From:  (Jim Propp)
Newsgroups:     sci.math.research
Subject:        Euler characteristic (again)
Date:           6 Jul 1995 16:53:56 -0400
Organization:   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

From:  (Jim Propp)
Date:           27 Jan 1997 12:24:59 -0500
Newsgroups:     sci.math.research
Subject:        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