Topology is a branch of mathematics that studies spaces that remain continuously invariant through distortion by theorizing invariance as a threshold of sameness within transformation. A classic example is the transformability of a doughnut into a coffee cup, and vice versa, accomplished by virtue of treating them both as surfaces enclosing spaces. Pulling and squishing one side of the doughnut, one can make a well deep enough to hold coffee, all the while preserving the common feature of a single hole: the handle of the mug and the center of the doughnut.

Anthropologists have drawn on topology to address methodological issues concerning comparison and generalization in order to both render and refine structure—searching for patterns and a system of internal relations—and to dynamize structuralist concerns of relation, continuity, and change. Edmund R. Leach (1961, 7) used an analogy with topology to describe the flexibility of networks of relations, advocating for the analysis of societies as “assemblages of variables.” Claude Lévi-Strauss (1969) moved beyond the algebraic logic developed through the study of kinship, producing in his work on mythology (Lévi-Strauss 1955) what became known as a canonical formula to capture the morphodynamism of myths as groups
of transformation. The terms of these explorations spread and splintered across various engagements with structuralism, whose proponents and critics have addressed the organizing logics of parts and wholes, insides and outsides, continuity and discontinuity, and totality and system, among a host of other themes.

In later developments, a multiplicity of engagements with theory and method have used topology as a conceptual language for understanding dynamicity, intensity, and transformation as other logics of relations and dynamics of structure. Alongside thinkers such as Marilyn Strathern (1991) and Bruno Latour (2005), who have used fractals and networks to look at questions of relationality and continuity, Gilles Deleuze (2004), in his characterization of structuralism, reaffirmed its scientific ambition as topological and relational, pointing to the structured nature of transformability through the figure of the spatium. In a seminal article, Annemarie Mol and John Law (1994) employed topology to frame the social as expressing the multiplicity and hybridity of spatial forms. Others have sought to identify a contemporary conjuncture, social formation, or material relation as intrinsically topological, carrying in itself a property or power of dynamicity, emergence, or indeterminacy (Lury, Parisi, and Terranova 2012).

Our interest in topology is not as an anthropology meant to replace prior ones: we take it as a set of techniques for abstraction, a method that foregrounds space as object and analytic, without implying a wholesale retheorization of space as such. The insight of mathematical topology is a classification of spaces through general properties of structure and their invariance without reference to distinctive measurements, qualities, and appearances relating to shape. In such a view, space is no longer a medium where an object having a certain shape is found. Rather, space can be treated like the surface of an object: a manifestation of structure that characterizes an object’s extent and distribution as itself a spatial form. Topology contrasts with conventional typologies based on simple commonalities or a rigid sameness of form. Though in anthropology it is often important to register difference, say, between a doughnut and a coffee cup, their equivalence under topological generalization may also highlight abstractions that afford new analytical perspectives on the spatial forms of the anthropological phenomena we work with.

This collection is a partial product of an exploratory workshop and is as much a demonstration of the possibilities of topology for framing various objects of inquiry as a case for the productivity of non-rigid spaces for methodological innovation. These papers deploy topological notions to highlight, register, or inform specifically spatial structures, employing space as an instrument of analysis across a wide variety of cases and themes.

As such, there are many paths through the papers, different orders, figures, and common themes; we provide one such formulation: Khashayar Beigi and Sarah Green explore the interaction of topological and geometric configurations of borders, as sites of passage, impasse, and differentiation, in their discussions of the pedagogy of HIV prevention in Russia with Tajik migrants and the experience of location between Turkey and Greece. Philippe Bonnin and William F. Stafford describe urban spaces produced as transit by logics, infrastructures, and sedimented experiences of the city as a space of routes, through discussions of the evolution of paths trodden and the interior design of autorickshaws in Delhi. Stéphane Gros and Susanne Küchler describe the work of surfaces as mediators of transmission, which differentiate and contain space and information, by examining facial tattoos in China and textiles and navigational tools in Melanesia. Franck Billé and Ishani Saraf focus on the persons and things instrumental to grounding the continuities and discontinuities that produce spaces of inclusion and circulation, through accounts of jurisdictional logics of exclusion in Russia and infrastructures of agreement in the global scrap metal trade. Terra Edwards and Marius Ionescu, Kamala Russell, and Paul Kockelman trace attenuations of contiguity, thresholds of transition, and breaking points as structural modulations of continuity, regarding obstacles in the navigation of Deaf-Blind life, the ethical attention to spatial relations in face-to-face interaction in Oman, and the common constraints on linguistic judgments across time, space, and truth.

It is our hope that navigating the variety of these engagements with topology will go some way toward showing the generative potential of mathematical thinking when pursued with the “rigor of inexactness” that characterizes the social sciences (Phillips 2013), and in particular the history of experiments with method in anthropology.