Have there been any computations of the higher homotopy groups of $MO(2)$, the Thom space of the universal $O(2)$-bundle? Thom himself noted in his landmark 1954 paper that
$$
\pi\_1(MO(2))=0,\quad \pi\_2(MO(2))=\mathbb{Z}/2,\quad\pi\_3(MO(2))=0,\quad \pi\_4(MO(2))=\mathbb{Z}.
$$
By the Pontrjagin-Thom construction $\pi\_n(MO(2))$ is the group of cobordism classes of embeddings of closed $(n-2)$-manifolds in $\mathbb{R}^n$, where a cobordism is an embedding in $\mathbb{R}^n\times [0,1]$.



Homotopy groups of $MO(2)$
There are a couple of references which show that $\pi\_5(MO(2))=\mathbb{Z}/2$.


*Suzuki, H.*, [**On the realization of the Stiefel-Whitney characteristic classes by submanifolds**](http://dx.doi.org/10.2748/tmj/1178244748), Tohoku Math. J., II. Ser. 10, 91-115 (1958). [ZBL0107.17001](https://zbmath.org/?q=an:0107.17001).


The proof uses the Postnikov tower. The conclusion is on page 108.


*Zvagel’skij, M. Yu.*, [**Cobordisms of embeddings with codimension two**](http://dx.doi.org/10.1023/A:1011325713206), J. Math. Sci., New York 104, No. 4, 1276-1282 (2001); translation from Zap. Nauchn. Semin. POMI 252, 40-51 (2001). [ZBL1052.57042](https://zbmath.org/?q=an:1052.57042).


The proof here is geometric, and (for me) harder to follow. This paper also contains a proof that $\pi\_6(MO(2))=0$.


