Contractibility of infinite Stiefel manifolds

Proof. Consider the space \(\widetilde {V}_k(\C ^{\infty })\) of all linear injections \(\C ^k \hookrightarrow \C ^{\infty }\) which are not necessarily isometries. Equivalently, \(\widetilde {V}_k(\C ^{\infty })\) is the topological space of \(k\)-tuples \((v_1, \dots , v_k)\) of linearly independent vectors in \(\C ^{\infty }\) which are not necessarily orthogonal to each other. Observe that the inclusion \(V_k(\C ^{\infty }) \hookrightarrow \widetilde {V}_k(\C ^{\infty })\) is a homotopy equivalence: the Gram-Schmidt orthogonalization procedure provides a deformation retract \(\widetilde {V}_k(\C ^{\infty }) \to V_k(\C ^{\infty })\).

It will thus suffice to show that \(\widetilde {V}_k(\C ^{\infty })\) is contractible. To this end, consider the ‘shift map’ \(T^k\colon \C ^{\infty } \to \C ^{\infty }\) given on basis vectors by \(T^k(e_n) := e_{n+k}\). This induces a map \(T^k\colon \widetilde {V}_k(\C ^{\infty }) \to \widetilde {V}_k(\C ^{\infty })\). Note that this map is homotopic to the constant map on \((e_1, \dots , e_k)\) via the straight-line homotopy: \[ H(v,t) := (1-t)(e_1, \dots , e_k) + t T^k(v). \] This is well-defined since the first \(k\) components of the vectors \(T^k(v_i)\) are all zero. Similarly, the map \(T^k\) is homotopic to the identity on \(\widetilde {V}_k(\C ^{\infty })\), again via the straight-line homotopy: \[ H(v,t) := (1-t)v + t T^k(v). \] To see that this is well-defined, assume for contradiction that for some time \(t\) the vectors are no longer linearly independent. Then there are scalars \(\lambda _1, \dots , \lambda _k\), not all zero, such that \(\sum _{i=1}^k \lambda _i ((1-t)v_i + t T^k(v_i)) = 0\). But then the nonzero vector \(w := \sum _{i=1}^k\lambda _i v_i\) satisfies \((1-t) w + t T^k(w) = 0\). For \(t<1\), this is impossible by considering the first nonzero coordinate of \(w\); for \(t=1\), it is impossible because \(T^k\) is injective. This contradicts linear independence of \(v\).

It follows that the identity on \(\widetilde {V}_k(\C ^{\infty })\) is homotopic to a constant map, and hence this space is contractible. □