# Topological obstructions for submanifolds in low codimension

## TL;DR

This paper establishes topological and geometric obstructions for certain submanifolds in Euclidean space and spheres, based on curvature bounds, Betti numbers, and pinching conditions.

## Contribution

It introduces new integral curvature bounds linked to Betti numbers and derives topological and intrinsic obstructions for specific classes of submanifolds.

## Key findings

- Integral curvature bounds in terms of Betti numbers for low codimension submanifolds.
- Topological obstructions for δ-pinched immersions.
- Intrinsic obstructions for minimal submanifolds with pinched second fundamental form.

## Abstract

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $\delta$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.