From:propp@math.mit.edu (Jim Propp)Newsgroups:sci.math.researchSubject:Euler characteristic (again)Date:6 Jul 1995 16:53:56 -0400Organization:MIT Department of Mathematics

Let X be the direct sum (not the direct product!) of countably many copies of the interval [0,1]; that is, X is the set of infinite sequences (x_1, x_2, x_3, ...) such that x_i is in [0,1] for all i, and such that all but finitely many of the x_i's vanish. I am trying to develop (or find in the literature) a theory of "infinite- dimensional polytopes" like X. In particular, I would like to be able to show that in at least one natural sense (and more likely in several natural senses), the Euler characteristic of X is 1. Define X_n as the subset of X containing all sequences (x_1, x_2, ...) for which all terms past the nth term vanish (so that X is the union of the X_n's), define a 0-cell in X as a point in X, and for k > 1 define a k-cell in X as a regular open subset of some X_n that is homeomorphic to the interior of a k-ball. Suppose X is written as a disjoint union of 0-cells, 1-cells, 2-cells, etc., such that for all k > 0, the boundary of any k-cell in the decomposition can be written as a union of the 0-cells, 1-cells, ..., and (k-1)-cells in the decomposition. Define the Poincare series associated with this decomposition as the power series in t in which the coefficient of t^k is the number of k-cells in the decomposition. * Must this Poincare series have an analytic continuation to a neighborhood * of t=-1, taking the value 1 at the point itself? Example: Let D_n denote the boundary of X_n and I_n denote the interior of X_n. We can decompose X into X_0, D_1-X_0, I_1, D_2-X_1, I_2, D_3-X_2, I_3, etc., yielding a decomposition with 2 0-cells, 2 1-cells, 2 2-cells, etc. The Poincare series is thus 2 + 2t + 2t^2 + ..., whose analytic extension to the plane (punctured at t=+1) takes the value 1 at t=-1. I have framed my question somewhat narrowly, but I'm interested in other questions of the same sort, and I'd like to know about any relevant work that's been done. Jim Propp Department of Mathematics MIT

From:propp@math.mit.edu (Jim Propp)Date:27 Jan 1997 12:24:59 -0500Newsgroups:sci.math.researchSubject:a loopy calculation

Here's a somewhat screwy way to calculate the Euler characteristic of the loop space of the sphere. Does anyone know of a theory that explains why this calculation gives the right answer (1), and that explains under what circumstances the low-tech combinatorial/analytical approach sketched below gives the correct Euler characteristic for other mapping-spaces? 1) Let X be the sphere, written as a disjoint union of 2 points, 2 open intervals, and 2 open disks; the closures of these six sets will be called cells. The cells are ordered by inclusion, forming a finite poset that we'll call P. We put the order topology on P so that the closed subsets of P are just the order-ideals of P; that is, the closure of an element of P is the set of all cells that are subsets of that cell. (Motivation: we will look at the loop space of P instead of the loop space of X, which is a win since P is finite.) 2) Write S^1 as [0,1] with its endpoints identified. Choose a point x in one of the open disks in X, and let F be the set of smooth maps from S^1 to X that take 0 to x. We want to calculate the Euler characteristic of F, but F is too complicated. So... 3) Consider the map c that sends each point in X to the smallest closed cell containing it. If we compose any map f in F with the map c, we get a map g from S^1 to P (a simpler sort of map than f, since the range is a finite set). Moreover, the new map g is continuous relative to the order topology on P. The map from F to G is sort of a retraction, so we "reason" that the Euler characteristic of F equals the Euler characteristic of G. 4) Say that a map in G has a jump at t (0 < t < 1) if the map is not constant on any neighborhood of t. Each map in G has only finitely many jumps (because each map in F was smooth). Let G_n(t_1,t_2,...,t_n) (with t_1 < t_2 < ... < t_n) be the set of maps in G that have jumps at t_1,t_2,...,t_n and no other points; this is a finite set whose cardinality (call it a_n) depends only on n, not on the t_i's. (To specify a map with jumps at certain locations, one need only specify the value of the map at the jumps and between the jumps, subject to the continuity constraint and to the condition that there be genuine jumps at those locations.) 5) Let G_n be the set of maps in G with exactly n jumps (the union of the aforementioned sets as the t_i's vary, with n fixed). Then G_n fibers over an open n-simplex (namely the set of all (t_1,t_2,...,t_n) with t_1 < t_2 < ... < t_n), and the open simplex has combinatorial Euler characteristic (-1)^n; since all the fibers have cardinality (and Euler characteristic) a_n, it is reasonable to say that G_n has Euler characteristic (-1)^n a_n. 6) The sum a_0 - a_1 + a_2 - a_3 + ... does not converge. However, if one determines the a_n's explicitly one finds that the power series a_0 - a_1 t + a_2 t^2 - a_3 t^3 + ... is the Taylor series of the rational function (1+8t-8t^2)/(1+12t-12t^2); and, plugging in the value t=1, we get the final answer 1, as expected. Jim Propp Department of Mathematics MIT