The decomposition and classification of radiant affine 3-manifolds

TL;DR

This paper decomposes radiant affine 3-manifolds into simpler geometric pieces, proving they are homeomorphic to Seifert fibered spaces or torus bundles, and confirms a conjecture about their structure.

Contribution

It introduces a decomposition method for radiant affine 3-manifolds into convex and concave parts, and proves they admit a total cross section, confirming Carrière's conjecture.

Findings

Decomposition into convex and concave affine manifolds.

Existence of a total cross section in radiant affine 3-manifolds.

Homeomorphism to Seifert fibered spaces or torus bundles.

Abstract

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformations fixing a common fixed point. We decompose an orientable closed radiant affine 3-manifold into radiant 2-convex affine manifolds and radiant concave affine 3-manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective $n$-manifolds developed earlier. Then we decompose a 2-convex radiant affine manifold into convex radiant affine manifolds and concave-cone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine 3-manifold admits a total cross section, confirming a conjecture of Carri\`ere, and hence every radiant affine 3-manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a torus.