# 03/21/2019: Complex Cobordism and Stable Homotopy Groups of Spheres (Douglas C. Ravenel)
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Last Annotation: 03/21/2019
## Highlights
- Now a Hopf algebra, such as B in 1\.3\.6, is a cogroup object in the category of commutative rings R, (Douglas C. Ravenel 40)
- which is to say that Hom\(B, R\) = GR is a group-valued (Douglas C. Ravenel 40)
- functor (Douglas C. Ravenel 41)
- For a p-typical analog of 1\.3\.6 we need to replace b by cogroupoid object in the category of commutative Z\(p\) -algebras K\. Such an object is called a Hopf algebroid \(A1\.1\.1\) and consists of a pair \(A, Γ\) of commutative rings with appropriate structure maps so that Hom\(A, K\) and Hom\(Γ, K\) are the sets of objects and morphisms, respectively, of a groupoid\. (Douglas C. Ravenel 41)
- To get at this question we use the spectrum J, which is the fibre of a certain map bu → Σ2 bu, where bu is the spectrum representing connective complex K-theory, i\.e\., the spectrum obtained by spectrum representing connective complex K-theory, i\.e\., the spectrum obtained by delooping the space Z × BU \. (Douglas C. Ravenel 51)
- spectra and the stable homotopy category as described, for example, in the first few sections of Adams [4]\. (Douglas C. Ravenel 61)
- Recall that H ∗ \(X\) is a module over the mod \(p\) Steenrod algebra A, to be described explicitly in the next chapter (Douglas C. Ravenel 61)
- 2\.1\.10\. Definition\. Given a sequence of spectra and maps f 1 f2 f3 X 0 ←− X1 ←− X2 ←− X3 ←− · · · , (Douglas C. Ravenel 64)
- m Xi , is the fiber of the map ←− Y g: whose ith component is the difference Y Xi ence be Y Xi → between Y Xi en the projection pi : Xj → Xi and the composite (Douglas C. Ravenel 65)
- This lim is not a categorical inverse limit \(Mac Lane [1, Section III\.4] because a ←− compatible collection of maps to the Xi , does not give a unique map to lim Xi \. For compatible collection of maps to the Xi , does not give a unique map to lim Xi \. Fo ←− this reason some authors \(e\.g\., Bousfield and Kan [1]\) denote it instead by holim (Douglas C. Ravenel 65)
- The Adams spectral sequence of 2\.2\.3 is useful for computing π∗ \(X\), i\.e\., [S 0 , X]\. With additional assumptions on E one can generalize to a spectral sequence for computing [W, X]\. (Douglas C. Ravenel 73)
- Throughout this book, P \(x\) will denote a polynomial algebra \(over a field which will be clear from the context\) on one or more generators x, and E\(x\) will denote the exterior algebra on same (Douglas C. Ravenel 79)
- We start by describing the dual Steenrod algebra A∗ (Douglas C. Ravenel 79)
- In this section we will consider four spectra \(M O, M U , bo, and bu\) in which the change-of-rings isomorphism of A1\.1\.18 can be used to great advantage (Douglas C. Ravenel 80)
- H ∗ \(M U ; Z\) = Z[b1 , b2 , \. \. \. ], where bi ∈ H2i \. H/\(p\) denote the mod \(p\) Eilenberg–M (Douglas C. Ravenel 81)
- π ∗ \(M U \) = Z[x1 , x2 , \. \. \. ] with xi ∈ π2i \(M U \)\. (Douglas C. Ravenel 81)
- A theorem of Milnor and Moore [3] says that every graded primitively generated Hopf algebra is isomorphic to the universal enveloping algebra of a restricted Lie algebra (Douglas C. Ravenel 88)
- The lambda algebra Λ is an associative differential bigraded algebra whose cohomology, like that of the cobar complex, is Ext (Douglas C. Ravenel 97)
- \. Its greatest attraction, which will not be exploited here, is that it contains for each n > 0 a subcomplex Λ\(n\) whose cohomology is the E2 -term of a spectral sequence converging to the 2-component of the unstable homotopy groups of S n \. In other words Λ\(n\) is the E 1 -term of an unstable Adams spectral sequence (Douglas C. Ravenel 97)
