"Η αναζήτηση της γυναίκας"
Translation:The woman's search
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That's right. The genitive can refer to either the subject or the object of an implied verb.
Similarly in English - if you talk about the love of a parent, then the parent is the one who loves (the subject of loving), but when you talk about a love of music, then the music is the thing which gets loved (the object of loving).
So here too, η αναζήτηση της γυναίκας can have the genitive της γυναίκας referring to either the subject or object of the implied verb αναζητώ "search": the fact that the woman searched for something, or the fact that someone searched for the woman.