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A multiple loop space.
A grouplike E-k algebra in Top.
An iterated loop space object in Top.
See at May recognition theorem.
For $A$ a pointed homotopy type, hence an ∞-groupoid equipped with a base point $\ast \xrightarrow{ pt_A } A$, then for $n \,\in\, \mathbb{N}$, the n-fold loop space of $A$ is the homotopy fiber of the basepoint-evaluation map on the mapping space from the homotopy type of the n-sphere:
We may present the sequence in the classical model structure on topological spaces or the classical model structure on simplicial sets, in the latter case we may assume that $A$ is presented by a Kan complex, so that, in either case, it is a fibrant object.
In either case, the canonical model for the iterated loop space is evidently the ordinary 1-category-theoretic fiber of the evaluation map out of the internal hom:
Moreover, the evaluation map is equivalently the image of the point inclusion under the internal hom-functor
Since either model category is a cartesian closed monoidal model category, hence an enriched model category over itself (this Exp.) and since the canonical model for $\ast \to ʃ S^n$ is a cofibration, in either case, the pullback-power axiom implies that $ev_\ast$ is a fibration. Therefore its ordinary fiber above models the homotopy fiber, and the claim follows.
The homotopy groups of the mapping space $Maps(ʃ S^n ,\, A)$ out of an n-sphere form a long exact sequence with those of $A$, of the following form:
This is the long exact sequence of homotopy groups applied to the homotopy fiber sequence from Prop. .
If $A \,\in\, Grp_\infty$ is n-truncated, then the evaluation map out of the mapping space from the (n+2)-sphere into it is a weak homotopy equivalence:
By assumption, the long exact sequence from Cor. collapses to exact segments of the form
(rational cohomology of iterated loop space of the 2k-sphere)
Let
(hence two positive natural numbers, one of them required to be even and the other required to be smaller than the first) and consider the D-fold loop space $\Omega^D S^n$ of the n-sphere.
Its rational cohomology ring is the free graded-commutative algebra over $\mathbb{Q}$ on one generator $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:
(Kallel-Sjerve 99, Prop. 4.10)
See at homology of iterated loop spaces.
(iterated loop spaces equivalent to configuration spaces of points)
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
the electric field map/scanning map constitutes a homotopy equivalence
between
the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. )
the d-fold loop space of the $d$-fold reduced suspension of the quotient space $Y / \partial Y$ (regarded as a pointed topological space with basepoint $[\partial Y]$).
In particular when $Y = \mathbb{D}^k$ is the closed ball of dimension $k \geq 1$ this gives a homotopy equivalence
with the d-fold loop space of the (d+k)-sphere.
(May 72, Theorem 2.7, Segal 73, Theorem 3, see Bödigheimer 87, Example 13)
(stable splitting of mapping spaces out of Euclidean space/n-spheres)
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
there is a stable weak homotopy equivalence
between
the suspension spectrum of the configuration space of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. )
the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. ).
Combined with the stabilization of the electric field map/scanning map homotopy equivalence from Prop. this yields a stable weak homotopy equivalence
between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the $d$-fold reduced suspension of $Y / \partial Y$.
(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)
In fact by Bödigheimer 87, Example 5 this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. :
Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. Volume 83, Number 4 (1977), 456-494. (euclid:bams/1183538891)
Infinite loop space theory revisited (pdf)
John Frank Adams, Infinite loop spaces, Hermann Weyl lectures at IAS, Princeton University Press (1978) (ISBN:9780691082066, doi:10.1515/9781400821259)
Peter May, The uniqueness of infinite loop space machines, Topology, vol 17, pp. 205-224 (1978) (pdf)
Jacob Lurie, Section 5.2.6 of Higher Algebra
On the Morava K-theory of iterated loop spaces of n-spheres:
In relation to configuration spaces of points:
Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf, doi:10.1007/BFb0067491)
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973) 213-221. MR 0331377 [pdf, doi:10.1007/BF01390197]
Victor Snaith, A stable decomposition of $\Omega^n S^n X$, Journal of the London Mathematical Society 7 (1974), 577 - 583 (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)
Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)
On ordinary cohomology of iterated loop spaces in relation to configuration spaces of points (see also at graph complex):
On the rational cohomology:
Last revised on March 28, 2023 at 15:43:05. See the history of this page for a list of all contributions to it.