A reader has recommended me - and other readers - the following essay by Sergey Novikov from 2000:
Novikov discusses many technical details of topology but also more philosophical issues about the right way to do mathematics.
He identifies the period 1950-1970 as the Golden Age Period. The years 1970-1980 represent the Period of Decay. The era since 1980 is the Period of Recovery.
After 1970, many people left topology for other fields which was probably the primary sociological reason of the decay. However, the internal reason behind the decay is that the mathematicians were not able to fully settle the proofs and disseminate the information about the incomplete and flawed proofs - something that used to work instantly in the better periods. No one was too interested in the full perfectionist proofs even if they were available.
Novikov thinks that the free exchange of information and its clear presentation is essential - and it is a gift that the Ancient Greeks gave us.
The period of recovery in topology started because of the influence of physics in the 1980s: quantum field theory, string theory, and some topological notions relevant for condensed matter physics (such as topology of Fermi surfaces): a new generation of leaders of theoretical physics was suddenly ready to care about the implications of their discoveries for mathematics itself instead of the previous generation's overemphasis on the application of all their ideas for experiments.
Novikov argues that in contrast to the bad and messy tendencies in the 1970s, physics didn't cause any problems in topology because the theoretical physicists never claim that their new statement is completely proven - they view their broader mathematical assertions (beautiful heuristic mathematics, as Novikov calls it) as "predictions" and their complete proof may be counted as a sort of experimental verification.