simple spine of D 3 has at least one vertex. However, a reasonable complexity must be zero on discs and spheres.To gain more flexibility, Matveev defined in 1988 the more general class of almost simple spines [28; 29] of 3-manifolds. An almost simple polyhedron is a compact polyhedron that can be locally embedded in a simple one. This more general definition allows one to use very natural objects as spines, such as a point for D 3 or a circle for D 2 S 1 : a point is not a vertex by definition, ...

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“…The pictures of cochains are a special case of the geometric cochains in stratified objects defined in [7]. See also [15], [16], [23] and [25]. Using geometric pictures of cochains and Rules I and II, the technique for calculating Massey products in a 2-dimensional CW complex is as follows.…”

Section: Figure 5amentioning

confidence: 99%

“…The proof of Theorem 2 is based on a geometric interpretation of cup products and coboundaries of cochains motivated by R. M. Goresky's geometric description of the algebraic topology of stratified objects, [7]. See also [15], [16], [23], [24] and [25]. I am indebted to W. S. Massey for suggesting that Goresky's viewpoint be applied to the problem of calculating Massey products, and for several very helpful conversations.…”

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confidence: 99%