  The space of static finite-energy configurations of the electroweak theory
admits a ${\bf Z}_{2}$ topological structure. Odd-parity configurations with
odd pure-gauge behavior at spatial infinity (S sphaleron, W-Z strings,
multisphalerons) mediate rapid $B+L$ violation in the Early Universe.
Configurations with even pure-gauge behavior (S$^{*}$ sphalerons, W-Z strings,
multisphalerons) are trivial and topologically equivalent to the vacuum.
