2026/07/22 by Souvik Mandal, Ankur Sarkar
#math.AT
We compute the relative smooth surgery structure sets of the thickenings \mathbbOP2×\mathbbDk of the Cayley projective plane \mathbbOP2 for every k≥ 1 with k≡ 0\pmod 4, by determining the corresponding normal invariants and surgery obstruction map. We show that the latter is not surjective and determine the 2-adic valuation of the generator of its image. As applications, we construct infinitely many pairwise non-homeomorphic closed smooth manifolds of dimension 16+k, homotopy equivalent to \mathbbOP2×\mathbbSk and distinguished by their total Pontryagin numbers; we compute the rational homotopy groups of the block diffeomorphism group \widetildeDiff(\mathbbOP2) in every degree congruent to 3 modulo 4; and we construct smooth \mathbbOP2-bundles over \mathbbS4, \mathbbS8, and \mathbbS12 whose total spaces have non-vanishing \widehat\mathfrakA-genus. These bundles yield elements of infinite order in the homotopy groups of the spaces of metrics of positive sectional, Ricci, and scalar curvature on \mathbbOP2 in degrees 3, 7, and 11.