New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Kuhn, Nicholas J.

doi:10.1007/978-3-0348-8312-2_14

Cited by 4 publications

(4 citation statements)

References 31 publications

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“…Hom U (H * (K (V, n)), H * (K (W, n))) ∼ = This result is an affirmative response to the Question raised as [10,Question 10.3], although the methods do not appear to shed any light upon the generalization to unstable Ext groups which is proposed as [10,Question 10.2].…”

Section: Theorem 1 For N a Positive Integer And V W Finitely-generatmentioning

confidence: 57%

“…10. The results of this paper involve an analysis of the equalizer diagrams, exploiting standard techniques for calculating spaces of homomorphisms in the functor category F.…”

Section: The Strategy and Main Steps Of The Proofmentioning

confidence: 99%

“…For the applications to stable splittings of the suspension spectra of EilenbergMacLane spaces, ∞ K (V, n), the reader is referred to [10].…”

Section: Corollary 116 There Is An Isomorphism Of Vector Spacesmentioning

confidence: 99%

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Endomorphisms of H*(K(V, n); ) in the category of unstable modules

Powell

2006

Math. Z.

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Section: Theorem 1 For N a Positive Integer And V W Finitely-generatmentioning

confidence: 57%

“…10. The results of this paper involve an analysis of the equalizer diagrams, exploiting standard techniques for calculating spaces of homomorphisms in the functor category F.…”

Section: The Strategy and Main Steps Of The Proofmentioning

confidence: 99%

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Endomorphisms of H*(K(V, n); ) in the category of unstable modules

Powell

2006

Math. Z.

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“…The existence of a close relationship between the mod 2 reduced cohomology of RP ∞ and the cohomology of the dual Brown-Gitler spectra T (k) was exhibited algebraically by Miller [Mil84] and Kuhn observed [Kuh01] that this has a topological counterpart, namely RP ∞ is a stable summand of hocolim{T (1) → T (2) → . .…”

Section: Ext Calculationsmentioning

confidence: 91%

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

Powell

2015

Topology and its Applications

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A structure theorem for bounded-below modules over the subalgebra A (1) of the mod 2 Steenrod algebra generated by Sq 1 , Sq 2 is proved; this is applied to prove a classification theorem for a family of indecomposable A (1)-modules. The action of the A (1)-Picard group on this family is described, as is the behaviour of duality.The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A (1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown-Gitler spectrum).2000 Mathematics Subject Classification. 19L41; 55S10. Key words and phrases. Steenrod algebra, indecomposable module, truncated projective space, Brown-Gitler spectra.The author is very grateful to an anonymous referee for remarks which have lead to an improvement in the presentation of the paper.

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The action of Young subgroups on the partition complex

Arone

Brantner

2021

Publ.math.IHES

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We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ | Π n | , which is the $\Sigma_{n}$ Σ n -space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$ { 1 , … , n } .We express the space of fixed points $|\Pi_{n}|^{G}$ | Π n | G in terms of subgroup posets for general $G\subset \Sigma_{n}$ G ⊂ Σ n and prove a formula for the restriction of $|\Pi_{n}|$ | Π n | to Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ Σ n 1 × ⋯ × Σ n k . Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of $|\Pi_{n}|$ | Π n | , commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of $|\Pi_{n}|$ | Π n | in terms of “atoms” $|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}$ | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for $\ell$ ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 to $\mathbf {F}_{p}$ F p for $p$ p an odd prime.

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New relationships among loopspaces, symmetric products, and Eilenberg Maclane spaces

Cited by 4 publications

References 31 publications

Endomorphisms of H*(K(V, n); ) in the category of unstable modules

Endomorphisms of H*(K(V, n); ) in the category of unstable modules

Truncated projective spaces, Brown–Gitler spectra and indecomposable A(1)-modules

The action of Young subgroups on the partition complex

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