WordNet sense 1 (a usually secretive or illicit sexual relationship):

affair, amour, intimacy, liaison, affaire

WordNet sense 2 (the condition of sharing in common with others (as fellows or partners etc.)):

participation

WordNet sense 3 (a sense of concern with and curiosity about someone or something):

interest

WordNet sense 4 (the act of sharing in the activities of a group):

engagement, involution