On the space of injective linear maps from $\bbR^d$ into $\bbR^m$

Abstract

In this short note, we investigate some features of the space $\Inject{d}{m}$ of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold $V_{m,d}$, viewed, in this context, as the set of orthonormal systems of $d$ vectors in $\bbR^m$. Finally, we show that the Stiefel manifold $V_{m,d}$ is a deformation retract of $\Inject{d}{m}$. One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in $\bbR^m$: in fact, the existence of the aforementioned deformation retraction is the key tool for showing a vanishing lemma for configuration space integrals {à} la Bott--Taubes (see \cite{BT} for the 3-dimensional results and \cite{CR1}, \cite{C} for a first glimpse into higher-dimensional knot invariants).