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39
The polyhedral product functor: a method of computation for momentangle complexes, arrangements and related spaces
, 2008
"... This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the ..."
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Cited by 28 (7 self)
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This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in GoreskyMacPherson [20], Hochster[22], Baskakov [3], Panov [36], and BuchstaberPanov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized momentangle complex applies for any homology theory. This decomposition gives an additive decomposition for the StanleyReisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of DenhamSuciu in [16].
On loop spaces of configuration spaces
 Trans. Amer. Math. Soc
"... Abstract. This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as “twistings ” between the factors. The main homological results a ..."
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Cited by 25 (0 self)
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Abstract. This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as “twistings ” between the factors. The main homological results are given in terms of extensions of the “infinitesimal braid relations ” or “universal YangBaxter Lie relations”. 1.
Homotopical dynamics II: Hopf invariants, smoothings and the Morse complex
 Ann. Scient. Ecole Norm. Sup
"... Abstract. The ambient framed bordism class of the connecting manifold of two consecutive critical points of a MorseSmale function is estimated by means of a certain Hopf invariant. Applications include new examples of nonsmoothable Poincaré duality spaces as well as an extension of the Morse compl ..."
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Cited by 11 (3 self)
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Abstract. The ambient framed bordism class of the connecting manifold of two consecutive critical points of a MorseSmale function is estimated by means of a certain Hopf invariant. Applications include new examples of nonsmoothable Poincaré duality spaces as well as an extension of the Morse complex. 1. Introduction. Let M be a smooth, compact, riemannian manifold and let f: M − → R be a smooth MorseSmale function, regular and constant on ∂M. The flow γ: M × R − → M used below is induced by −∇f. Assume that P and Q are consecutive critical points of f (this means that f(P)> f(Q) and that
On the LusternikSchnirelmann category of maps
, 1998
"... We give conditions when cat(f g) < cat(f) + cat(g). We apply our result to show that under suitable conditions for rational maps f, mcat(f) < cat(f) is equivalent to cat(f) = cat(fidSn). Many examples with mcat(f) < cat(f) satisfying our conditions are constructed. We also resolve one op ..."
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Cited by 7 (2 self)
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We give conditions when cat(f g) < cat(f) + cat(g). We apply our result to show that under suitable conditions for rational maps f, mcat(f) < cat(f) is equivalent to cat(f) = cat(fidSn). Many examples with mcat(f) < cat(f) satisfying our conditions are constructed. We also resolve one open case of Ganea’s conjecture by constructing a space X such that cat(X S1) = cat(X) = 2. In fact for every Y 6 ’ , cat(X Y) cat(Y) + 1 < cat(Y) + cat(X). We show that this same X has the property cat(X) = cat(XX) = cl(XX) = 2. Finally we give an example of a CW complex Z such that cat(Z) = 2
2PRIMARY ANICK FIBRATIONS
"... Abstract. Cohen conjectured that for r ≥ 2 there is a space T 2n+1 (2 r) and a homotopy fibration sequence Ω 2 S 2n+1 Ω 2 S 2n+1 ϕr − → S 2n−1 − → T 2n+1 (2 r) − → ΩS 2n+1 with the property that the composition ϕr − → S2n−1 E2 − → Ω2S2n+1 is homotopic to the 2rpower map. We positively resolve this ..."
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Cited by 6 (4 self)
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Abstract. Cohen conjectured that for r ≥ 2 there is a space T 2n+1 (2 r) and a homotopy fibration sequence Ω 2 S 2n+1 Ω 2 S 2n+1 ϕr − → S 2n−1 − → T 2n+1 (2 r) − → ΩS 2n+1 with the property that the composition ϕr − → S2n−1 E2 − → Ω2S2n+1 is homotopic to the 2rpower map. We positively resolve this conjecture when r ≥ 3. Several preliminary results are also proved which are of interest in their own right. 1.
Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms
"... In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs.f1;f2 / of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbe ..."
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Cited by 6 (5 self)
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In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs.f1;f2 / of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC.f1;f2 / (and MC.f1;f2/, resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to.f1;f2/. Furthermore, we deduce finiteness conditions for MC.f1;f2/. As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E.f1;f2 / into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf–Ganea homomorphisms which turn out to yield finiteness obstructions for MC. 55M20, 55Q25, 55S35, 57R90; 55N22, 55P35, 55Q40 1
Classifying spaces and fibrations of simplicial sheaves
 J. Homotopy Relat. Struct
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Homotopy Localization Nearly Preserves Fibrations
"... this paper. We then prove Theorem A in the second section and (C),(D) and (E) in sections three and four. The last section concludes with the proof of (B). We work in the category of pointed CWcomplexes and in particular all function complexes are spaces of pointed maps. ..."
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Cited by 5 (1 self)
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this paper. We then prove Theorem A in the second section and (C),(D) and (E) in sections three and four. The last section concludes with the proof of (B). We work in the category of pointed CWcomplexes and in particular all function complexes are spaces of pointed maps.
L.: Embeddings up to homotopy of twocones in Euclidean space
 Trans. Amer. Math. Soc
"... Abstract. We say that a finite CWcomplex X embeds up to homotopy in a sphere Sn+1 if there exists a subpolyhedron K ⊂ Sn+1 having the homotopy type of X. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when X is a simply ..."
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Cited by 5 (0 self)
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Abstract. We say that a finite CWcomplex X embeds up to homotopy in a sphere Sn+1 if there exists a subpolyhedron K ⊂ Sn+1 having the homotopy type of X. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when X is a simplyconnected twocone (a twocone is the homotopy cofibre of a map between two suspensions). We give different applications of this result: we prove that if X is a twocone then there are no rational obstructions to embeddings up to homotopy in codimension 3. We give also a description of the homotopy type of the boundary of a regular neighborhood of the embedding of a twocone in a sphere. This enables us to construct a closed manifold M whose LusternikSchnirelmann category and conelength are not affected by removing one point of M.