Stiefel-Whitney classes and immersions of orientable and Spin manifolds

TL;DR

This paper provides explicit formulas and bounds for the largest Euclidean space into which orientable and Spin manifolds can be immersed, using Stiefel-Whitney classes and ko-homology calculations.

Contribution

It derives a simple formula for nonimmersion bounds of orientable manifolds and completes the bounds for Spin manifolds in specific dimensions, improving on historical results.

Findings

Explicit formula for orientable manifolds' nonimmersion bounds

Complete bounds for Spin manifolds in dimensions less than 24 and at 33,34

Use of ko-homology of Eilenberg-MacLane spaces for calculations

Abstract

We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for n<24 and n=33,34. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results for Spin manifolds require detailed calculations of ko-homology groups of mod-2 Eilenberg-MacLane spaces.