  We apply a large-scale summation of Feynman diagrams, including the class of
parquet-diagrams {\em plus} important contributions outside the parquet class,
for calculating effective pairing interactions and subsequently the superfluid
gap in P-wave pairing in neutron matter. We employ realistic nucleon-nucleon
interactions of the $v_8$ type and perform calculations up to a Fermi momentum
of $1.8\,$fm$^{-1}$. We find that many-body correlations lead to a strong
reduction of the spin-orbit interaction, and, therefore, to an almost complete
suppression of the $^3$P$_2$ and $^3$P$_2$-$^3$F$_2$ gaps. We also find pairing
in $^3$P$_0$ states; the strength of the pairing gap depends sensitively on the
potential model employed. Our results for triplet pairing are relevant for
assessing superfluidity in neutron star interiors, whose presence can affect
the cooling of neutron stars.
