Nilpotency of the Bauer-Furuta stable homotopy Seiberg-Witten invariants

We prove a nilpotency theorem for the Bauer-Furuta stable homotopy Seiberg-Witten invariants for smooth closed 4-manifolds with trivial first Betti number.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants

We show the "non-existence" results are essential for all the previous known applications of the Bauer-Furuta stable homotopy Seiberg-Witten invariants. As an example, we present a unified proof of the adjunction inequalities. We also show that the nilpotency phenomenon explains why the Bauer-Furuta stable homotopy Seiberg-Witten invariants are not enough to prove 11/8-conjecture.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

A new approach to the characterization of closed forms in the nongradient method (Stochastic Analysis on Large Scale Interacting Systems)

"Stochastic Analysis on Large Scale Interacting Systems". October 26～29, 2015. edited by Ryoki Fukushima, Tadahisa Funaki, Yukio Nagahata, Makoto Nakashima, Hirofumi Osada and Yoshiki Otobe. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We announce our recent result on the characterization of closed forms on configuration spaces associated to interacting particle systems. In the context of the study of hydrodynamic limit, closed forms on an infinite configuration space in a L2 space are well studied and their characterization theorem plays an essential role if our model is non-gradient. In this article, we report that closed forms in the set of local functions can be characterized by a similar way as L2 functions but its proof is very simple and completely different from that for L2 functions. With this new observation, we also have an alternative proof of the original characterization theorem in the L2 space, which does not require the sharp estimate of the spectral gap, for the class of lattice gases that are reversible under the Bernoulli measures. Moreover, we extend these characterization theorems for the models in a crystal lattice from Zd