The Homotopy Type of MSU

Pengelley, David

doi:10.2307/2374085

Cited by 29 publications

(15 citation statements)

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“…Passing to colimits, we obtain f and a homotopy h : q ≃ p • f n . Since the homology groups of BoP are concentrated in even degrees [15], we can replace it by a minimal complex, with cells only in even degrees. This allows us to reverse the roles of X and BoP to construct g.A similar argument proves the following result.…”

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confidence: 99%

Minimal atomic complexes

Baker

May

2004

Topology

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Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of M U considered as a commutative S-algebra, although it is a core of M U considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, eo 2 , and BoP at the prime 2, all BP n at any prime p, and the indecomposable wedge summands of Σ ∞ CP ∞ and Σ ∞ HP ∞ at any prime p are minimal atomic.1991 Mathematics Subject Classification. Primary 55P15, 55P42, 55P60. Key words and phrases. atomic space, atomic spectrum, nuclear space, nuclear spectrum. L CJ 2n−1Since J 2n−1 is a wedge of (2n − 1)-spheres and π 2n−1 (F p) = 0, [J 2n−1 , F p] = 0. A standard result, given in just this form in [14, Lemma 1], shows that there are maps f n+1 and h n+1 that make the diagram commute. Passing to colimits, we obtain f and a homotopy h : q ≃ p • f n . Since the homology groups of BoP are concentrated in even degrees [15], we can replace it by a minimal complex, with cells only in even degrees. This allows us to reverse the roles of X and BoP to construct g.A similar argument proves the following result.

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confidence: 99%

Minimal atomic complexes

Baker

May

2004

Topology

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“…If i = 0 then our spectrum has the same homology as the spectrum BoP, constructed by David Pengelly in [10]. Using the methods of his paper (p.1116-1120) it is easy to prove that in this case BiP is equivalent to BoP.…”

Section: Using Induction We Build a Chain Of Morphismsmentioning

confidence: 85%

On bordism rings with principal torsion ideal

Vershinin¹

1991

Lecture Notes in Mathematics

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1 Introduction Bordism spectra have long been known [12] as a rich source of examples of homology comodules over the dual of the Steenrod algebra, and introducing singularities [2] allows us to vary these structures further. Such computations were pioneered for the unitary case MU in [3], and more recently for the symplectic case MSp in [ 13,14].In this paper we introduce some new sequences of singularities into MSp, and study the homology and homotopy properties of the resulting spectra. We prove that they are multiplicative, characterise their homology comodules, and show that their homotopy rings have torsion ideals generated by a single element in dimensions of the form 2 i -3. Our spectra axe mainly of interest at the prime 2, where they lie in the poorly mapped territory between MSp and BP.I owe many thanks to Nigel Ray, for much helpful advice on improving this paper. Homotopy and homology computationsNigel Ray's well known family of elements ¢i {11] lie in the symplectic bordism ring MSp.. Each ¢h i _> 1, is indecomposable in MSpsi-3 and has 2¢i = 0. It is also convenient to write ¢0 for the class 0x of the non-trivially framed circle in MSpa.Consider the following sequences of elements

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“…Let A be the mod 2 Steenrod algebra and Q i the Milnor primitives, [Mil58]. From [Pen82] we know the mod 2 cohomology…”

Section: Review Of Bopmentioning

confidence: 99%

The Omega spectrum for Pengelley’s $BoP$

Wilson

2020

Homology, Homotopy and Applications

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We compute the homology of the spaces in the Omega spectrum for BoP . There is no torsion in H * (BoP i ) for i ≥ 2, and things are only slightly more complicated for i < 2. We find the complete homotopy type of BoP i for i ≤ 6 and conjecture the homotopy type for i > 6. This completes the computation of all H * (M SU * ). CONTEXTThere are several standard (co)bordism theories and related spectra. We have, for example, unoriented (co)bordism, MO, oriented (co)bordism, MSO, Spin (co)bordism, MSpin, complex (co)bordism, MU, and special unitary cobordism, MSU. These theories are associated with Omega spectra classifying them, i.e. M k (X) = [X, M k ] with ΩM k+1 = M k . We are only interested in the p = 2 versions so all spectra should be considered localized at 2. We will suppress the notation and let M = M (2) .Thom, [Tho54, 1954], computed the coefficients MO * and gave the stable homotopy type of the spectrum MO. It is just the product of a lot of mod 2 Eilenberg-MacLane spectra. This gives the complete homotopy type of the MO k as well. From Serre's computation, [Ser53, 1953], of the cohomology of the Eilenberg-MacLane spaces, we also know H * (MO k ) (mod 2). Wall, [Wal60, 1960], showed MSO (at 2) is the stable product of integral and mod 2 Eilenberg-MacLane spectra. Again, this gives the complete homotopy type of the MSO k and, by Serre, the homology of these spaces. Anderson, Brown, and Peterson, [AEBP67, 1967], computed MSpin * and gave the stable homotopy type, and consequently, the unstable homotopy type of MSpin k . Stably, this is copies of the mod 2 Eilenberg-MacLane spectra and connected covers of the spectrum bo. Stong, [Sto63, 1963], computed the cohomology of (most of) the unstable connected covers for bo, giving (most of) H * (MSpin k ). At p = 2, Milnor, [Mil60, 1960], and Novikov, [Nov62, 1962], both computed MU * . With the construction of the Brown-Peterson spectrum, [BP66, 1966], the stable homotopy of MU was described as a product of BP spectra. In 2010 Mathematics Subject Classification. 55N10,55N22,55P15.

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The Homotopy Type of MSU

Cited by 29 publications

References 16 publications

Minimal atomic complexes

Minimal atomic complexes

On bordism rings with principal torsion ideal

The Omega spectrum for Pengelley’s $BoP$

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