Thom Spectra Research Articles

Bordism groups and shape theory

The THOM isomorphism Theorem (1) allows an immediate extension to the strong shape category of compacta: The shape homology TBG n (X) with coefficient in a THOM spectrum turns out to be isomorphic to the bordism group Ω G n(X) which is defined like Ω G n (X) but with strong shape morphisms replacing continuous mappings (Theorem 1.2).

Equivariant RO(G)-graded bordism theories

Let G be a finite group. The RO( G)-graded bordism theories of Pulikowski [7] and Kosniowski [3] are studied. Representing equivariant Thom spectra are constructed, and the relevant transversality results proved. New methods for splitting away from the order of G are described, and behavior in the presence of a gap hypothesis is examined.

An algebraic filtration of H*(MO;ℤ2)

Let 2* denote the dual of the mod two Steenrod algebra. In [5] an algebraic filtration B*(n) of H*(BO; ℤ2) was constructed such that each B*(n) is a bipolynomial sub Hopf algebra and sub 2*-comodule of H*(BO; ℤ2). In Lemma 3.1 we prove that the Thom isomorphism determines a corresponding filtration of H*(MO;ℤ2) by polynomial subalgebras and sub 2*-comodules M*(n). Let (n) denote the subalgebra of 2 generated by Sq2k, 0 ≦ k < n, and let *(n) be its dual, a quotient Hopf algebra of 2*. In Section 3 we construct a polynomial algebra and *(n)-comodule R(n) such that M*(n)≃2*□*(n)R(n) as algebras and 2*-comodules. Here □ denotes the cotensor product defined in [9, §2]. Dually it will follow that M*(n) has a sub (n)-module and subcoalgebra T(n) such that M*(n)≃2⊗n)T(n) as coalgebras and 2-modules. We also show that M*(n) can not be realised as the homology of a spectrum for n≧4. Of course M*(0)=H*(MO;ℤ2), M*(1)=H*(MSO;ℤ2), M*(2)=H* (MSpin;ℤ2) and M*(3)=H*(MO<8>;ℤ2). Moreover, it follows from [4; Thm. 2.10, Cor. 2.11] that M*(n)=Images[H*(MO<ϕ(n)>;ℤ2)→H*(MO;ℤ2)] and M*(n) ≃ Image [H*(MO;ℤ2)→ H*(MO<ϕ(n)>;ℤ2)]. Here MO<k> id the Thom spectrum of BO<k>, the (k−1)-connected covering of BO, and ϕ(n)=8s + 2t where n = 4s + t, 0≦t≦3. In Section 4 we sketch the odd primary analogue—a filtration pM*(n) of H*(MUp, 0;ℤp) for p an odd prime. MUp, 0 is the Thom spectrum of the (2p-3)-connected factor of the Adams splitting [2] of BU(p).

The splitting of 𝐵𝑂⟨8⟩∧𝑏𝑜 and 𝑀𝑂⟨8⟩∧𝑏𝑜

Let B O ⟨ 8 ⟩ BO\left \langle 8 \right \rangle denote the classifying space for vector bundles trivial on the 7 7 -skeleton, and M O ⟨ 8 ⟩ MO\left \langle 8 \right \rangle the associated Thom spectrum. It is proved that, localized at 2 2 , B O ⟨ 8 ⟩ ∧ b o BO\left \langle 8 \right \rangle \wedge \,bo and M O ⟨ 8 ⟩ ∧ b o MO\left \langle 8 \right \rangle \wedge \,bo split as a wedge of familiar spectra closely related to b o bo , where b o bo is the spectrum for connective K O KO -theory.

