# Sum of embedded submanifolds

## TL;DR

This paper constructs explicit representatives for the sum of embedded submanifolds in manifolds, extending to codimension-2 cases and analyzing intersection properties, advancing understanding of submanifold homology classes.

## Contribution

It provides explicit constructions for the sum of embedded submanifolds and explores the intersection bounds for oriented codimension-1 submanifolds.

## Key findings

- Explicit construction of sum representatives for homology classes.
- Extension of construction to codimension-2 co-oriented submanifolds.
- Lower bounds on intersection components based on homology.

## Abstract

In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We construct such a representative explicitly. We describe the analogous construction for codimension-2 co-oriented submanifolds, and examine the special case of oriented and/or co-oriented submanifolds. We also give a lower bound for the number of connected components of the intersection of two oriented codimension-1 submanifolds in terms of the homology classes they represent.