ICMS

The Kervaire invariant and stable homotopy theory

Apr 25, 2011 - Apr 29, 2011

15 South College Street, Edinburgh EH8 9AA

Organisers

Name	Institution
Greenlees, John	University of Sheffield
Hopkins, Mike	Harvard University
Ranicki, Andrew	University of Edinburgh
Strickland, Neil	University of Sheffield

Michel Kervaire (1927-2007)

At the 2009 Atiyah80 ICMS Workshop Hill, Hopkins and Ravenel announced a proof that there do not exist manifolds with Kervaire invariant 1 in dimensions 2^k-2 for any k≥8. This solved the long standing open Kervaire invariant problem (except for dimension 126, the only dimension which now remains open), which has been of central importance in homotopy theory for over 40 years.

The principal objective of the workshop is to disseminate the methods used in the solution of the Kervaire invariant conjecture, and the use of equivariant and motivic methods more generally. It is also hoped to assess the prospects for using the methods to prove structural results about stable homotopy, and to report on progress which will have taken place by the time of the meeting.

Doug Ravenel has a website dedicated to the Kervaire invariant problem, with further information about the 2009 announcement and subsequent developments. 

Invited speakers:
P.Akhmetev (Izmiran, Troitsk)
M.Atiyah   (Edinburgh)
R.Bruner   (Wayne State)
M.Hill     (Virginia)
M.Hopkins  (Harvard)
J.Jones    (Warwick)
J.Lannes   (Ecole Polytechnique)
T.Lawson   (Minnesota)
M.Mahowald (Northwestern)
J.Morava   (Johns Hopkins)
E.Pedersen (Copenhagen)
D.Ravenel  (Rochester)
S.Schwede  (Bonn)

Arrangements

Participation
All places have been allocated.

Venue
The workshop will take place in the Newhaven lecture theatre at ICMS, 15 South College Street, Edinburgh, Scotland.

Lecture theatre facilities
The lecture theatre is fitted with data projection, visualiser (the new generation of over-head), smart-board and blackboard. Speakers may wish to note that since the lectures will be filmed, it is inadvisable to use blackboards as presentation medium simply because they don't come out clear on film. Please use alternative methods. Please also note that blackboard and data projection cannot be used simultaneously. More information about this is available form Helene Frossling.

Accommodation
Accommodation has been booked for all delegates who are not based in Edinburgh. Participants who wish to make use of this booking must inform Helene Frossling of their dates of travel. Accommodation booked by ICMS will be paid for by the workshop grants, apart from any costs incurred by changing bookings from single to double occupancy. DELEGATES WILL BE INFORMED OF THE NAME AND ADDRESS OF THEIR ALLOCATED ACCOMMODATION SOMETIME BETWEEN 7 AND 12 APRIL.

Posters
All participants are welcome to present posters during the workshop. Poster boards will be available throughout the week in the Chapterhouse where all catering will be served. It will therefore be easy to display posters as well as provide feedback on other people's posters throughout the week. Please note that the poster boards are big enough for sizes A0 PORTRAIT or A1 LANDSCAPE. Velcro is the only option for hanging the posters but there is plenty on site. 

Visas
Attendees may need to apply for a visa to enter the UK. This government website gives more information. 

Conference Co-ordinator
Helene Frossling is the conference co-ordinator for this workshop.

Programme

Programme published 1 April 2011.

 

Monday 25 April

09.00 - 09.55	Registration and coffee/tea
09.55 - 10.00	Welcome & Health and Safety information
10.00 - 11.00	Michael Atiyah (University of Edinburgh) The magic square and the Kervaire invariant
11.00 - 11.45	Coffee and tea served in the Chapterhouse
11.45 - 12.45	Michael Hopkins (Harvard University) The Kervaire invariant: guide to the obstructions
12.45 - 14.30	Lunch served in the Chapterhouse
14.30 - 15.30	John Jones (University of Warwick) The search for framed manifolds with Kervaire invariant one
15.30 - 16.30	Coffee and tea served in the Chapterhouse
16.30 - 17.30	Erik Pedersen (University of Copenhagen) H-spaces and the Kervaire invariant
18.00 - 19.00	Wine reception at the Informatics Forum

Tuesday 26 April

09.30 - 10.30	Jack Morava (Johns Hopkins University) HHR's equivariant cohomology theory and Abelian varieties Link to slides (external)
10.30 - 11.30	Coffee and tea served in the Chapterhouse
11.30 - 12.30	Doug Ravenel (University of Rochester) Outline of the proof Link to slides or Link to concise version of slides
12.30 - 14.00	Lunch
14.00 - 15.00	John Greenlees (University of Sheffield) Equivariant stable homotopy theory and ordinary cohomology
15.15 - 16.15	Mark Mahowald (Northwestern University) The role of KI classes in homotopy
16.15 - 17.00	Coffee and tea served in the Chapterhouse
17.00 - 18.00	Jean Lannes (École Polytechnique) Remakes of Browder's paper on the Kervaire invariant Link to slides

Wednesday 26 April

09.30 - 10.30	Mike Hill (University of Virginia) The slice filtration
10.30 - 11.30	Coffee and tea served in the Chapterhouse
11.30 - 12.30	Peter Akhmetev (Izmiran) A geometric approach to stable homotopy groups of spheres: The Kervaire invariant Link to slides
12.30 - 14.00	Lunch
	Free afternoon
17.30	ICMS closing time

