Psychotopologies: closing the circuit between psychic and material space

In this paper we present an argument for a psychoanalytic understanding of space. While Freud struggled to move away from his own early prepsychoanalytic attempts at mapping the psyche through cerebral localization, he nevertheless found himself compelled to use spatial language and topographical models throughout his career. In his ambivalence, Freud emphasized that the space of the psyche should be read as no more than metaphorical. We argue that the topographical models that Freud struggled with were constrained by the metrics of Euclidean space. The psyche is spatial, just not in topographical terms. For Jacques Lacan many of the psychic operations that Freud described (such as the transference) are better understood in terms of topological operations. Lacan uses such figures as the torus, the cross-cap, and the Mobius strip to demonstrate how the subject is formed through internal exclusions and external inclusions. Using Freud's famous case of the Rat Man, we argue that the neurotic's journey shows him seeking to overcome a psychospatial problem, one that resembles a topological conundrum. Through the Rat Man's story, we demonstrate how Lacan's topology of the subject (the R-schema as cross-cap) accounts for the Mobius twist that allows the neurotic to situate people, events, and places that are apparently separated in time and space in the same place. Ultimately, we consider the usefulness of topology over Euclidean space for building an understanding of space that is at once psychic and material.