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Products in the Adams Spectral Sequence 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 1-chain complexes and 1-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 2 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 2 Hjaj+jbj+jc|−1(X ) where dx = bc and dy = ab: Colin Aitken Products in the Adams Spectral Sequence 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 2 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 2 Hjaj+jbj+jc|−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 jai |−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 jai |−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 = 3 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 = 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 jhi j = 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 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 jhi j = 2 . 2 Ext has basis elements hi hj , except that hi hi+1 = 0: 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.
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