Geometric dimension of bundles on real projective spaces

Totally skew embeddings of manifolds

“…It has been thoroughly studied using various topological methods [1,[6][7][8]10,26,27]. In particular, [27] Note that there are significant jumps in the lower bounds in the above table whenever n is a power of 2.…”

“…It has been thoroughly studied using various topological methods [1,[6][7][8]10,26,27]. In particular, [27] Note that there are significant jumps in the lower bounds in the above table whenever n is a power of 2. Further this table suggests that N (2 p ) ≥ 3(2 p ) for all p. The next Corollary of Theorem 1.4, which is based on the vanishing of Stiefel-Whitney classes of (n + q)ξ n−1 in dimensions q and higher, shows that this is the case: Corollary 1.6 Suppose that N (n) = 2n + q, with q ≤ n, and n + q = 2 r + m with 0 ≤ m < 2 r .…”

Vector bundles of low geometric dimension over real projective spaces

“…Since 4 divides 60, the map (60α) + further extends to RP 14 11 → BSp(2), again denoted by (60α) + . We now take an extension RP 15 11 = RP 14 11 ∨ S 15 (60α )…”

“…using a map g: S 15 → BSp(2) representing a certain element [g] ∈ π 15 (BSp(2)) ≈ π 14 (Sp(2)) = Z/1680, to be specified later. The attaching map π: S 15 → RP 15 11 for the top-dimensional cell of RP 16 11 is the composition of the double covering S 15 → RP 15 with the collapsing map RP 15 c − → RP 15 11 . Homotopically, we can wedge decompose π as π 0 ∨ 2ι with π 0 : S 15 → RP 14 11 being the first component and the second component 2ι: S 15 → S 15 a map of degree 2.…”

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

“…The group KO(P n ) of equivalence classes of stable vector bundles over real projective space is a finite cyclic 2-group generated by the Hopf line bundle ξ n . Many papers (e.g., [1], [22], [23], [24]) have been devoted to computing the geometric dimension of multiples kξ n of the Hopf bundles, in part because certain cases are equivalent to determining whether P n can be immersed in a certain Euclidean space (e.g., [10]). …”