= {(*u*_{i}, *v*_{i}) | *u*_{i}^{2} = *v*_{i}^{2} = 1 and *u*_{i}*v*_{i} = 0}.

Hence, the projection to the For any subset

- (iii)
- If
*n*is*odd*, then for*i*2*n*- 1, we have a short exact sequence0(*S*^{n})/2()_{2}(*S*^{n + 1})0. - (iv)
- If
*n*is*even*, then we have a split exact sequence for each*i*0(*S*^{n})()(*S*^{n + 1})0.

[] = 1 + (- 1)^{n + 1} (*S*^{n})

where
1 (

- (v)
- multiplication by
*d*if*i*< 2*n*- 1 - (vi)
- multiplication by
*d*on the torsion subgroup, for*i*= 2*n*- 1 (and in particular on , if*n*is odd) - (vii)
- multiplication by
*d*^{2}on (*S*^{n}) .