Piecewise linear structures on topological manifolds

On the classification of certain piecewise linear and differentiable manifolds in dimension eight and automorphisms of $#_{i=1}^b(S^2\times S^5)$

In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan Müller. In order to understand the structure of these manifolds, we will analyze their minimal handle presentations and describe explicitly to what extent these handle presentations are determined by the cohomology ring and the characteristic classes. It turns out that the cohomology ring and the characteristic classes do not suffice to reconstruct a manifold …

On piecewise smooth cohomology of locally trivial Lie groupoids

Mishchenko-Oliveira proved the piecewise smooth cohomology and Lie algebroid cohomology of a Lie algebroid on a combinatorial compact manifold are isomorphic. In this paper, we describe an application of that result locally trivial Lie groupoids on combinatorial manifolds to obtain an isomorphism between de Rham cohomology and piecewise de Rham cohomology of left invariants forms.

Free fermions on a piecewise linear four-manifold. I: Exotic chain complex

Recently, an algebraic realization of the four-dimensional Pachner move 3--3 was found in terms of Grassmann--Gaussian exponentials, and a remarkable nonlinear parameterization for it, going in terms of a $\mathbb C$-valued 2-cocycle. Here we define, for a given triangulated four-dimensional manifold and a 2-cocycle on it, an `exotic' chain complex intimately related to the mentioned parameterization, thus providing a basis for algebraic realizations of all four-dimensional Pachner moves.

Simplicial Complexity: piecewise linear motion planning in robotics

Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity of the realization $\|K\|$. Our approach is well suited for designing and implementing algorithms that search for optimal motion planners for autonomous systems in real-life applications.

Free fermions on a piecewise linear four-manifold. II: Pachner moves

This is the second in a series of papers where we construct an invariant of a four-dimensional piecewise linear manifold $M$ with a given middle cohomology class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work, multiplied by a correcting factor. Here we find this factor by studying the behavior of our construction under all four-dimensional Pachner moves, and show that it can be represented in a multiplicative form: a product of same-type multipliers over all 2-faces, multiplied by a product …

双星体簇代数与分段线性不变量

Bistellar Cluster Algebras and Piecewise Linear Invariants

本研究受簇代数研究启发，构建了不满足经典簇代数可交换公理的双星体簇代数，并以此结合组合拓扑技术证明2维或4维分段线性同胚流形对应正向系的极限为PL不变量。