The Homotopy Type of MSU

0. Introduction. This paper examines the homotopy type of the Thom spectrum MSU associated with special unitary cobordism. For odd primes p, standard methods show that the p-localization MSU(p) is equivalent to a wedge of suspensions of the Brown-Peterson spectrum BP. Forp = 2, however, this is not the case, and our work is devoted to determining the 2-primary homotopy type of MSU. This involves a new indecomposable spectrum, and our main results are the following. There is an indecomposable 2-local spectrum, which we call BoP, such that MSU(2) is equivalent to a wedge of suspensions of BoP and BP. Under the equivalence, the Thom class lies in a BoP summand. As a comodule over the dual Steenrod algebra A [11], H*(BoP; Z/2) is a sum of 4 '2 2 suspensions of B = Z/2[?1, t2, ..., j, . ... C A, where tj is the conjugate of Milnor's generator (j. There is one suspension of B beginning in each nonnegative dimension divisible by 8. BoP bears strong similarities to BP and the (1)-connected K-theory spectra bo and bu. In particular, in Section 6 we show there is a map BoP bo(2) inducing an epimorphism v* of homotopy groups. In fact, v* induces an isomorphism of torsion subgroups, and its torsion free kernel is concentrated in even dimensions. A brief summary of our methods is as follows. In Sections 1 and 2 we describe the Adams spectral sequence for 7r*MSU(2), including a computation of the differentials, with particular attention paid to the product structure. Anderson, Brown, and Peterson [4] gave a computation for these differentials, but their proof requires some correction, and in any case we will need the more extensive knowledge of the product structure. In Sections 3 to 5, we construct BoP and show it is indecomposable. To produce BoP, first the Sullivan-Baas construction is applied to MSU, yielding a spectrum representing a bordism theory of SU-manifolds with

$K(Z,O)$ and $K(Z_{2},O)$ as Thom spectra

$K(Z,O)$ and $K(Z_{2},O)$ as Thom spectra

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

K(Z/2) as a Thom Spectrum

Dyer-Lashof operations are used to give a simple proof that K(Z/2) is a Thom spectrum. The purpose of this note is to give a simple proof of Mahowald's striking observation that the Eilenberg-Mac Lane spectrum K(Z/2) is a Thom spectrum. Our proof follows closely a proposed (but incorrect) proof by Madsen and Milgram [2]. Although Mahowald has given another proof [3], the idea of [2] is so appealing that it seems useful to record a corrected version. Let 'q: S1 BO represent the generator of T1 BO = Z/2. Since BO is a double loop space there is an induced map 2Q2S I = Q222(s I n Q2222BO-BO Let M(y) denote the Thom spectrum associated with y (localized at 2). THEOREM (MAHOWALD). M(y)K(Z/2). PROOF. We recall that the Dyer-Lashof operation Q1 is defined for double loop spaces and that H*22S3 = Z/2[xl, x2, ... , Xk,...] where xk = Q, * * * Qltl the (k 1) fold iterate of Q1 applied to the fundamental class 1 (Z/2 coefficients are used throughout). Also H*K(Z/2) = Z/2[t1, (2' * * * I (k* ... ] = A* the dual of the mod 2 Steenrod algebra A. Let a: A -* H*MO denote evaluation on the Thom class U, i.e. a(a) = aU, al* is an algebra morphism in homology. Finally, let I': M(y) -> MO denote the map of Thom spectra induced by y; r'* is also an algebra morphism in homology. Since deg xk = 2k 1 = deg (k, H*22S3 H*M(y) and A* have the same rank in each dimension and so the theorem follows from LEMMA. * a H*M(y) H*MO A* is surjective. PROOF. Consider the commutative diagram Received by the editors August 1, 1977. AMS (MOS) subject classifications (1970). Primary 55B20; Secondary 57D75. 'Supported in part by NSF Grant MPS76-07051. ?D American Mathematical Society 1978

𝐾(𝑍/2) as a Thom spectrum

Dyer-Lashof operations are used to give a simple proof that K ( Z / 2 ) K({\mathbf {Z}}/2) is a Thom spectrum.

Generalized Thom spectra and transversality for spherical fibrations

Generalized Thom spectra and transversality for spherical fibrations

On Complex-Spin Manifolds

BU BSO in which w is a fibration, the maps in question being given by classification of the universal bundles. Let B denote the total space of the induced fibration over BU. Then BY is a classifying space for stable bundles having both complex and Spin structures. Thus a complex-spin manifold M is a manifold together with an equivalence class of liftings to BE of the classifying map for the stable normal bundle of M. By applying the Thom construction [8], one sees immediately that the cobordism groups of complex-spin manifolds, SC&S, are isomorphic to the stable homotopy groups, r*(Ml), of the Thom spectrum associated to B,. Corresponding to the diagram of classifying spaces

Notes on Homotopy of Thom Spectra

Notes on Homotopy of Thom Spectra