Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

📝 Abstract

With a compact PL manifold X we associate a category T(X). The objects of T(X) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence BT (X) \approx BPL(X) holds, where PL(X) is the simplicial group of PL-homeomorphisms. Thus the space BT(X) is a canonical countable (as a CW-complex) model of BPL(X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a T(X)-coloring of some triangulation K of B. The vertices of K are colored by objects of T(X) and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: BT(S^{n-1}) \approx BO(n) for n=1,2,3,4. The result is proved in a sequence of results on similar models of B\PL(X). Special attention is paid to the main noncompact case X=R^n and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on “fragmentation of a fiberwise homeomorphism”, a generalization of the folklore lemma on fragmentation of an isotopy.

💡 Analysis

With a compact PL manifold X we associate a category T(X). The objects of T(X) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence BT (X) \approx BPL(X) holds, where PL(X) is the simplicial group of PL-homeomorphisms. Thus the space BT(X) is a canonical countable (as a CW-complex) model of BPL(X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a T(X)-coloring of some triangulation K of B. The vertices of K are colored by objects of T(X) and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: BT(S^{n-1}) \approx BO(n) for n=1,2,3,4. The result is proved in a sequence of results on similar models of B\PL(X). Special attention is paid to the main noncompact case X=R^n and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on “fragmentation of a fiberwise homeomorphism”, a generalization of the folklore lemma on fragmentation of an isotopy.

📄 Content

1. Let X be a compact PL manifold. There is a natural generalization of piecewise linear triangulations of X, namely, the structures of piecewise linear regular cell (or “ball”) complexes 1 on X. The set of all regular PL ball complexes on X is partially ordered by subdivision. We denote this poset by R(X). It is convenient to consider a subdivision Q 0 Q 1 of ball complexes as a morphism of “geometric assembly” with source Q 0 and target Q 1 . By forgetting the geometry, to a geometric PL ball complex Q we can associate an abstract PL ball complex P(Q) (an “abstract PL ball complex” is a natural generalization of the notion of an abstract simplicial complex). The correspondence P sends the poset R(X) to some new category R(X) whose objects are abstract PL ball complexes and morphisms are “abstract assemblies.” One may imagine an abstract assembly Q 0 Q 1 of abstract ball complexes as a way of gluing together the abstract balls of Q 0 into larger balls so as to obtain the complex Q 1 . This way of gluing may be not unique. Figure 1 should give an idea of a unique geometric assembly of two particular geometric ball complexes, and Fig. 2 should give an idea of three possible combinatorial assemblies

With the functor P we associate a cellular map of classifying spaces

BP –→ BR(X).

The map BP has a description in terms of an action of the group of PL homeomorphisms on PL ball complexes on X. Namely, the natural action of PL homeomorphisms on the set R(X) can be extended to a cellular action of a discrete group PL δ (X) on BR(X):

(1) PL(X) δ × BR(X) -→ BR(X).

1 For exact definitions, see Sec. 2 on page 15. One may imagine something like the boundary complex of a convex 3-polytope as a “ball complex” and a planar 3-connected graph as an “abstract ball complex.” Or one may simply think about geometric triangulations instead of “ball complexes” and about combinatorial manifolds instead of “abstract ball complexes.”

Then the cellular space of orbits BR(X)/PL(X) δ coincides with BR(X), and BR(X) BP –→ BR(X) is a projection to the space of orbits. We should mention that the action (1) is highly nonfree.

The category R(X) is an object of the classical combinatorial topology of the manifold X. For example, Alexander’s theorem on combinatorial manifolds [2] is the assertion that the space BR(X) will remain connected if we restrict the class of all morphisms to the more tame class of “stellar assemblies.”

Denote by PL(X) the simplicial group of PL homeomorphisms of X. Denote by |PL(X)| the cellular topological group that is the geometric realization of PL(X). In statistical models of topological quantum field theory, the simple fact is known that (2) π 1 BR(X) ≈ π 0 |PL(X)|.

The group π 0 |PL(X)| is the mapping class group of the manifold X. We prove the following generalization of (2).

Theorem A. The spaces BR(X) and BPL(X) are homotopy equivalent.

Thus the category R(X) is a discrete category that represents a delooping of the simplicial group PL(X). Generally, for any topological (or simplicial) group there exists a delooping using a discrete category or even a discrete monoid [24]. The source of problems about discrete categories representing deloopings of classical spaces is the algebraic K-theory of topological spaces, starting with the famous Hatcher’s paper [12]. The closest assertion to our theorems is probably Steinberger’s theorem [32, Theorem 1, p. 12], which is a refinement of Hatcher’s conjecture [12,Proposition 2.5,p. 109]. Steinberger’s theorem says that the discrete category of ordered simplicial complexes whose morphisms are monotone maps with contractible preimages of simplices classifies Serre PL bundles.

Let EPL(X) be the contractible total space of the universal principal bundle for the group |PL(X)|, let (3) |PL(X)| × EPL(X) -→ EPL(X)

be the canonical free action, and let EPL(X) -→ BPL(X) be the projection to the space of orbits. Informally speaking, we prove that the nonfree action (1) of the discrete group PL δ (X) on the contractible space BR(X) can be deformed to the canonical free action (3) by a homotopy. In this form, our results are relatives of Levitt’s models for BPL (see [19]) presenting BPL as orbit spaces. But in our case we are able to eliminate geometry completely.

The first situations where Theorem A can be regarded as already known appear when dim X = 1 and X is the interval I or the circle S 1 . Here R(I) op is the category whose objects are all finite ordinals and morphisms are generated by injective monotone maps and the additional map of inverting the order; R(S 1 ) op is the category whose objects are all cyclically ordered finite sets and morphisms are generated by injective monotone maps and the additional map of inverting the order. The category R(S 1 ) op is closely related to Connes’ cyclic category. In these cases, BR(I) ≈ BPL(I) ≈ BO (1) and BR(S 1 ) ≈ BPL(S 1 ) ≈ BO (2).

The last assertion is a close relative of the theorem on the homotopy type of the c

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