jump has 24 distinct synonyms spread across 17 of its 21 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
make a sudden physical attack on
“The muggers jumped the woman in the fur coat”
increase suddenly and significantly
“Prices jumped overnight”
enter eagerly into
“He jumped into the game”
start (a car engine whose battery is dead) by connecting it to another car's battery
a sudden and decisive increase
“a jump in attendance”
an abrupt transition
“a successful leap from college to the major leagues”
(film) an abrupt transition from one scene to another
the act of jumping; propelling yourself off the ground
“he advanced in a series of jumps”
Synonyms, antonyms and the broader and narrower relations for jump come from Princeton University’s WordNet. Full sources and licences.