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The Cohomotopy charge map is the function that assigns to a configuration of points their total charge as measured in Cohomotopy-cohomology theory.
This is alternatively known as the “electric field map” (Salvatore 01 following Segal 73, Section 1, see also Knudsen 18, p. 49) or the “scanning map” (Kallel 98).
For $D \in \mathbb{N}$ the Cohomotopy charge map is the continuous function
from the configuration space of points in the Euclidean space $\mathbb{R}^D$ to the $D$-Cohomotopy cocycle space vanishing at infinity on the Euclidean space(which is equivalently the space of pointed maps from the one-point compactification $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold iterated based loop space of the D-sphere), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit charge to their total charge as measured in Cohomotopy-cohomology theory (Segal 73, Section 3).
This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to equivariant Cohomotopy. The following graphics illustrates the Cohomotopy charge map on G-space tori for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-equivariant Cohomotopy:
graphics grabbed from SS 19
In some situations the Cohomotopy charge map is a weak homotopy equivalence and hence exhibits, for all purposes of homotopy theory, the Cohomotopy cocycle space of Cohomotopy charges as an equivalent reflection of the configuration space of points.
(group completion on configuration space of points is iterated based loop space)
from the full unordered and unlabeled configuration space (here) of Euclidean space $\mathbb{R}^D$ to the $D$-fold iterated based loop space of the D-sphere, exhibits the group completion (here) of the configuration space monoid
(Cohomotopy charge map is weak homotopy equivalence on sphere-labeled configuration space of points)
For $D, k \in \mathbb{N}$ with $k \geq 1$, the Cohomotopy charge map (1)
is a weak homotopy equivalence from the configuration space (here) of unordered points with labels in $S^k$ and vanishing at the base point of the label space to the $D$-fold iterated loop space of the D+k-sphere.
The May-Segal theorem generalizes from Euclidean space to closed smooth manifolds if at the same time one passes from plain Cohomotopy to twisted Cohomotopy, twisted, via the J-homomorphism, by the tangent bundle:
Let
$X^n$ be a smooth closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
the J-twisted (n+k)-Cohomotopy space of $X^n$, hence the space of sections of the $(n + k)$-spherical fibration over $X$ which is associated via the tangent bundle by the O(n)-action on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$
the configuration space of points on $X^n$ with labels in $S^k$.
(Bödigheimer 87, Prop. 2, following McDuff 75)
In the special case that the closed manifold $X^n$ in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:
Let
$X^n$ be a parallelizable closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
$(n+k)$-Cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere
the configuration space of points on $X^n$ with labels in $S^k$.
The concept goes back to
Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 0331377 (pdf)
Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)
with comprehensive review in
See also:
Sadok Kallel, Particle Spaces on Manifolds and Generalized Poincaré Dualities (arXiv:math/9810067)
Paolo Salvatore, Configuration spaces with summable labels, In: Aguadé J., Broto C., Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 (arXiv:math/9907073)
Ben Knudsen, Configuration spaces in algebraic topology (arXiv:1803.11165)
Hisham Sati, Urs Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)
Last revised on October 18, 2019 at 17:53:01. See the history of this page for a list of all contributions to it.