An $omega$-meson in motion with respect to a nuclear medium can couple to a $sigma$-meson through a particle-hole excitation. This coupling is large. We investigate its consequences for the width of $omega$-mesons in matter and for the s-wave annihilation of pions into lepton pairs which can take place in relativistic heavy ion collisions. We find that the two pion decay of $omega$-mesons, resulting from the $omega-sigma$ transition and the subsequent $2pi$ decay of the $sigma$-meson, leads to a substantial broadening of $omega$-mesons in matter and possibly to an observable effect in experiments measuring the $e^+e^-$ decay of vector mesons produced in nuclei and in relativistic heavy-ion collisions. The inverse process, the s-wave annihilation of pions into $omega$-mesons decaying into $e^+e^-$ pairs, has in general a much smaller cross section than the corresponding p-wave annihilation through $rho$-mesons and is expected to contribute rather little to the total $e^+e^-$ pair production in relativistic heavy ion collisions.
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