Products in the Adams Spectral Sequence

Colin Aitken

May 20, 2021

Colin Aitken Products in the Adams Spectral Sequence Outline

(Note: adding in all the signs made the formulas unreadable, so the equations I write down will only be true in characteristic 2.) What are Toda Brackets and Massey Products? The Adams Spectral Sequence Gugenheim-May ∞-chain complexes and ∞-Gugenheim-May

Colin Aitken Products in the Adams Spectral Sequence Prologue: a universal property of suspensions

Proposition (Ancient History) The space of maps ΣX → Y is equivalent to the space of maps X → Y with two chosen nullhomotopies.

Colin Aitken Products in the Adams Spectral Sequence Prologue: a universal property of suspensions

Proposition (Ancient History) The space of maps ΣX → Y is equivalent to the space of maps X → Y with two chosen nullhomotopies.

Colin Aitken Products in the Adams Spectral Sequence 1 hg is nullhomotopic, so we have a homotopy

hgf = (hg)f → (0)f = 0

2 gf is nullhomotopic, so we have a homotopy

hgf = h(gf ) → h(0) = 0 By the previous slide, we obtain a map ΣX → W .

Consequence: Toda Brackets

Suppose we have maps

g X −→f Y −→ Z −→h W

such that gf and hg are both nullhomotopic. Idea (Toda) The composition hgf : X → W is nullhomotopic for two reasons:

Colin Aitken Products in the Adams Spectral Sequence Consequence: Toda Brackets

Suppose we have maps

g X −→f Y −→ Z −→h W

such that gf and hg are both nullhomotopic. Idea (Toda) The composition hgf : X → W is nullhomotopic for two reasons: 1 hg is nullhomotopic, so we have a homotopy

hgf = (hg)f → (0)f = 0

2 gf is nullhomotopic, so we have a homotopy

hgf = h(gf ) → h(0) = 0 By the previous slide, we obtain a map ΣX → W .

Colin Aitken Products in the Adams Spectral Sequence Toda Brackets

In real life, this is slightly more complicated: we assumed that hg is nullhomotopic, but the same function can have more than one nullhomotopy. Deﬁnition g Let X −→f Y −→ Z −→h W be such that each composition is nullhomotopic. The Toda Bracket hf , g, hi is the set of all maps ΣX → W coming from all possible nullhomotopies of hg and gf . If X is a spectrum, the Toda Bracket will be a coset of the indeterminacy subgroup Hom(ΣY , W )f + h Hom(ΣX , Z). Proposition If a paper using Toda brackets contains a mistake, it almost certainly involves overlooked indeterminacies.

Colin Aitken Products in the Adams Spectral Sequence Exercise Check that d(ax + yc) = 0, so that ax + yc does in fact give a homology class.

Massey Products

The same structure appears when we consider the homology of differential graded algebras (“chain complexes where you can multiply the elements”). We will work over F2 to avoid signs. Definition Let X be a differential graded F2-algebra, and choose a, b, c ∈ H∗(X ) such that ab = bc = 0 in H∗(X ). The Massey product ha, b, ci is the set of all possible values of

ax + yc ∈ H|a|+|b|+|c|−1(X )

where dx = bc and dy = ab.

Colin Aitken Products in the Adams Spectral Sequence Massey Products

ax + yc ∈ H|a|+|b|+|c|−1(X )

where dx = bc and dy = ab.

Exercise Check that d(ax + yc) = 0, so that ax + yc does in fact give a homology class.

Colin Aitken Products in the Adams Spectral Sequence Why should I care about Massey products?

One of the hardest parts of working with spectral sequences is computing diﬀerentials, which often reduces to a bunch of ad hoc methods thrown together. We can reduce the work if there is a multiplicative structure: Theorem Many spectral sequences satisfy the Leibniz Rule

dr (ab) = adr (b) + dr (a)b. Many times we can’t write a given element as a product of two smaller elements, but we can get lucky and write it as a Massey product: Theorem In many spectral sequences (under good conditions)

dr ha, b, ci ⊂ hdr (a), b, ci + ha, dr (b), ci + ha, b, dr (c)i.

