English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

There is considered the image of the symplectic cobordism ring fl|p in the unoriented cobordism ring Nt. A polynomial subalgebra of N* is exhibited, with all generators in dimensions divisible by 16, such that the image is contained in the polynomial subalgebra. The methods combine the A-theory characteristic numbers as used by Stong with the use of the Landweber-Novikov ring. Let AT be a closed smooth manifold of dimension k. For n large, AT can be embedded in Sin + k in essentially only one way. Let v denote the normal bundle of the embedding. We say that M is quaternionic if v has the structure of a quaternionic vector bundle. Stong [11] proved that in dimensions k<24 all StiefelWhitney numbers of a quaternionic manifold AT" vanish. In his thesis, David Segal [8] extended this to dimensions k<32. He also showed the existence of a quaternionic AT32 unorientedly cobordant to [FF(2)]16. To put the situation more generally, there is the unoriented cobordism ring N* = 1,Nk, and there is a quaternionic cobordism group Q|p = 2 ^l" [10]. There isa natural homomorphism Q|p -> TV*, and Im (D|p -> TV*) is the subalgebra of TV* consisting of all cobordism classes represented by quaternionic manifolds M. 1 am not able to compute this image, but I do give an upper bound to the image and now turn to an outline of the method. In their more widely ranging studies of symplectic cobordism, both Porter [14] and Segal have recently obtained very closely related results. Let A' denote a graded vector space over Z2 with basis consisting of all sR, where F ranges over all sequences R = (rlf r2,...) of nonnegative integers with 2 rk<co and where degiB= ^krk. Landweber [5] and Novikov [7] put a Hopf algebra structure on A'. The product can be defined as composition of operators. Namely each sR acts on any polynomial algebra Z2[xx,..., xm} by s0(xt) = xh sA«(xù=4 + 1 where A, = (0,..., 0, 1, 0,...), sR(Xi) = 0 if F # 0 and F + Ak, sB(yz) = 2 sBi(y)sR2(z). R1 + B2=R Received by the editors March 26, 1970. AMS 1968 subject classifications. Primary 5710, 5732; Secondary 5730.

Stiefel-Whitney numbers of quaternionic and related manifolds