- 3\.4\.4\. Theorem\. \(a\) \(Browder [1]\)\. For p = 2 h2j is a permanent cycle iff there is a framed manifold of dimension 2j+1 − 2 with Kervaire invariant one\. Such are known to exist for j ≤ 5\. For more discussion see 1\.5\.29 and 1\.5\.35\. (Douglas C. Ravenel 107)
- We do not know how to make this computation directly\. (Douglas C. Ravenel 112)
- We do not know the image of the map in 3\.4\.19 (Douglas C. Ravenel 113)
- n Section 1 we made some easy Ext calculations and thereby computed the homotopy groups of such spectra as M U and bo\. The latter involved the cohomology of A\(1\), the subalgebra of the mod \(2\) Steenrod algebra generated by Sq 1 and Sq 2 \. (Douglas C. Ravenel 114)
- The use of the Adams spectral sequence in computing cobordism rings is becoming more popular\. The spectra M O, M SO, M SU , and M Spin were originally (Douglas C. Ravenel 114)
- analyzed by other methods \(see Stong [1] for references\) but in theory could be analyzed with the Adams spectral sequence (Douglas C. Ravenel 115)
- The spectrum M Oh8i \(the Thom spectrum associated with the 7-connected cover of BO\) has been investigated by Adams spectral sequence methods in Giambalvo [2], Bahri [1], Davis [3, 6], and Bahri and Mahowald [1]\. In Johnson and Wilson [5] the Adams spectral sequence is used to compute the bordism ring of manifolds with free G-action for an elementary abelian p-group G\. (Douglas C. Ravenel 115)
- The most prodigious Adams spectral sequence calculation to date is that for the symplectic cobordism ring by Kochman [1, 2, 3]\. (Douglas C. Ravenel 115)
- In Section 2 we described the May SS\. The work of Nakamura [1] enables one to use algebraic Steenrod operations \(A1\.5\) to compute May differentials\. (Douglas C. Ravenel 115)
- The May SS is obtained from an increasing filtration of the dual Steenrod algebra A∗ \. (Douglas C. Ravenel 115)
- The Adams spectral sequence was used in the proof of the Segal conjecture for Z/\(2\) by Lin [1] and Lin et al\. [2]\. (Douglas C. Ravenel 120)
- Finally, we must mention the Whitehead conjecture\. The n-fold symmetric product Spn \(X\) of a space X is the quotient of the n-fold Cartesian product by the action of the symmetric group Σn \. Dold and Thom [1] showed tha (Douglas C. Ravenel 120)
- Sp ∞ \(X\) = lim Spn \(X\) is a product of Eilenberg–Mac Lane spaces whosw homo←− topy is the homotopy of X\. Symmetric products can be defined on spectra and we have Sp∞ \(S 0 \) = HJ, the integer Eilenbergh–Mac Lane spectrum\. After localizing at the prime p one considers S 0 → Spp \(S 0 \) → Spp 2 \(S 0 \) → · · · and \(3\.5\.16\) H ← S 0 ← Σ−1 Spp \(S 0 \)/S 0 ← Σ−2 Spp 2 \(S 0 \)/Spp \(S 0 \) ← · · · \. Whitehead conjectured that this diagram induces an long exact sequence of homotopy groups\. In particular, the map Σ−1 Spp \(S 0 \)/S 0 → S 0 shouls induce a surjection in homotopy in positive dimensions; this is the famous theorem of Kahn surjection in homotopy in positive dimensions; this is the famous theorem of Kahn and Priddy (Douglas C. Ravenel 121)
- In Section 4 we set up the Adams–Novikov spectral sequence and use it to compute the stable homotopy groups of spheres through a middling range of dimensions, namely ≤ 24 for p = 2 and ≤ 2p3 − 2p − 1 for p > 2\. (Douglas C. Ravenel 123)
- The main results are Quillen’s theorem 4\.1\.6, which identifies π∗ \(M U \) with the Lazard ring L \(A2\.1\.8\); the Landweber–Novikov theorem 4\.1\.11, which describes M U∗ \(M U \); the Brown– Peterson theorem 4\.1\.12, which gives the spectrum BP ; and the Quillen–Adams theorem 4\.1\.19, which describes BP∗ \(BP (Douglas C. Ravenel 123)