Colin Aitken Products in the Adams Spectral Sequence Idea (Matric Massey Products) We can also deﬁne massey products not just of elements, but of matrices of elements! The spectral sequence idea from before still applies to this type of Massey product (but with even more indeterminacy horror).

Generalizations: Higher and Matric

There are two ways to generalize the ideas we’ve built up: Idea (Higher Massey Products)

Suppose we have a1, ··· , an in H∗(X ) such that:

1 ai ai+1 = 0 in H∗(X ) for all i.

2 hai , ai+1, ai+2i contains 0 for all i. 3 (etc.)

We can deﬁne ha1, a2, ··· , ani ⊂ HΣi |ai |−n+2(X ).

Colin Aitken Products in the Adams Spectral Sequence Generalizations: Higher and Matric

There are two ways to generalize the ideas we’ve built up: Idea (Higher Massey Products)

Suppose we have a1, ··· , an in H∗(X ) such that:

1 ai ai+1 = 0 in H∗(X ) for all i.

2 hai , ai+1, ai+2i contains 0 for all i. 3 (etc.)

We can deﬁne ha1, a2, ··· , ani ⊂ HΣi |ai |−n+2(X ).

Idea (Matric Massey Products) We can also deﬁne massey products not just of elements, but of matrices of elements! The spectral sequence idea from before still applies to this type of Massey product (but with even more indeterminacy horror).

Colin Aitken Products in the Adams Spectral Sequence The Ext group looks scary, but with modern technology it’s reasonably computable – known up through t − s = 210 or so. The differentials come from higher cohomology operations, which are not too hard to define but absolutely miserable to compute. Upshot: differentials are hard and there are a whole bunch of them.

The Adams Spectral Sequence

Let’s look more closely at a speciﬁc spectral sequence. Theorem (Adams) There is a spectral sequence converging to the (p-completion of the) stable homotopy groups of spheres, with:

s,t ∧ E2 = ExtAp (Fp, Fp) ⇒ πt−s (Sp )

Colin Aitken Products in the Adams Spectral Sequence The Adams Spectral Sequence

Let’s look more closely at a speciﬁc spectral sequence. Theorem (Adams) There is a spectral sequence converging to the (p-completion of the) stable homotopy groups of spheres, with:

s,t ∧ E2 = ExtAp (Fp, Fp) ⇒ πt−s (Sp )

The Ext group looks scary, but with modern technology it’s reasonably computable – known up through t − s = 210 or so. The differentials come from higher cohomology operations, which are not too hard to define but absolutely miserable to compute. Upshot: differentials are hard and there are a whole bunch of them.

Colin Aitken Products in the Adams Spectral Sequence Adams Spectral Sequence for p = 3

Colin Aitken Products in the Adams Spectral Sequence Adams Spectral Sequence for p = 2

Colin Aitken Products in the Adams Spectral Sequence Counterexample 3,11 There is an element c0 in E2 which is not a product of copies of hi .

Adams Spectral Sequence for p = 2

This is very complex. But the ﬁrst couple of values of s (the y-axis) are understandable: Exercise (hard!) s,• s Compute the values of E2 = ExtA(F2, F2) for small s: 0 Ext has a single copy of F2 in degree 0. 1 Ext has basis elements h0, h1, h2, h3, ··· , which live in degree i |hi | = 2 . 2 Ext has basis elements hi hj , except that hi hi+1 = 0.

Do these elements generate the whole Adams E2 page?

Colin Aitken Products in the Adams Spectral Sequence Adams Spectral Sequence for p = 2

Do these elements generate the whole Adams E2 page? Counterexample 3,11 There is an element c0 in E2 which is not a product of copies of hi .

