What typically happens when a research field becomes "dead"?
Topology is surely not dead. Maybe geometric topology, or its specific parts that were targeted at the Poincaré conjecture?

Even then, there are several inequivalent definitions of deadness. One meaningful watershed is when advisers stop suggesting the field to their grad students because they don't believe the students can build their career on it anymore; a lot of fields are dead by these standards. Another, more restrictive, criterion is when editors can no longer find referees for papers on the field.

None of these things dooms the field to eternal dormance (in fact, many dead fields have recovered, some many times), and in fact people do sometimes work on dead fields: some because they don't care about building their career (e.g., since they already have a steady job, or since they are too idealistic to care), some because they feel like they can revitalize the subject, some because they hope to find things of use for other fields, and some simply because they don't know that it's dead.