James Maps, Segal Maps, and the Kahn-Priddy Theorem

Caruso, J.; Cohen, F. R.; May, J. P.; Taylor, L. R.

doi:10.2307/1999533

Cited by 17 publications

(33 citation statements)

References 10 publications

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“…Q.X^Y / (see Caruso, Cohen, May and Taylor [5]) defined for pointed spaces X and Y by passage to the limit over the smash product maps^W k † k X ` †`Y ! kC` †kC`X^Y .…”

Section: T T T T T T T T T T T T T Tmentioning

confidence: 99%

Bordism groups of immersions and classes represented by self-intersections

Eccles

Grant

2007

Algebr. Geom. Topol.

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A well-known formula of R J Herbert's relates the various homology classes represented by the self-intersection immersions of a self-transverse immersion. We prove a geometrical version of Herbert's formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert's formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalise the formula to other homology theories. The proof is based on Herbert's but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.

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“…Q.X^Y / (see Caruso, Cohen, May and Taylor [5]) defined for pointed spaces X and Y by passage to the limit over the smash product maps^W k † k X ` †`Y ! kC` †kC`X^Y .…”

Section: T T T T T T T T T T T T T Tmentioning

confidence: 99%

Bordism groups of immersions and classes represented by self-intersections

Eccles

Grant

2007

Algebr. Geom. Topol.

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“…(Recall that each of these spaces is a homotopy commutative H-space since each is the 0-space of a special Γ-space. The homotopy-theoretic group completion of an H-space is defined in [8]. )…”

Section: Group Completions Via Mapping Telescopesmentioning

confidence: 99%

Semi-Topological K-Homology and Thomason's Theorem

Walker¹

2002

K-Theory

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Abstract. In this paper, we introduce the "semi-topological K-homology" of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an . From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the "Bott inverted" semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X an . In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that "Bott inverted" algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients.

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“…See [3, 1.4.2]. For any Ex ring space, the general formulas of [3,II,[1][2][3] determine the multiplicative Pontryagin product and homology operations on elements decomposable under the additive Pontryagin product and homology operations. Although the details have yet to be worked out, it is clear from the geometric diagrams on hand that such formulas must also exist for £" ring spaces.…”

Section: (6 A) Kxd(q X)mentioning

confidence: 99%

“…We originally planned to say more about this here but have decided to defer the discussion to [8], where the foundational work necessary to make it rigorous is carried out. This is a sequel to [2], but only § §2-4 of that paper are essential preliminaries to this one. 1.…”

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confidence: 99%

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James maps and 𝐸_{𝑛} ring spaces

Cohen¹,

May²,

Taylor³

1984

Trans. Amer. Math. Soc.

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Abstract. We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James mapis an En ring space and j is a S"-map, where 6" is the little n-cubes operad. This implies thaty has an n-fold delooping with domain 2"X It also implies an algorithm for the calculation of j and thus of each (jq) on mod p homology. When « = oo and p = 2, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture.In [2, §4], we analyzed the behavior with respect to pairings of the James maps j : CX -> QDq(G, X) used in [4] to obtain the stable splittings of spaces CA". In particular, we proved that if the given coefficient system G has a sum, then the product of the targets is a ring space and the product of the/ 's is an exponential 77-map. In fact, this is only a fragment of the full structure present in the most interesting examples.In the first two sections, we give a parametrized elaboration of this analysis. We assume the given coefficient system G is a "module" over an operad § and we deduce, essentially, thatis a S-map. Here Uq>0 denotes the weak infinite product (all but finitely many coordinates of each point are the basepoint). Technically, we shall replace the target by its equivalent Q( V i>0 Dq(G, A")) and § by an equivalent, but larger, operad.This analysis requires the introduction of general notions of additive and multiplicative actions of operads on coefficient systems, and these generalizations of the definitions of an operad and of an action of one operad on another give considerable added flexibility to the theory of iterated loop spaces developed in [9][10][11][12][13].When G is G(R"), our results imply that (jq) has an Ai-fold delooping 2"A"-£"( II QDq(R",X)\. \ q»\ '

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James Maps, Segal Maps, and the Kahn-Priddy Theorem

Abstract: Abstract. The standard combinatorial approximation C(R", X) to Q"2"X is a filtered space with easily understood filtration quotients DJ

Cited by 17 publications

References 10 publications

Bordism groups of immersions and classes represented by self-intersections

Bordism groups of immersions and classes represented by self-intersections

Semi-Topological K-Homology and Thomason's Theorem

James maps and 𝐸_{𝑛} ring spaces

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