The Space of Linear Maps into a Grassmann Manifold

TL;DR

This paper characterizes the space of degree-one holomorphic maps from the Riemann sphere into Grassmann manifolds, revealing it as a sphere bundle over a flag manifold, and computes related homology groups.

Contribution

It provides a complete description of the space of such maps, including unparameterized maps, and details the case of maps into quadric Grassmannians.

Findings

Space of maps is a sphere bundle over a flag manifold

Homology groups of the map spaces are computed

Detailed analysis of maps into quadric Grassmannians

Abstract

We show that the space of all holomorphic maps of degree one from the Riemann sphere into a Grassmann manifold is a sphere bundle over a flag manifold. Using the notions of "kernel" and "span" of a map, we completely identify the space of unparameterized maps as well. The illustrative case of maps into the quadric Grassmann manifold is discussed in details and the homology of the corresponding spaces computed.