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The proposal made 50 years ago by Schulman (1968), Laidlaw & Morette-DeWitt(1971) and Dowker (1972) to decompose the propagator according to the homotopyclasses of paths was a major breakthrough: it showed how Feynman functionalintegrals opened a direct window on quantum properties of topological origin inthe configuration space. This paper casts a critical look at the argumentsbrought by this series of papers and its numerous followers in an attempt toclarify the reason why the emergence of the unitary linear representation ofthe first homotopy group is not only sufficient but also necessary.

Path integrals in a multiply-connected configuration space (50 years after)