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Differential Topology and Homology

Unbeknownst to most outsiders, theoretical physics underwent a significant transformation -- albeit not yet a true Kuhnian paradigm shift -- in the 1970's and 80's: the traditional tools of mathematical physics (real and complex analysis), which deal with the space-time manifold only locally, were supplemented by topological approaches (more precisely, methods from differential topologygif) that account for the global (holistic) structure of the universe. This trend was seen in the analysis of anomalies in gauge theoriesgif; in the theory of vortex-mediated phase transitionsgif; and in string and superstring theories. gif Numerous books and review articles on ``topology for physicists'' were published during these years. gif

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. gif gif
As Althusser rightly commented, ``Lacan finally gives Freud's thinking the scientific concepts that it requires''. gif More recently, Lacan's topologie du sujet has been applied fruitfully to cinema criticismgif and to the psychoanalysis of AIDS. gif In mathematical terms, Lacan is here pointing out that the first homology groupgif 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. gif 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 polynomialgif) from three-dimensional Chern-Simons quantum field theory. gif

Analogous topological structures arise in quantum gravity, but inasmuch as the manifolds involved are multidimensional rather than two-dimensional, higher homology groups play a role as well. These multidimensional manifolds are no longer amenable to visualization in conventional three-dimensional Cartesian space: for example, the projective space tex2html_wrap_inline1407 , which arises from the ordinary 3-sphere by identification of antipodes, would require a Euclidean embedding space of dimension at least 5. gif Nevertheless, the higher homology groups can be perceived, at least approximately, via a suitable multidimensional (nonlinear) logic. gif gif



next up previous
Next: Manifold Theory: (W)holes and Up: Transgressing the Boundaries: Towards Previous: Quantum Gravity: StringWeave



Daniel Sleator
Thu Jun 6 15:34:37 EDT 1996