From Math to Geography – What is Topology?

Defining Topology

Originating from mathematics, topology has been somewhat ‘appropriated’ by geography, emerging as a key theory within post-structuralist spatial thinking (Martin & Secor, 2014). In both fields, it looks at features not affected by distortion, whether geometrically or spatially (Murdoch, 2006). Adapted to analyze change and relations in space, the uses of this theory in geography are diverse – ranging from borders and territory, to analyses of cities, bodies or even memory (Martin & Secor, 2014). Broadly, topology is understood as the antithesis to topography, moving away from the description of entities in contained surfaces, to the relations and interactions of those entities ‘below’ the surface (Murdoch, 2006). In geography, topology can be defined as the theory which conceptualizes space as relational, multiple, and in a continual state of becoming. The two approaches to topology that geographers mainly engage with are the topologies of Actor-Network theorists and Gilles Deleuze (Martin & Secor, 2014). Both approaches use topology to examine space-times outside of fixed, linear and measurable coordinates, but in slightly different ways. 

For Actor-Network theorists, mainly John Law and Annemarie Mol, topology consists of localizing stable objects in variable coordinate systems as they are displaced through space (Mol & Law, 1994; Law, 1999). They first describe regional spaces – absolute spaces of ‘here’ and ‘there’, defined by boundaries and fixed coordinates (Mol & Law, 1994; Murdoch, 2006). Second, there are network spaces of stabilized ‘actor-networks’ – sets of heterogeneous entities and their relations (Mol & Law, 1994; Murdoch, 2006). Network spaces are plainly relational, with relations making up the network, and the strength of those relations defining their distance and size (Mol & Law, 1994). Regional spaces and network spaces are inextricably linked, leading to the topological feature of multiplicity. This is best represented through Law’s (2000) discussion of Latour’s ‘immutable mobiles’, where he states that an object can occupy regional and network space simultaneously. Because objects are only stable so long as their networks are stable, they must remain ‘immutable’ within network space, but can move across Cartesian coordinates in regional space without changing their shape (Law, 2000). Another characteristic of this topological space is that it is ‘performed’ by objects. As objects are brought into being and stabilized, they concurrently perform spatialities into existence (Law, 2000). If an object, or the relations forming that object, change, so will the space it has originated from. These notions of performativity are explicitly present in Mol and Law’s (1994) description of fluid spaces – spaces absent of boundaries and consisting mainly of gradients where relations are constantly being added incrementally. Through the description of these regional, network, and fluid spaces, Law and Mol characterize a topology that is relational, multiple and continually becoming. 

The other main topology that has been applied by many geographers is Gilles Deleuze’s topology (Martin & Secor, 2014). The main difference in this topology is that the relational, multiple and continually becoming elements of space are more linked, each vital to this conceptualization of topological space. This topology is one of transformation (Murdoch, 2006), and how space is formed by these changes. Transformations in topological space involve the conversion and reversal of entities and relations into each other, and forming spaces in result (Murdoch, 2006). This continual conversion and undoing of points to create space contributes to its ‘becoming’, where space is on an ‘emergent trajectory’ subject to transformative changes (Murdoch, 2006). Similar to the actor-network topology, space is not an ‘a priori given’ but co-emergent with entity transformation and an outcome of relations (Murdoch, 2006). This space is multiple in nature, emerging singularly but joining other spaces, creating more multiplicity or resulting in some sort of ‘unity’ (Murdoch, 2006). In Deleuze’s topology space is understood sequentially as relations being continually transformed, resulting in emergent spaces that then combine in multiplicities. 

Vocabulary

In addition to a definition of space, the study of topology also utilizes a ‘borrowed’ vocabulary from mathematics, (Allen, 2011). Following is a brief topological vocabulary describing four key terms: folding, translation, transduction and mutation. 

The first term in this vocabulary is folding – a process of bringing two or more disparate features together in space and time. Space is folded when two distinct sites are brought together topologically, where their absolute locations remain the same, but they are connected through a network (Mol & Law, 1994). In network spaces, landscapes are continually folded by the underlying network, connecting relations in space (Murdoch, 2006). In a reciprocal process, relations and objects can be folded into each other to produce more unified spaces, while simultaneously having spaces folded into those relations and objects (Law & Mol, 2001). In Deleuze’s topology there is also the concept of a ‘virtual continuum’, where virtual processes create an actual topological ‘structure’ (Martin & Secor, 2014). This mutual constitution of the virtual and actual have a temporal, as well as spatial aspect, topologically folding the past, present and future into each other (Martin & Secor, 2014; Shields, 2013)

The next commonly used topological terms are translation and transduction. These terms are similar because they both describe how something is ‘transferred’ between states (Murdoch, 2006; Kitchin & Dodge, 2011) The term translation originated in Actor-Network Theory, referring to the process of bringing something into a network by being ‘interested’ in that network (Murdoch, 2006). Transduction on the other hand, is defined as “the constant making anew of a domain in reiterative and transformative practices” (Kitchin & Dodge, 2011, p. 263). Both translation and transduction exemplify the relationality and changing natures of topological space, both relying on relations between objects to constantly create and bring new spaces into being. 

Finally, we can expand on the notion of change in topological space by looking at the term mutation. Defined biologically, mutation describes a change in a genetic sequence by internal processes or external environmental factors (Loewe, 2008). This concept is easily applicable to topology which is concerned with how spaces are ‘continually becoming’, and how relations change. Mutation can be extended to the multiplicity of space as well, conceptualizing how things mutate through multiple, interconnected spaces.

This topological vocabulary further identifies topology as a theory concerned with relationality, multiplicity and change. Challenging rigid topographical thinking, topology allows for the conceptualization of space-times as non-linear, and emergent through relations. Topology is  an exceedingly useful tool for analyzing space in an open, dynamic way, and geography should continue to explore and engage with this space. 

References 

Allen, J. (2011). Topological twists: Power’s shifting geographies. Dialogues in Human Geography, 1(3): 283-298.

Kitchin, R., & Dodge, M. (2011). Code/Space. Massachusetts Institute of Technology. 

Law, J. (1999). After ANT: Complexity, naming and topology. The Sociological Review, 47(1): 1-14.

Law, J. (2000). Objects, Spaces and Others. Centre for Science Studies.

Law, J., & Mol, A. (2001). Situating technoscience: An inquiry into spatialities. Environment and Planning D: Society and Space, 19(5): 609-621. 

Loewe, L. (2008). Genetic mutation. Nature Education, 1(1): 113.

Martin, L., & Secor, A.J. (2014). Towards a post-mathematical topology. Progress in Human Geography, 38(3), 420-438.

Mol, A., & Law, J. (1994). Regions, networks and fluids: Anaemia and social topology. Social Studies of Science, 24(4): 641-671. 

Murdoch, J. (2006). Post-structuralist geography. Sage. 

Shields, R. (2013). Spatial questions: Cultural Topologies and Social Spatialisations. Sage. 

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