homotopy theory, (∞,1)-category theory, homotopy type theory
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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 pointed homotopy type, hence an ∞-groupoid equipped with a base point , then for , the n-fold loop space of 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 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 is a cofibration, in either case, the pullback-power axiom implies that is a fibration. Therefore its ordinary fiber above models the homotopy fiber, and the claim follows.
The homotopy groups of the mapping space out of an n-sphere form a long exact sequence with those of , of the following form:
This is the long exact sequence of homotopy groups applied to the homotopy fiber sequence from Prop. .
If 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 of the n-sphere.
Its rational cohomology ring is the free graded-commutative algebra over on one generator of degree and one generator of degree :
(Kallel-Sjerve 99, Prop. 4.10)
See at homology of iterated loop spaces.
(iterated loop spaces equivalent to configuration spaces of points)
For
, a natural number with denoting the Cartesian space/Euclidean space of that dimension,
the electric field map/scanning map constitutes a homotopy equivalence
between
the configuration space of arbitrary points in vanishing at the boundary (Def. )
the d-fold loop space of the -fold reduced suspension of the quotient space (regarded as a pointed topological space with basepoint ).
In particular when is the closed ball of dimension 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
, a natural number with denoting the Cartesian space/Euclidean space of that dimension,
there is a stable weak homotopy equivalence
between
the suspension spectrum of the configuration space of an arbitrary number of points in vanishing at the boundary and distinct already as points of (Def. )
the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in , vanishing at the boundary and distinct already as points in (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 -fold reduced suspension of .
(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)
In fact by Bödigheimer 87, Example 5 this equivalence still holds with treated on the same footing as , hence with on the right replaced by 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 , 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.