nLab principal SO(8)-bundle

Idea

A principal SO(8)-bundle $P$ fulfills:

w_2^2(P) \equiv p_1(P) \mod 2;

w_4^2(P) \equiv p_2(P) \mod 2;

w_6^2(P) \equiv p_3(P) \mod 2;

w_8^2(P) \equiv p_4(P) \mod 2.

(In general, a principal $SO(n)$ -bundle $P$ fulfills $w_{2k}^2(P)\equiv p_k(P) mod 2$ for $2k\leq n$ .)

Let $X$ be an orientable 8-manifold (with fundamental class $[X]\in H^8(X,\mathbb{Z})\cong\mathbb{Z}$ ) and $P=Fr_{SO}TX$ be the frame bundle of its tangent bundle (which doesn’t change characteristic classes). Using Hirzebruch's signature theorem connects one of its Stiefel-Whitney numbers with its signature:

w_4^2[X]-w_2^4[X] =(p_2-p_1^2)[X] mod 2 =\sigma(X) mod 2

A principal SO(8)-bundle $P$ fulfills:

p_4(P) =e^2(P).

(In general, a principal $SO(2n)$ -bundle $P$ fulfills $p_n(P)=e^2(P)$ .)

The two previous propositions together imply $w_8^2(P)\equiv e^2(P) mod 2$ and one even has:

A principal SO(8)-bundle $P$ fulfills:

w_8(P) \equiv e(P) mod 2.

(In general, a principal $SO(n)$ -bundle $P$ fulfills $w_n(P)\equiv e(P) mod 2$ .)

One has $S^n\cong SO(n+1)/SO(n)$ , hence there is a principal SO(8)-bundle $SO(9)\twoheadrightarrow S^8$ . Such principal bundles are classified by:
$\pi_8B SO(8) \cong\pi_7 SO(8) \cong\mathbb{Z}^2.$

John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)
Robert Gompf and András Stipsicz, 4-Manifolds and Kirby Calculus (1999), Graduate Studies

in Mathematics, Volume 20 [ISBN: 978-0-8218-0994-5, doi:10.1090/gsm/020]
Allen Hatcher: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]

Last revised on March 12, 2026 at 13:18:08. See the history of this page for a list of all contributions to it.