e–one, two, three inﬁnity - university of notre damewgd/drafts/haynesslides.pdfambushes before...

E–One, Two, Three . . . Infinity†

A tour through part of the mathematical world ofHaynes R. Miller

June 25, 2008, Bonn

† Sorry, George

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Charge to the speaker

Remain unstable!

Charge to the speaker

Remain unstable!

Nothing chromatic!

A brief review of spectral sequencesThings to watch out for in what’s coming

Spectral sequence: a machine for calculation, which can. . .

vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out

All this drama (and more) takes place in papers by Haynes.



vanish

collapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out




vanishcollapse

spawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out




vanishcollapsespawn an heir

suffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out




vanishcollapsespawn an heirsuffer a localization

impersonate a rivaldivulge a (partial) secretwork absolutely everything out




vanishcollapsespawn an heirsuffer a localizationimpersonate a rival

divulge a (partial) secretwork absolutely everything out




vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secret

work absolutely everything out




vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out


The unstable Ad Spec Seq for generalized homologyBendersky, Curtis, Miller

X a space; R = MU, BP, connective ring spectrum

CX = Ω∞(R ∧ X ) X // CX//// C2X · · ·oo

E2¬ab(X ) =⇒ π∗X

E2-page: spawned by another SS (R, X nice)

Untori Primj R∗X =⇒ E2¬ab(X ) =⇒ π∗X

Partial secret revealed for R = BP1-line for X = odd sphere

On relations between Adams spectral sequenceswith an application to the stable homotopy of the Moore space

Fancy footwork for Moore spectrum M

E2BP(E2

Z/pM) Mahowald +3

May

E2Z/pM

Adams

E2BPM

Novikov+3 π∗M

Algebra in (May) gives topology in (Adams).E3[1/φ] = E∞[1/φ]

π∗M[1/φ] & LK (1)M

A localization theorem in homological algebraAlgebraic background for last slide

A = S⊗E , S =coalgebra, E =exterior Hopf algebra

CotorE(k , k) = k [q]

CotorA(k , M) = module over k [q]

Compute q−1 CotorA(k , M)

q−1E2M =

Eroded by dMay2 = dAdams

2

Unstable credentials: generalized by Hess & Levi (2007)

C∗Ω(homotopy fibre)

Spec seq for homology of an infinite delooping

X an ∞-loop space

QX = Ω∞Σ∞X X QXoo // Q2X · · ·oooo

E2¬ab(X ) =⇒ H∗B∞X

E2-page: spawned by another SS

Untori Indecj H∗X =⇒ E2¬ab(X ) =⇒ H∗B∞X

Indecj = 0, j 6= 0, 1“Λ-algebra” complex for Untor∗

Collapse and work absolutely everything out X = ZDyer-Lashof =⇒ Steenrod

The Segal conjecture for elementary abelian p-groupsAdams, Gunawardena, Miller

Segal conjecture G finite

S0 ⇐⇒∐

BΣn

Map(BG, S0)?⇐⇒ Map(BG,

∐BΣn)

Map(BG, S0)?∼ ∨O Σ∞B Aut(O)

Reductions1 Enough: G a p-group (May-McClure)2 Enough: G elementary abelian, equivariantly (Carlsson)3 G elem abel: Ext-calculation (AGM) + glue (May-Priddy)

Ext-calculation V = (Z/p)n, S = β(H1)

E2( S−1H∗BV ) ∼= p(p−1)2 E2( Σ−nZ/p )

Unstable!ttiiiiii

BG

The Sullivan conj. on maps from classifying spaces

Sullivan conjecture G finite, X fin. dim.

Map(BG, X )?∼ X

Enough: G = Z/p, π1X = ∗

X 1-connected, V = Z/p

CX = Ω∞(HZ/p ∧ X ) X // CX//// C2X · · ·oo

E2¬ab(X ) =⇒ π∗ Map(BV , X )

E2-page: spawned by another SS H = H∗BV

Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map

Sullivan conjecture: proof


X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishes

Dualize, use homotopy theory of simpl. commutative algebrassuspensionreverse unstable Adams Spectral Sequence

No maps (finite dim’l)→ H∗BZ/p

Unexti vanishes for i > 0

Dualize, prove H∗BZ/p is injective as an unstable moduleretract of limit of dual Brown-Gitler modules



X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishesDualize, use homotopy theory of simpl. commutative algebras

suspensionreverse unstable Adams Spectral Sequence


Unexti vanishes for i > 0

Dualize, prove H∗BZ/p is injective as an unstable moduleretract of limit of dual Brown-Gitler modules



X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishes

Dualize, use homotopy theory of simpl. commutative algebrassuspensionreverse unstable Adams Spectral Sequence


Unexti vanishes for i > 0Dualize, prove H∗BZ/p is injective as an unstable module

retract of limit of dual Brown-Gitler modules



X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishesDualize, use homotopy theory of simpl. commutative algebras

suspensionreverse unstable Adams Spectral Sequence


Unexti vanishes for i > 0Dualize, prove H∗BZ/p is injective as an unstable module

retract of limit of dual Brown-Gitler modules

Generalized Sullivan conjecture

Homotopy fixed points G acts on X

X G = MapG( ∗ , X )

X hG = MapG(EG, X ) = EG ×G X // BGvv R_l

Sullivan fixed point conjecture X finite, G = Z/p

X hG ?∼p X G

Trivial action ⇐⇒ Sullivan conjecture

Theorem: OK mod π1 subtlety

E2 over Ap⊗H∗B =⇒ π∗( E // Bww Q_m

)

SS for π∗X hG impersonates SS for π∗X G

And more, and more, and more

An Incomplete List

1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds


An Incomplete List

1 Vanishing lines (Miller-Wilkerson)

2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds


An Incomplete List

1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)

3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds


An Incomplete List

1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)

4 Double suspension (Harper-Miller)5 Stiefel manifolds


An Incomplete List

1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)

5 Stiefel manifolds


An Incomplete List

1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds

Quote I for the dayfrom Massey-Peterson towers and maps from classifying spaces

. . .the Massey-Peterson theory should beregarded as a piece of light artillery, with whichone can move quickly and execute smallambushes before wheeling in 〈heavier guns〉.

Quote II for the dayfrom The Sullivan conjecture on maps from classifying spaces

In summary, the proof proceeds by constructinga long chain of spectral sequences and thenshowing that, miraculously, the initial term of theinitial spectral sequence is trivial.

Just another example of. . .

e–one, two, three inﬁnity - university of notre damewgd/drafts/haynesslides.pdfambushes before...

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