- We begin by informally defining the spectrum M U \. For more details see Milnor and Stasheff [5]\. Recall that for each n ≥ 0 the group of complex unitary n × n matrices U \(n\) has a classifying space BU \(n\)\. It has a complex n-plane bundle γn over it which is universal in the sense that any such bundle ξ over a paracompact space X is the pullback of γn , induced by a map f : X → BU \(n\)\. Isomorphism classes of such bundles ξ are in one-to-one correspondence with homotopy classes of classes of such bundles ξ are in one-to-one correspondence with homotopy classes of maps from X to BU \(n\)\. Any Cn -bundle ξ has an associated disc bundle D\(ξ\) and sphere bundle S\(ξ\)\. The Thom space T \(ξ\) is the quotient D\(ξ\)/S\(ξ\)\. Alternatively, for compact X, T \(ξ\) is the one-point compactification of the total space of ξ\. (Douglas C. Ravenel 123)
- M U \(n\) is T \(γn \), the Thom space of the universal n-plane bundle γn over BU \(n\)\. (Douglas C. Ravenel 123)
- It follows from the celebrated theorem of Thom [1] that π∗ \(M U \) is isomorphic to the complex cobordisrn ring \(see Milnor [4]\) which is defined as follow (Douglas C. Ravenel 124)
- A stably complex manifold is one with a complex structure on its stable normal bundle\. \(This notion of a complex manifold is weaker than others, e\.g\., algebraic, analytic, and almost complex\.\) All such manifolds are oriented\. Two closed stably complex manifolds M1 and M2 are cobordant if there is a stably complex manifold W whose boundary is the disjoint union of M1 \(with the opposite of the given orientation\) and M2 \. (Douglas C. Ravenel 124)
- Milnor and Novikov’s calculation of π∗ \(M U \) \(3\.1\.5\) implies that two such manifolds are cobordant if they have the same Chern numbers\. (Douglas C. Ravenel 124)
- On the other hand, the connection with formal group laws \(A2\.1\.1\) discovered by Quillen [2] \(see 4\.1\.6\) is essential to all that follows\. This leads one to suspect that there is some unknown formal group theoretic construction of M U or its associated infinite loop space\. For example, many well-known infinite loop spaces have been constructed as classifying spaces of certain types of categories \(see Adams [9], section 2\.6\), but to our knowledge no such description exists for M U \. (Douglas C. Ravenel 124)
- 4\.1\.1\. Definition\. Let E be an associative commutative ring spectrum\. A complex orientation for E is a class 1 E\(CP \) xE ∈ E e ≃E e 2 \(S e 2 \(CP ∞ \) whose restriction to E = π0 \(E\) is 1, where CP e E\(CP 1 \)≃Ee 2 \(S 2 \) ∼ = π0 \(E\) n denotes n-dimensional complex projective space (Douglas C. Ravenel 124)
- Recall that M U is built up out of Thom spaces M U \(n\) of complex vector bundles over BU \(n\) and that the map BU \(n\) → M U \(n\) is an equivalence when n = 1\. The composition is an equivalence when n = (Douglas C. Ravenel 124)
- Alternatively, xM U could be defined to be the first Conner–Floyd Chern class of the canonical complex line bundle over CP ∞ (Douglas C. Ravenel 125)
- Hence a complex orientation xE leads to a formal group law FE over E ∗ \(pt\.\)\. (Douglas C. Ravenel 125)
- The spectrum BP is named after Brown and Peterson, who first constructed it via its Postnikov towe (Douglas C. Ravenel 128)
- Brown and Peterson [1] also showed that BP can be constructed from H \(the integral Eilenberg–Mac Lane spectrum\) by killing all of the torsion in its integral homology with Postnikov fibration (Douglas C. Ravenel 129)
- More recently, 0 Priddy [1] has shown that BP can be constructed from S\(p\) by adding local cells to kill off all of the torsion in its homotopy\. (Douglas C. Ravenel 129)
- BP bears the same relation to p-typical formal group laws \(A2\.1\.17\) that M U bears to formal group laws as seen in 4\.1\.6 (Douglas C. Ravenel 129)
- Cartier’s theorem A2\.1\.18, which states that any formal group law over a Z\(p\) -algebra is canonically isomorphic to p-typical one\. (Douglas C. Ravenel 129)