Applications of Topology to Psychoanalysis
and Cinema Criticism

At about the same time, in the social and psychological sciences Jacques Lacan pointed out the key role played by differential topology:
  This diagram [the Möbius strip] can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. 57,58
As Althusser rightly commented, "Lacan finally gives Freud's thinking the scientific concepts that it requires". 59 More recently, Lacan's topologie du sujet has been applied fruitfully to cinema criticism 60 and to the psychoanalysis of AIDS. 61 In mathematical terms, Lacan is here pointing out that the first homology group 62 of the sphere is trivial, while those of the other surfaces are profound; and this homology is linked with the connectedness or disconnectedness of the surface after one or more cuts. 63 Furthermore, as Lacan suspected, there is an intimate connection between the external structure of the physical world and its inner psychological representation qua knot theory: this hypothesis has recently been confirmed by Witten's derivation of knot invariants (in particular the Jones polynomial 64 ) from three-dimensional Chern-Simons quantum field theory. 65

57 Lacan (1970, 192-193), lecture given in 1966. For an in-depth analysis of Lacan's use of ideas from mathematical topology, see Juranville (1984, chap. VII), Granon-Lafont (1985,1990), Vappereau (1985) and Nasio (1987,1992); a brief summary is given by Leupin (1991). See Hayles (1990, 80) for an intriguing connection between Lacanian topology and chaos theory; unfortunately she does not pursue it. See also Zizek (1991, 38-39, 45-47) for some further homologies between Lacanian theory and contemporary physics. Lacan also made extensive use of concepts from set-theoretic number theory: see e.g. Miller (1977/78) and Ragland-Sullivan (1990).

58 In bourgeois social psychology, topological ideas had been employed by Kurt Lewin as early as the 1930's, but this work foundered for two reasons: first, because of its individualist ideological preconceptions; and second, because it relied on old-fashioned point-set topology rather than modern differential topology and catastrophe theory. Regarding the second point, see Back (1992).

59 Althusser (1993, 50): "Il suffit, à cette fin, reconnaitre que Lacan confère enfin à la pensée de Freud, les concepts scientifiques quélle exige". This famous essay on "Freud and Lacan" was first published in 1964, before Lacan's work had reached its highest level of mathematical rigor. It was reprinted in English translation in 1969 (New Left Review).

60 Miller (1977/78, especially pp. 24-25). This article has become quite influential in film theory: see e.g. Jameson (1982, 27-28) and the references cited there. As Strathausen (1994, 69) indicates, Miller's article is tough going for the reader not well versed in the mathematics of set theory. But it is well worth the effort. For a gentle introduction to set theory, see Bourbaki (1970).

61 Dean (1993, especially pp. 107-108).

62 Homology theory is one of the two main branches of the mathematical field called algebraic topology . For an excellent introduction to homology theory, see Munkres (1984); or for a more popular account, see Eilenberg and Steenrod (1952). A fully relativistic homology theory is discussed e.g. in Eilenberg and Moore (1965). For a dialectical approach to homology theory and its dual, cohomology theory, see Massey (1978). For a cybernetic approach to homology, see Saludes i Closa (1984).

63 For the relation of homology to cuts, see Hirsch (1976, 205-208); and for an application to collective movements in quantum field theory, see Caracciolo et al. (1993, especially app. A.1).

64 Jones (1985).

65 Witten (1989).

Works Cited

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Back, Kurt W. 1992. "This business of topology." Journal of Social Issues 48(2): 51-66.

Bourbaki, Nicolas. 1970. "Théorie des Ensembles." Paris: Hermann.

Caracciolo, Sergio, Robert G. Edwards, Andrea Pelissetto and Alan D. Sokal. 1993. "Wolff-type embedding algorithms for general nonlinear oe-models." Nuclear Physics B 403: 475--541.

Dean, Tim. 1993. "The psychoanalysis of AIDS." October 63: 83-116.

Eilenberg, Samuel and John C. Moore. 1965. "Foundations of Relative Homological Algebra." Providence, R.I.: American Mathematical Society.

Eilenberg, Samuel and Norman E. Steenrod. 1952. "Foundations of Algebraic Topology." Princeton, N.J.: Princeton University Press.

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Hayles, N. Katherine. 1990. "Chaos Bound: Orderly Disorder in Contemporary Literature and Science." Ithaca: Cornell University Press.

Hirsch, Morris W. 1976. "Differential Topology." New York: Springer.

Jameson, Fredric. 1982. "Reading Hitchcock." October 23: 15-42.

Jones, V.F.R. 1985. "A polynomial invariant for links via Von Neumann algebras." Bulletin of the American Mathematical Society 12: 103-112.

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Lacan, Jacques. 1970. "Of structure as an inmixing of an otherness prerequisite to any subject whatever." In "The Languages of Criticism and the Sciences of Man," pp. 186-200, edited by Richard Macksey and Eugenio Donato. Baltimore: Johns Hopkins Press.

Leupin, Alexandre. 1991. "Introduction: Voids and knots in knowledge and truth." In "Lacan and the Human Sciences," pp. 1-23, edited by Alexandre Leupin. Lincoln, Neb.: University of Nebraska Press.

Lewin, Kurt. 1936. "Principles of topological psychology." New York: McGraw-Hill.

Massey, William S. 1978. "Homology and Cohomology Theory." New York: Marcel Dekker.

Miller, Jacques-Alain. 1977/78. "Suture (elements of the logic of the signifier)." Screen 18(4): 24-34.

Nasio, Juan-David. 1987. "Les Yeux de Laure: Le Concept d'Objet a dans la Théorie de J. Lacan. Suivi d'une Introduction `a la Topologie Psychanalytique." Paris: Aubier.

Nasio, Juan-David. 1992. "Le concept de sujet de l'inconscient." Texte d'une intervention realisée dans le cadre du séminaire de Jacques Lacan "La topologie et le temps", le mardi 15 mai 1979. In "Cinq Lecons sur la Th&eoacute;rie de Jacques Lacan." Paris: Éditions Rivages.

Ragland-Sullivan, Ellie. 1990. "Counting from 0 to 6: Lacan, 'suture', and the imaginary order." In "Criticism and Lacan: Essays and Dialogue on Language, Structure, and the Unconscious," pp. 31-63, edited by Patrick Colm Hogan and Lalita Pandit. Athens, Ga.: University of Georgia Press.

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Vappereau, Jean Michel. 1985. "Essaim: Le Groupe Fondamental du Noeud." Psychanalyse et Topologie du Sujet. Paris: Point Hors Ligne.

Witten, Edward. 1989. "Quantum field theory and the Jones polynomial." Communications in Mathematical Physics 121: 351-399.