Norman Levitt

The existence of combinatorial formulae for characteristic classes

Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on a E M depends only on the ordered star st(a, M). For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class. For some time it has been known that there is a simple combinatorial representation for the homology duals of the Stiefel-Whitney classes of a combinatorial manifold (Whitney (8), cf. Cheeger (1), Halperin and Toledo (2)). It is natural to ask whether there is an analogous result for other characteristic classes. For instance, can one give a simple combinatorial formula for the Pontrjagin classes or for their homology duals? What is being sought is a formula which depends only on the local structure of the combinatorial manifold K (as a simplicial complex). In this paper we prove a theoretical result. We establish that formulae of this type exist for all characteristic classes and for their homology duals. But the method of proof makes it extremely difficult to actually give such a formula explicitly. Our formulae depend, in general, on local ordering of the complex, but for rational classes (such as the rational Pontrjagin classes) this dependence disappears. Miller (4) has shown that the rational characteristic numbers of K are in fact the only numerical invariants of K which admit formulae in terms of the local (unordered) structure of K. Thus, for a general characteristic class, some other datum such as our local ordering is necessary. One corollary to the existence of local formulae is that any manifold which can be triangulated so that the links of q-simplexes admit orientation revers-

The euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes

If a numerical homotopy invariant of finite simplicial complexes has a local formula, then, up to multiplication by an obvious constant, the invariant is the Euler characteristic. Moreover, the Euler characteristic itself has a unique local formula.

The science of conjecture: Evidence and probability before pascal

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MATHEMATICS AS THE STEPCHILD OF CONTEMPORARY CULTURE

M Y TITLE WILL DOUBTLESS strike some as impertinent, others as paranoid. Mathematicians in particular will hardly be likely to view themselves as pariahs skulking at the margins of academic and intellectual life. Mathematical research, partly because it is incredibly cheap by the standards of experimental physics, chemistry, or biology, tends to be resistant to the ups and downs of funding policy. It flourishes with exponential vigor. Every year, hundreds of thousands of papers appear in thousands of journals, and journals themselves proliferate at the rate of dozens per year. Math libraries groan with the burden of incoming publications. Math department bulletin boards are scarcely able to hold all the conference announcements that arrive with “please post” requests. In that sense, mathematics and related fields can hardly be thought of as the starveling outcasts of intellectual life. And yet, I will argue, mathematics itself occupies an ambiguous place in the larger world of ideas. It is as much resented as admired. Even worse, there is a widespread strategy, among humanist intellectuals, of cheerfully conceding that one is mystified and paralyzed by mathematics, with the clear corollary that this defect, like lack of perfect pitch, is to be regretted mildly or not at all. Implicitly, one categorizes mathematics, and those deeply involved with its ideas and methods, as an arcane cult, mostly harmless, sometimes useful, but, all in all, inexplicable and extraneous to one’s deepest interests, whatever those might be. Occasionally, the affectation of disinterest falls away, and comments on mathematics and mathematicians become positively envenomed. Yet even discounting this occasional spitefulness, and concentrating solely on the rather more widespread combination of ignorance and indifference, we find a deep fault line in the culture of those who take ideas seriously, or pretend to. This is the theme I want to explore. At the same time, I propose to understand the gulf between mathematics-and, as will be seen, I am content to use that term rather broadly-and humanistic cuZture as something rather novel in Western intellectual life, if one takes the long view. This exclusion of mathematics from the general traffic in ideas is not only recent, but goes

Was Hobbes a crank

Much as I appreciate any attempt to explore the fact that an encounter with mathematics can turn on the light of reason, I was astonished – and even a little alarmed – by Robert P Creases choice of Thomas Hobbes as an exemplar (January p15). Hobbes, indeed, presents us with the other side of the coin. His famous discovery of the art of mathematical inference (assuming that his biographer John Aubrey got the facts straight) is a perfect illustration of how, for a person with more enthusiasm than talent, such an encounter can produce not a mathematician, but a crank.

A higher superstition? A reply to Steve Fuller's review

We will not abuse the privilege acknowledged with sincere gratitude of responding to Steve Fuller. Not for us the comforts of essay form, of historical stories and speculations, of elaborating the obvious difficulty, say, of building a supercomputer, even with all the ’specialized knowledge’ to hand, without social input and support. Facts should suffice to deal with the troubling assertions that spice Fuller’s potage, without running to quite his generous length. In what follows we try to address the addressable points, seriatim. So, first, to mean spirits. Of the reviews printed to date, nearly all have been favorable; none of those detects mean-spiritedness, although one reviewer smells anger. Very few have been outright hostile: one of these is by a sociologist (who is interested in ’ideological work’) writing in Science; another by a reviewer for the Times Higher Education Supplement, whose knowledge of our views seems to have been acquired without reading the book. Only the sociologist reported mean-spititedness (that, presumably, which sociologists do not display towards science or objectivity). It is fair to note that some of the magnanimous spirits who follow Internet ’Sci-Tech-Studies’ also judge us ungenerous to a fault, and that

Dynamical polysystems and vector bundles

Abstract Let M n be a smooth manifold with smooth vector fields v 1 , v 2 . The 1-parameter groups defined by these vector fields combine to define an action of the free product R ∗ R on M n . For suitable choice of v 1 , v 2 , the isotropy group L of some basepoint is of the same homotopy type as the loop space of M n . Moreover, the natural linear representation of L into O( n ) defined by the L -action on the tangent space at the basepoint deloops to the tangent bundle of M n . This observation can be amplified: k -dimensional vector bundles over M n are in 1-1 correspondence with equivalence classes of smooth representations of L into O( k ). Consequently, for any CW complex C homotopy equivalent to a finite dimensional manifold, k -vector bundles over C may be identified with k -dimensional representations of L for some suitable subgroup L of R ∗ R .