Unravelling the Dodecahedral Spaces

The Weber-Seifert dodecahedral space is non-Haken

In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new heuristic pruning techniques, and a new theoretical test that extends the seminal work of Jaco and Oertel.

十二面体L-空间与双曲4-流形

Dodecahedral L-spaces and hyperbolic 4-manifolds

本文证明了由4个或更少直角双曲十二面体镶嵌的29个有理同调3-球面中恰好有6个是L-空间，并据此构造了Seiberg-Witten不变量消失的双曲4-流形显式例子，回答了Agol和Lin的问题。

Monopole Floer homology, eigenform multiplicities and the Seifert-Weber dodecahedral space

We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $\mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.

The residual gravity acceleration effect in the Poincare dodecahedral space

In a flat space, the global topology of comoving space can induce a weak acceleration effect similar to dark energy. Does a similar effect occur in the case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish the Poincare space from other well-proportioned spaces? The residual acceleration effect in the Poincare space is studied here using a massive particle and a nearby test particle of negligible mass, in S^3 embedded in R^4. The weak limit gravitational attraction on a test particle at distance r is set \propto [r_C \sin(r/r_C)]^{-2}, where r_C = curvature radius, in ord…

Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincare dodecahedral space S3/I*, the binary octahedral space S3/O*, the binary tetrahedral space S3/T*, the prism manifolds S3/D_m* and the lens spaces L(p,1).

On Isosystolic Inequalities for T^n, RP^n, and M^3

If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology has the maximal cup-length, then for any riemannian metric g on M, we show that the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g) are related by the following isosystolic inequality: Sys(M,g)^n \leq n! Vol(M,g). The inequality can be regarded as a generalization of Burago and Hebda's inequality for closed essential surfaces and as a refinement of Guth's inequality for closed n-manifolds whose Z/2Z-cohomology has the maximal cup-length. We also establish the same inequality in the context of …