On the Space of Injective Linear Maps from R into R

In this short note, we investigate some features of the space I ( R ,R ) of linear injective maps from R d into R; in particular, we discuss in detail its relationship with the Stiefel manifold Vm,d , viewed, in this context, as the set of orthonormal systems of d vectors in R. Finally, we show that the Stiefel manifold Vm,d is a deformation retract of I ( R ,R ) . One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in R: 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 [4] for the 3-dimensional results and [6], [12] for a first glimpse into higher-dimensional knot invariants).