# K-theory and immersions of spatial polygon spaces

## TL;DR

This paper computes the algebraic K-theory of spatial polygon spaces and uses it to identify when these spaces can or cannot be immersed or embedded in Euclidean spaces, revealing new geometric constraints.

## Contribution

It determines the algebra K(N(ell)) for polygon spaces and applies obstruction theory to establish immersion and non-immersion results for various side length configurations.

## Key findings

- Computed algebra K(N(ell)) for spatial polygon spaces.
- Identified non-immersibility in certain Euclidean spaces.
- Established exact conditions for immersibility in specific dimensions.

## Abstract

For ell a generic n-tuple of positive numbers, N(ell) denotes the space of isometry classes of oriented n-gons in R^3 with side lengths specified by ell. We determine the algebra K(N(ell)) and use this to obtain nonimmersions of the 2(n-3)-manifold N(ell) in Euclidean space for several families of ell. We also use obstruction theory to tell exactly when N(ell) immerses in R^{4n-14} for two families of ell.