Topology beyond application: Drawing social and mathematical worlds into rhythm

For more than sixty years, topology has provided geographers with tools for lifting the veils of Euclideanism and showing how spaces are relational, agential, embodied, and ontologically multiple. A field of mathematics since the nineteenth century, topology has been reworked and mobilised by geographers in diverse ways: from spatial science, through networks and assemblages, to the influences of Law and Mol, Deleuze and Guattari, Agamben, Barad, and others. Recently, geographers have called for employing topology with greater critical discernment, while continuing to develop its promising and subversive potential. In this article, we aim to help enrich topology's presence in geography and the social sciences more broadly. Following a brief historical introduction, we examine the bodily basis of mathematical reasoning in order to undo the qualitative/quantitative binary and outline a creative, affective approach to abstraction. This lays the ground for the main contribution of the article, where we detail, with several diagrams, key concepts in mathematical topology which critically engage and augment geographers’ theories of space. The potential for an unlikely alliance emerges between geography and mathematics, with implications for both disciplines – provided the inseparability of object and subject, and of knowledge and social order, are carefully attended to.