  Chas and Sullivan introduced string homology, which is the equivariant
homology of the loop space with the $S^1$ action on loops by rotation. Craig
Westerland computed the string homology for spheres with coefficients in
$\mathbb{Z} /2\mathbb{Z}$, and in Somnath Basu's dissertation, he computes the
string homology and string bracket for spheres over rational coefficients, and
he finds that the bracket is trivial. In this paper, we compute string homology
and the string bracket for spheres with integer coefficients, treating the odd-
and even-dimensional cases separately. We use the Gysin sequence and
Leray-Serre spectral sequence to aid in our computations. We find that over the
integers, the string Lie algebra bracket structure is more interesting, and not
always zero. The string bracket turns out to be non-zero on torsion coming from
string homology. We also make some computations of the Goldman Lie algebra
structure, and more generally, the string Lie algebra structure of closed,
orientable surfaces.
