Pontryagin algebras of some momentanglecomplexes
Abstract
We consider the problem of describing the Pontryagin algebra (loop homology) of momentangle complexes and manifolds. The momentangle complex Z_K is a cell complex built of products of polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural torus action and plays an important role in toric topology. In the case when K is a triangulation of a sphere, Z_K is a topological manifold, which has interesting geometric structures. Generators of the Pontryagin algebra H_*(\Omega Z_K) when K is a flag complex have been described in the work of Grbic, Panov, Theriault and Wu. Describing relations is often a difficult problem, even when K has a few vertices. Here we describe these relations in the case when K is the boundary of a pentagon or a hexagon. In this case, it is known that Z_K is a connected sum of products of spheres with two spheres in each product. Therefore H_*(\Omega Z_K) is a onerelator algebra and we describe this one relation explicitly, therefore giving a new homotopytheoretical proof of McGavran's result. An interesting feature of our relation is that it includes iterated Whitehead products which vanish under the Hurewicz homomorphism. Therefore, the form of this relation cannot be deduced solely from the result of McGavran.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.00283
 Bibcode:
 2015arXiv151200283V
 Keywords:

 Mathematics  Algebraic Topology
 EPrint:
 10 pages