Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism

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.

💡 Research Summary

The paper introduces a purely combinatorial framework for PL (piecewise‑linear) fiber bundles. For a fixed compact PL manifold X the author defines a category T(X) whose objects are all combinatorial manifolds PL‑homeomorphic to X and whose morphisms are “combinatorial assemblies”, i.e. PL‑maps that glue one triangulation of X into another. The nerve of this category yields a classifying space BT(X), which is shown to be homotopy equivalent to the classical classifying space BPL(X) of the simplicial group of PL‑homeomorphisms of X. Consequently BT(X) provides a canonical countable CW‑complex model of BPL(X), in contrast to the usual infinite‑dimensional constructions.

The core of the proof is a family of “fragmentation of a fiberwise homeomorphism” lemmas. Classical fragmentation states that any isotopy can be broken into a finite product of isotopies supported in small neighborhoods. The author generalizes this to the fiberwise setting: given a PL‑homeomorphism of a bundle with fiber X over a base B, one can subdivide B finely enough so that the homeomorphism can be expressed as a composition of homeomorphisms each supported over a single simplex of the base. This decomposition respects the combinatorial structure of the fibers and stays within the countable category T(X). The lemmas guarantee that the required subdivisions and local adjustments can be performed uniformly, which is essential for constructing a homotopy inverse between BT(X) and BPL(X).

With the equivalence in hand, the author describes an explicit combinatorial model for a PL‑bundle over a PL polyhedron B. Choose a triangulation K of B. A T(X)‑coloring of K assigns to each vertex an object of T(X) (i.e. a specific triangulated copy of X) and to each edge a morphism in T(X) (an assembly map between the two vertex‑objects). The 2‑skeleton of K must satisfy the obvious commutativity condition: around each triangle the three edge‑maps compose to the identity. Such a coloring encodes precisely a PL‑bundle with fiber X and base B; the total space is obtained by gluing the fiber copies according to the edge‑maps. Thus the whole bundle is described by a finite amount of combinatorial data (vertices, edges, and their labels).

The paper also works out concrete low‑dimensional examples. For X = S^{n‑1} the author proves that BT(S^{n‑1}) is homotopy equivalent to the real Grassmannian BO(n) for n = 1, 2, 3, 4, thereby recovering classical results in a purely combinatorial language. The non‑compact case X = ℝⁿ is treated with special care, leading to combinatorial models of the tangent bundle and the Gauss functor of a combinatorial manifold. These constructions illustrate that the framework works uniformly for compact and non‑compact fibers.

In summary, the paper achieves three major advances: (1) it identifies a countable, purely combinatorial classifying space BT(X) for PL‑homeomorphisms; (2) it provides an explicit, functorial coloring model for PL‑fiber bundles over any PL base; and (3) it establishes a powerful fragmentation technique for fiberwise PL‑homeomorphisms, which underlies the homotopy equivalence. This work opens the door to algorithmic and computational approaches to PL‑bundles, to combinatorial descriptions of characteristic classes, and to further extensions such as higher‑dimensional Grassmannians or PL‑structures on exotic manifolds.