In an example of Enlightenment ‘engaged research' and public intellectual practice, Euler established the basis of topology and graph theory through his solution to the puzzle of whether a stroll around the seven bridges of 18th-century Königsberg was possible without having to cross any given bridge twice. This ‘Manifesto' argues that, born in a form of cultural studies, topology offers 21st-century researchers a model for mapping the dynamics of time as well as space, allowing the rigorous description of events, situations, (...) changing cultural formations and social spatializations. Law and Mol's network spaces, Serre's folded time, Massey’s ‘power geometries’, Lefebvre's ‘production of space’ and ‘rhythmanalysis’ can be developed through a cultural topological sensitivity that allows time to be understood as not only progressive but cyclical, relationships and the ‘reach’ of power can be understood through ‘knots', and a topology of experience to model the ‘plushness of the Real' via extra- and over-dimensioned time-spaces that capture nuance while drawing on systematic conceptual resources. (shrink)