The Topology of Stiefel Manifolds

Abstract

Stiefel manifolds are an interesting family of spaces much studied by algebraic topologists. These notes, which originated in a course given at Harvard University, describe the state of knowledge of the subject, as well as the outstanding problems. The emphasis throughout is on applications (within the subject) rather than on theory. However, such theory as is required is summarized and references to the literature are given, thus making the book accessible to non-specialists and particularly graduate students. Many examples are given and further problems suggested.

On the Adams Filtration of a Generator of the Free Part of $π^s_*(Q_n)$

“…[11]). Then it is a 2-connected CW-complex having one 4f-1 dim cell for each i with l<Li<n. It is known that the order of the cokernel of the stable Hurewicz homomorphism h: ^In-i(On)->//in-i(On ; Z) is equal to a(n-l)-(2n-l}\ (see Theorem 3.10), where and throughout the paper a(i) denotes 1 if i is an even integer and 2 if i is an odd integer.…”

“…James (cf. [11]), and p: Qn^Qn/Qn-i-S* 71 ' 1 denote the collapsing map. Then 0 is an isomorphism onto the free part of K\ n -i(Qn\ and q* is identified with the composition p*°6: n^n-i(Sp(n))-^7i:t n -i(S* n~l ).…”

Odd primary homotopy decompositions of gauge groups

“…G such that H .G/ Š ƒ. z H .A// and { includes the generating set into the exterior algebra. For example, when G D SU.n/ we have A D †CP n 1 and when G D Sp.n/ we have A equal to the space Q n constructed by James [17]. Theorem 3.1 states some properties of the decomposition in [28] of the groups G in (1).…”

Lifting Problem of $η$ and Mahowald's Element $η_j$

“…Since Q n is a stable retract of Sp(n) [5] and since HP" is a stable retract of Q(U(2n + 2)/Sp(n + 1)) [2], these problems are closely related to the unstable lifting problem of r\ in the canonical Stiefel bundles: the problem with respect to Diagram (1) is related to the lifting problem of rj to the quaternionic Stiefel bundle A" njk -> S Remark. There is no unstable lift of rj to the usual bundle projection Sp(n) -> S For the proof of Theorem A we need the following lemmas;…”