Colin Aitken Products in the Adams Spectral Sequence Adams Spectral Sequence for p = 2

Colin Aitken Products in the Adams Spectral Sequence Generation by Massey Products

However, c0 can be generated from h0, h1, and h2 if we allow ourselves to use Massey Products. Theorem

In the E2 page of the Adams Spectral Sequence, we have

2 c0 = hh0, h2, h1i If we didn’t already know the answer, we could compute:

2 2 2 d2(c0) ⊆ hd2(h0), h2, h1i + hh0, d2(h2), h1i + hh0, h2, d2(h1)i

but the right hand side is zero, so the left hand side must be too! (See e.g. “Motivic Stable Homotopy and the Stable 51 and 52 Stems" by Isaksen and Xu for a real application of this method.)

Colin Aitken Products in the Adams Spectral Sequence Proof. ∗ Let U be the differential graded algebra ExtA(k, k), and let IU be the fiber of the augmentation map U → k. Observe: L The decomposables are in the image of IU ⊗U IU → IU. L By a fiber sequence, this is the kernel of IU → IU ⊗U k. Using a quick factorization, we see that every element in the L kernel of the map ΣIU → k ⊗U k is decomposable.

Gugenheim-May

If we generalize “products” to include matric Massey products, then the hi do generate the whole E2 page. Theorem (Gugenheim-May) Let k be a ﬁeld, and let A be a k-algebra. Then, every element of i ExtA(k, k) with i > 1 is decomposable by matric Massey products.

Colin Aitken Products in the Adams Spectral Sequence Gugenheim-May

Proof. ∗ Let U be the differential graded algebra ExtA(k, k), and let IU be the fiber of the augmentation map U → k. Observe: L The decomposables are in the image of IU ⊗U IU → IU. L By a fiber sequence, this is the kernel of IU → IU ⊗U k. Using a quick factorization, we see that every element in the L kernel of the map ΣIU → k ⊗U k is decomposable.

Colin Aitken Products in the Adams Spectral Sequence Gugenheim-May

Proof (continued).

Letting B• denote the bar construction B(A, A, k). Realize this as a map of bar constructions:

σ0 : k/ Hom (B , k) → Hom (B ⊗ A, k) ⊗L k. A • A • HomA(B•,k) The right hand side simpliﬁes to

(Hom (B , k) ⊗ A∗) ⊗L k. A • HomA(B•,k) which is of course just A∗, living entirely in ﬁltration zero! Therefore the irreducibles must live in ﬁltration zero of ΣIU, which 1 corresponds to ExtA(k, k)! Question: can we generalize this theorem? We need a more expansive version of Ext.

Colin Aitken Products in the Adams Spectral Sequence A few facts: There is a notion of a “derived ∞-category” Fun⊕(C, Sp), and it is a localization of Ch(C). If C is stable, equivalent data to a simplicial object of C. There is a notion of homology H∗(X ), and a spectral sequence 2 Ep,q = Hp(πq(X•)) ⇒ Hp+q(X ).

Chain Complexes with values in an ∞-category

Deﬁnition Let C be an additive (∞, 1) category. A chain complex in C is:

··· / 0 / 0 / 0 / 0 / 0 > > > >

··· / X4 / X3 / X2 / X1 / X0

where each Xi is in C.

Colin Aitken Products in the Adams Spectral Sequence Chain Complexes with values in an ∞-category

Deﬁnition Let C be an additive (∞, 1) category. A chain complex in C is:

··· / 0 / 0 / 0 / 0 / 0 > > > >

··· / X4 / X3 / X2 / X1 / X0

where each Xi is in C. A few facts: There is a notion of a “derived ∞-category” Fun⊕(C, Sp), and it is a localization of Ch(C). If C is stable, equivalent data to a simplicial object of C. There is a notion of homology H∗(X ), and a spectral sequence 2 Ep,q = Hp(πq(X•)) ⇒ Hp+q(X ).

Colin Aitken Products in the Adams Spectral Sequence Proof: Gugenheim-May, but put “∞-” in front of all the words. Remark This only uses the symmetric monoidal stable ∞-category structure and the compactness of the sphere spectrum, so similar theorems hold equivariantly, motivically, etc.

