Searcharxiv⌕ Search

arXiv subjects

Hiro Lee Tanaka

Publications and source records attributed to Hiro Lee Tanaka.

24 records · Page 2Linked to original sources
arXiv2017

Local structures on stratified spaces

We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous space-valued sheaves on these conically smooth stratified spaces in terms of tangential data, and we similarly characterize 1-excisive invariants of stratified spaces. These results are based on the existence of open handlebody decompositions for conically smooth stratified spaces, an inverse function theorem, a tubular neighborhood theorem, an isotopy extension theorem, and functorial resolutions of singularities to smooth manifolds with corners.

math.AT↗
arXiv2016

The Fukaya category pairs with Lagrangian cobordisms exactly

We prove that a pairing between the Fukaya category and the oo-category of Lagrangian cobordisms respects mapping cones. This is another step toward constructing a lift of Fukaya categories to the level of spectra (in the sense of stable homotopy theory). As corollaries, we show that the map in our previous work from cobordism groups to Floer cohomology lifts to the level of spectra, and one also recovers some results of Biran and Cornea for what we call "vertically collared" cobordisms.

math.SG↗
arXiv2016

The Fukaya category pairs with Lagrangian cobordisms

Fix a suitably convex, exact symplectic manifold M. We consider the stable oo-category Lag(M) of non-compact Lagrangians whose (higher) morphisms are (higher) Lagrangian cobordisms between them. We show that this oo-category pairs with the Fukaya category Fuk(M) of compact branes. In fact, we also show that there is a subcategory of Lag(M) which pairs with the wrapped Fukaya category of M. This is a first step in a project to enrich wrapped Fukaya categories over cobordism spectra. As a corollary, we show that cobordant compact branes are equivalent in the Fukaya category. We will also mention several other applications (without proof) of the oo-categorical appraoch: One can realize Seidel's representation as a π_0-level consequence of a map of spaces; stable cobordism groups of non-compact branes map to Floer cohomology groups; some of Biran-Cornea's results can be recovered from the colored planar operad associated to the s-dot constructions of each category; and there is an Eoo map of spectra from exact Lagrangian cobordisms in Euclidean space to the integers.

math.SG↗
arXiv2016

Factorization homology of stratified spaces

This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg--Steenrod axioms; it can also be viewed as an analogue of the Baez--Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topological quantum field theory. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of stratified spaces -- this is a nonabelian version of the Poincaré duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with 1-dimensional submanifolds, these structure gives rise to knot and link homology theories.

math.AT↗
arXiv2014

Structured singular manifolds and factorization homology

We provide a framework for the study of structured manifolds with singularities and their locally determined invariants. This generalizes factorization homology, or topological chiral homology, to the setting of singular manifolds equipped with various tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Factorization homology theories for singular manifolds are characterized by a generalization of the Eilenberg-Steenrod axioms. Using these axioms, we extend the nonabelian Poincaré duality of Salvatore and Lurie to the setting of singular manifolds -- this is a nonabelian version of the Poincaré duality given by intersection homology. We pay special attention to the simple case of singular manifolds whose singularity datum is a properly embedded submanifold and give a further simplified algebraic characterization of these homology theories. In the case of 3-manifolds with 1-dimensional submanifolds, this structure gives rise to knot and link homology theories akin to Khovanov homology.

math.AT↗