Let U be the infinite unitary group and \(\lambda_ i\) a generator of the primitives in \(H_{2i+1}(U)\). It is well known that \(\delta_ i=(i!)\cdot \lambda_ i\) is stably spherical. For \(i\geq 1\), \(j\geq 1\), \(n\geq \max (i,j)+1\) and p an odd prime the authors show that elements \(x_{i,j}=(\delta_ i\cdot \delta_{j+p-1}-\delta_{i+p-1}\cdot \delta_ j)/p\) in \(H_*(U(n))\) are stably spherical and can be represented by hypersurfaces. They state that for \(n\leq 2p-1\) the elements \(\delta_ i\), \(x_{i,j}\) (1\(\leq i\leq n-1\), \(1\leq j\leq p-1)\) multiplicatively generate the subring of stably spherical elements in \(H_*(U(n);{\mathbb{Z}}_{(p)})\), thus computing \(\pi^ s_*(U(n))_{(p)}/tor\) in this case. As an application they describe \(\beta_ 1\), the first nonzero element in the 5-primary component of cok(J) by a framed hypersurface. (In a subsequent paper [A. Baker, N. Ray and the second author, ”Hypersurfaces framées et l’élément \(\beta_ 1\) de Toda” (preprint)] this is done for all odd primes.) The paper ends with a stable splitting of U. [For a more explicit splitting see H. Miller, Topology 24, 411-419 (1985)].