arXiv subjects

Leo Yoshioka

Publications and source records attributed to Leo Yoshioka.

4 recordsLinked to original sources

arXiv2025

On hidden face contributions of configuration space integrals for long embeddings

Configuration space integrals are powerful tools for studying the homotopy type of the space of long embeddings in terms of a combinatorial object called a graph complex. It is unknown whether these integrals give a cochain map due to potential obstructions called hidden faces. The purpose of this paper is to address these hidden faces by modifying configuration space integrals: we incorporate the acyclic bar complex of some dg algebra into the original graph complex, without changing its cohomology. Then, we give a cochain map from the new graph complex to the de Rham complex of the space of long embeddings modulo immersions, by combining the original configuration space integrals with Chen's iterated integrals. As the original complex, we choose quite a modified graph complex so that it is quasi-isomorphic to both the hairy graph complex and a graph complex introduced in the context of embedding calculus.

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arXiv2025

Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs

In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2-loop or 3-loop part of modified configuration space integrals using a modified Bott-Cattaneo-Rossi graph complex. We show the non-triviality of the cycles by pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2-loop part, we provide an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long embeddings of codimension two, which Budney--Gabai and Watanabe first established. We also show the non-finite generation of the 2(j-1)-th homotopy group by using the 3-loop part.

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arXiv2023

Two graph homologies and the space of long embeddings

Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings. Two graph homologies have been invented from two approaches to study the space of long embeddings: the hairy graph homology from embedding calculus, and BCR graph homology from configuration space integral. In this paper, we construct a monomorphism from the top hairy graph homology to the top BCR graph homology, though the latter graph homology is quite modified. This map and its left inverse are analogs of PBW isomorphism between the space of open Jacobi diagrams and the space of Jacobi diagrams on the unit circle, in the theory of Vassiliev knot invariants.

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arXiv2022

Cocycles of the space of long embeddings and BCR graphs with more than one loop

The purpose of this paper is to construct non-trivial cocycles of the space $Emb(\mathbb{R}^j, \mathbb{R}^{n})$ of long embeddings. We construct the cocycles by integral over configuration spaces, associated with Bott-Cattaneo-Rossi graphs with more than one loop. As an application, we give explicitly a non-trivial family of trivial long embeddings for odd $n,j$ with $n-j \geq 2$ and $j \geq 3$. This family (cycle) is constructed from a chord diagram on directed lines. The non-triviality is shown by cocycle-cycle paring, described by paring between graphs and chord diagrams.

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