On the space of injective linear maps from ᵈ into ᵐ

الأصل · EN

In this short note, we investigate some features of the space dm of linear injective maps from ᵈ into ᵐ; 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 ᵐ. Finally, we show that the Stiefel manifold Vₘ,d is a deformation retract of dm. One possible application of this remarkable fact lies in the study of perturbative invariants of higher-dimensional (long) knots in 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 BT for the 3-dimensional results and CR1, C for a first glimpse into higher-dimensional knot invariants).