Counting elements in homotopy sets

Switzer, Robert M.

doi:10.1007/bf01174773

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…πi(X)=0 for i^n) space, the square is a pull back, L = L(M, n+l)=EπX π K(M, n+l) where Mis a π-module, n=π x {X), PL->L is the path fibration [6] over and under B~Bτz-K{π y 1), k is the zero-section of the fibration k: L^>B, θ 0 is a fibration inducing the identity on π lf and θ is a map over B whose vertical homotopy class also will be denoted by 0eUP +1 (Z; M). The notation for mapping spaces is as in Switzer [10]. Aut#(Z), Z<^{X, Y\, can be thought of as a set of path-components of F λ (Z, * Z) write aut # (Z)c Fι(Z, * Z) for the subspace made up by these path-components such that 7Γ 0 aut # (Z)=Aut # (Z).…”

Section: Spaces Of Self-homotopy Equivalencesmentioning

confidence: 99%

Self-homotopy equivalences of $H_*(-;\bold Z/p)$-local spaces

Møller¹

1989

Kodai Math. J.

106

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Under certain finiteness conditions, ^-completion commutes with the formation of a certain group of self-homotopy equivalences. Introduction.Let X be a pointed, 0-connected topological space, Aut(Z) the group of based homotopy classes of based self-homotopy equivalences of X, and Aut # (Z) the kernel of the obvious homomorphism Aut(Z)-> τiAutπi(X)where we further assume either that X is a Cΐf-complex of dimension d or that 7r*(Z)=0 for *>d, l^d

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Section: Spaces Of Self-homotopy Equivalencesmentioning

confidence: 99%

Self-homotopy equivalences of $H_*(-;\bold Z/p)$-local spaces

Møller¹

1989

Kodai Math. J.

106

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“…It is easy to see that scanning has a stable analog st : SP ∞ (M + )− − →Γ(sτ + M) but harder to verify that st is a (weak) homotopy equivalence [14,19]. Note that SP ∞ (M + ) SP ∞ M × Z and SP ∞ (M) is Proposition 7.1 [38] • If A ⊂ X ⊂ X is a nested sequence of NDR pairs, and j : X → X the inclusion, then the induced map Γ u (X, A; E, B)− − →Γ uj (X , A; E, B) yields a fibration with Γ u (X, X ; E, B) as fibre. Finally and according to [10, page 29], if E− − →B is a Hurewicz fibration and s, t are two sections, then s and t are homotopic if and only if they are section homotopic.…”

Section: Scanning and Stabilitymentioning

confidence: 99%

“…7 Appendix: Some homotopy properties of section spaces All spaces below are assumed connected. We discuss some pertinent statements from Switzer [38]. Let p : E− − →B be a Serre fibration, i : A → X a cofibration (A can be empty) and u : X− − →E a given map.…”

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confidence: 99%

Symmetric products, duality and homological dimension of configuration spaces

Kallel¹

2008

Geometry and Topology Monographs

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We discuss various aspects of "braid spaces" or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated symmetric products. We finally refine and then apply a theorem of McDuff on the homological connectivity of a map from braid spaces to some spaces of "vector fields". 55R80; 55S15, 18G20To Fred Cohen on his 60th birthday 500 Sadok Kallel[8]). Its fundamental group written Br n (M) is the "braid group" of M . The object of this paper is to study the homology of braid spaces and the main approach we adopt is that of duality with the symmetric products. In so doing we take the opportunity to refine and elaborate on some classical material. Next is a brief content summary.Section 2 describes the homotopy type of braid spaces of some familiar spaces and discusses orientation issues. Section 3 introduces truncated products, as in Bödigheimer, Cohen and Milgram [6] and Milgram and Löffler [24], states the duality with braid spaces and then proves our first main result on the cohomological dimension of braid spaces. Section 4 uses truncated product constructions to split in an elementary fashion the homology of braid spaces for punctured manifolds. In Section 5 we prove our sharp connectivity result for reduced symmetric products of CW complexes which seems to be new and a significant improvement on work of Nakaoka and Welcher [42]. In Section 5.2 we make the link between the homology of symmetric and truncated products by discussing a spectral sequence introduced by Bödigheimer, Cohen and Milgram and exploited by them to study "braid homology" H * (B(M, n)). Finally Section 6 completes a left out piece from McDuff and Segal's work on configuration spaces [23]. In that paper, H * (B(M, n)), for closed manifolds M , is compared to the homology of some spaces of "compactly supported vector fields" on M and the main theorem there states that these homologies are isomorphic up to a range that increases with n. We make this range more explicit and use it for example to determine the abelianization of the braid groups of a closed Riemann surface. A final appendix collects some homotopy theoretic properties of section spaces that we use throughout.Below are precise statements of our main results which we have divided up into three main parts. Unless explicitly stated, all spaces are assumed to be connected. The n th symmetric group is written S n . Connectivity and cohomological dimensionFor M a manifold, we write H * (M, ±Z) for the cohomology of M with coefficients in the orientation sheaf ±Z; in other words H * (M, ±Z) is the homology ...

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“…Throughout this note, I use the notation of [8]: If (X, A) is a pair of spaces, p: Y->B a fibration, and u : X-^Y a continuous map, then F U (X, A Y, B) is the space, equipped with the compactly generated topology associated to the compact-open topology, of all maps v: X-+Y such that υ\A-u\A and pv=pu. An empty space in the A-entry or a one-point space in the 5-entry will be omitted; thus e.g.…”

Section: Introductionmentioning

confidence: 99%

Self-maps on twisted Eilenberg-Mac Lane spaces

Møller

1988

Kodai Math. J.

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Counting elements in homotopy sets

Cited by 8 publications

References 11 publications

Self-homotopy equivalences of $H_*(-;\bold Z/p)$-local spaces

Self-homotopy equivalences of $H_*(-;\bold Z/p)$-local spaces

Symmetric products, duality and homological dimension of configuration spaces

Self-maps on twisted Eilenberg-Mac Lane spaces

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