# Algebraic Topology of Special Lagrangian Manifolds

## TL;DR

This paper investigates the algebraic topology of the Grassmannian of oriented 3-planes in 6-space, computes its cohomology ring, and applies these results to embeddings of surfaces and 3-manifolds in Euclidean spaces.

## Contribution

It provides new topological insights into the Grassmannian of 3-planes in 6-space and applies these to study embeddings related to special Lagrangian geometry.

## Key findings

- Computed the cohomology ring of the Grassmannian of oriented 3-planes in 6-space
- Proved results on the topology of these Grassmannians
- Applied topological results to embeddings of surfaces and 3-manifolds

## Abstract

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.