Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bordered Floer homology for manifolds with torus boundary via immersed
  curves   This paper gives a geometric interpretation of bordered Heegaard Floer
homology for manifolds with torus boundary. If $M$ is such a manifold, we show
that the type D structure $\widehat{\mathit{CFD}}$ may be viewed as a set of
immersed curves decorated with local systems in $\partial M$. These
curves-with-decoration are invariants of the underlying three-manifold up to
regular homotopy of the curves and isomorphism of the local systems. Given two
such manifolds and a homeomorphism $h$ between the boundary tori, the Heegaard
Floer homology of the closed manifold obtained by gluing with $h$ is obtained
from the Lagrangian intersection Floer homology of the curve-sets. This
machinery has several applications: We establish that the dimension of
$\widehat{\mathit{HF}}$ decreases under a certain class of degree one maps
(pinches) and we establish that the existence of an essential separating torus
gives rise to a lower bound on the dimension of $\widehat{\mathit{HF}}$. In
particular, it follows that a prime rational homology sphere $Y$ with
$\widehat{\mathit{HF}}(Y)<5$ must be geometric. Other results include a new
proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a
complete characterisation of toroidal L-spaces in terms of gluing data; and a
proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an