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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.
(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 5.1.3 of Higher Algebra
In relation to configuration spaces of points:
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)
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 September 10, 2020 at 09:35:24. See the history of this page for a list of all contributions to it.