∞-Gugenheim-May

Theorem (C = Sp, for simplicity)

Let k be a ﬁeld spectrum (e.g. HFp or a Morava K-theory), and let A be an associative k-algebra spectrum. If the conditionally convergent spectral sequence

Extπ∗A(π∗k, π∗k) ⇒ π∗(HomA(k, k)).

converges, then π∗(HomA(k, k)) is generated under matric Toda brackets by elements coming from Ext0 and Ext1 .

Colin Aitken Products in the Adams Spectral Sequence ∞-Gugenheim-May

Theorem (C = Sp, for simplicity)

Let k be a ﬁeld spectrum (e.g. HFp or a Morava K-theory), and let A be an associative k-algebra spectrum. If the conditionally convergent spectral sequence

Extπ∗A(π∗k, π∗k) ⇒ π∗(HomA(k, k)).

converges, then π∗(HomA(k, k)) is generated under matric Toda brackets by elements coming from Ext0 and Ext1 . Proof: Gugenheim-May, but put “∞-” in front of all the words. Remark This only uses the symmetric monoidal stable ∞-category structure and the compactness of the sphere spectrum, so similar theorems hold equivariantly, motivically, etc.

Colin Aitken Products in the Adams Spectral Sequence Theorem

The Er page of the Adams spectral sequence is generated by the s = 0 and s = 1 lines by “matric Toda brackets”. (What does this mean in ordinary language???) Proof: as above, but restrict to a “quotient” ∞-category where all m maps HFp → Σ HFp are equivalent for m > r − 2.

Two special cases

Theorem (Slightly weaker form of Cohen’s theorem) ∧ Every element of π∗(Sp ) is generated (under matric Toda brackets) by Hopf maps and multiples of the identity.

Proof: set k = HFp and A = Hom(HFp, HFp).

Colin Aitken Products in the Adams Spectral Sequence Two special cases

Theorem (Slightly weaker form of Cohen’s theorem) ∧ Every element of π∗(Sp ) is generated (under matric Toda brackets) by Hopf maps and multiples of the identity.

Proof: set k = HFp and A = Hom(HFp, HFp). Theorem

Colin Aitken Products in the Adams Spectral Sequence Question Whatever these matric Toda brackets are, can we relate their values on the Er page to those on the Er+1 page to get a reﬁnement of Moss’s convergence theorem?

Question The generating hypothesis for a ring R states that, for perfect complexes of R-modules X and Y ,

HomD(R)(X , Y ) = HomR (H∗(X ), H∗(Y ))

and is typically false. The equivalent question for ﬁnite spectra is open and hard. Are there any tractable n-categorical versions?

Questions You Can Ask Me in Three Months

Question

Can we say that the “matric Toda brackets” of the Er page reduce to matric Massey products? (when r=2, yes!)

Colin Aitken Products in the Adams Spectral Sequence Question The generating hypothesis for a ring R states that, for perfect complexes of R-modules X and Y ,

HomD(R)(X , Y ) = HomR (H∗(X ), H∗(Y ))

and is typically false. The equivalent question for ﬁnite spectra is open and hard. Are there any tractable n-categorical versions?

Questions You Can Ask Me in Three Months

Question

Can we say that the “matric Toda brackets” of the Er page reduce to matric Massey products? (when r=2, yes!)

Question Whatever these matric Toda brackets are, can we relate their values on the Er page to those on the Er+1 page to get a reﬁnement of Moss’s convergence theorem?

Colin Aitken Products in the Adams Spectral Sequence Questions You Can Ask Me in Three Months

Question

Can we say that the “matric Toda brackets” of the Er page reduce to matric Massey products? (when r=2, yes!)

Question Whatever these matric Toda brackets are, can we relate their values on the Er page to those on the Er+1 page to get a reﬁnement of Moss’s convergence theorem?

Question The generating hypothesis for a ring R states that, for perfect complexes of R-modules X and Y ,

HomD(R)(X , Y ) = HomR (H∗(X ), H∗(Y ))

and is typically false. The equivalent question for ﬁnite spectra is open and hard. Are there any tractable n-categorical versions? Colin Aitken Products in the Adams Spectral Sequence