# Spectral numbers and manifolds with boundary

## TL;DR

This paper introduces spectral invariants for submanifolds with boundary within symplectic geometry, establishing their properties and limits related to Floer homology in the cotangent bundle.

## Contribution

It defines spectral invariants for submanifolds with boundary and explores their properties and limiting behavior in Floer homology.

## Key findings

- Spectral invariants are assigned to homology classes and Hamiltonians.
- Spectral invariants associated with the entire Floer homology are limits of nested open sets.
- The paper establishes properties of these spectral numbers in the context of manifolds with boundary.

## Abstract

We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^*M$. We also derive certain properties of spectral numbers, for example we prove that spectral invariants $c_\pm(H,N)$ associated to the whole Floer homology $HF_*(H,N:M)$ of the submanifold $N$, are limits of the decreasing nested family of open sets.