Thursday 28 April

09.30 - 10.30	Stefan Schwede (University of Bonn) Ultra-commutative ring spectra
10.30 - 11.30	Coffee and tea served in the Chapterhouse
11.30 - 12.30	Robert Bruner (Wayne State University) Classical approaches to Kervaire classes Link to slides (external)
12.30 - 14.00	Lunch
14.00 - 15.00	Tilman Bauer  (University of Amsterdam) The periodicity theorem
15.15 - 16.15	Hans-Werner Henn (Université de Strasbourg) The chromatic splitting conjecture; the case of the prime 3 and chromatic level 2
16.15 - 17.00	Coffee and tea served in the Chapterhouse
19.00 - 23.30	Conference dinner. Served in the Pentland suite at the John McIntyre Centre, Pollock Halls

Friday 29 April

09.30 - 10.30	Neil Strickland (University of Sheffield) Tambura functions
10.30 - 11.30	Coffee and tea served in the Chapterhouse
11.30 - 12.30	Hal Sadofsky (University of Oregon) The detection theorem Link to slides
12.30 - 14.00	Lunch
14.00 - 15.00	Doug Ravenel (University of Rochester) The gap theorem Link to slides
15.15 - 16.15	Tyler Lawson (University of Minnesota) Relative Picard and Brauer groups for S-algebras
16.15 - 17.00	Coffee and tea served in the Chapterhouse

Presentations:

Presentation Details
Akhmetev, Peter
Geometric approach to stable homotopy groups of spheres: The Kervaire invariant.
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A solution of the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on Z/2[2]--framed and on Z/2[3]--framed immersions, and quaternionic-cyclic structure on Z/2[4]--framed immersions. Using these concepts, we prove that for sufficiently large n, n = 2ℓ-2, an arbitrary skew-framed immersion in Euclidean n-space Rn has zero Kervaire invariant. arXiv:1011.5717v1 [math.AT]
Atiyah, Sir Michael
The magic square and the Kervaire invariant
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The relation between the Hopf invariant and the 4 classical projective planes has been very important. It seems likely that this relation may extend to the Kervaire invariant and the manifolds that appear in the Tits-Freudenthal "magic square". I will describe these manifolds and some of their topological properties.
Bauer, Tilman
The periodicity theorem
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The proof of the non-existence of Kervaire-invariant one elements by Hill, Hopkins, and Ravenel rests on three theorems about a ring spectrum Ω they construct: the Detection Theorem, which says that the Kervaire-invariant one elements map to non-zero classes in the homotopy of Ω; the Gap Theorem, which says that Ω has no homotopy in dimensions -1, -2, and -3; and the Periodicity Theorem, which says that Ω is 256-periodic (and therefore has zero homotopy in any dimension congruent to -1, -2, or -3 modulo 256). In my talk, I will present their proof of the Periodicity Theorem.
Bruner, Robert
Classical approaches to Kervaire classes
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We will review the classical inductive approach to the Kervaire classes and the evidence that it provides for the Finiteness Conjecture, that Sq0 families in Ext contain only a finite number of non-zero homotopy classes.
Greenlees, John
Equivariant stable homotopy theory and ordinary cohomology
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The talk will provide an introduction to some of the basic equivariant ingredients of the Hill-Hopkins-Ravenel proof. Orthogonal spectra, change of groups, geometric and categorical fixed points, ordinary homology and cohomology, Mackey functors.
Jones, John
The search for framed manifolds with Kervaire invariant one
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In view of the work of Hill, Hopkins and Ravenel we know that the only dimensions in which there are framed manifolds with Kervaire invariant one are 2, 6, 14, 30, 62 and possibly 126. In this talk I will survey what is known about constructing examples in these special dimensions and I will also give some examples of other interesting phenomena in differential topology which seem to be closely related to the Kervaire invariant.
Lannes, Jean
Remakes of Browder's paper on the Kervaire invariant
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I will describe two remakes of the celebrated paper by William Browder in which he has established the link between the Kervaire invariant and the Adams spectral sequence for mod 2 cohomology. The first one involves manifolds with boundary and the second one (collaboration with Haynes Miller) manifolds with a corner of codimension 2.
Lawson, Tyler
Relative Picard and Brauer groups
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Picard and Brauer groups capture information about invertible modules and central simple algebras. In the classical case the calculation of these objects is often aided by the use of Galois cohomology. We will discuss the generalizations to ring spectra, due to Hopkins-Mahowald-Sadofsky and Baker-Richter-Szymik respectively. We then discuss how to compute these using homotopy-fixed-point spaces as Galois cohomology in higher category theory. (Joint with David Gepner.)
Mahowald, Mark
The role of the KI classes in homotopy
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In the late 60’s I began a study of the homotopy elements in the sphere which were related to the image of the J-homomorphism. This led to a conjecture. Let  βm be the mth non-zero generator in the image of the J-homomorphism. (β1 = η, β2 = ν, ...) Let ϕ(m) = 8a + 2b for m = 4a + b and b < 3. Then βm ∈ π ϕ(m) − 1 S0. The conjecture is: The Whitehead product [ιn, βm] = 0 for n = 2m+1 − ϕ(m) − 1 if and only if hm2 represents a homotopy class in the Adams spectral sequence (see Memoir of AMS # 72 (1967)). In this lecture I will discuss the mathematics which led to this conjecture and the consequence of the failure of hm2 to represent a homotopy class.
Morava, Jack
HHR's equivariant cohomology theory and abelian varieties
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The equivariant spectrum used in Hill, Hopkins, and Ravenel’s work is the localization of a spectrum appearing naturally in Behrens and Lawson’s theory of topological automorphic forms (for the Shimura stack of abelian varieties of type U(1, 3)). In an entirely expository talk I will sketch some of the latter work, and how it relates to H, H & R.
Ravenel, Douglas
Talk 1: "Outline of the proof" Talk 2: "The gap theorem"
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Outline of the proof This is joint work with Mike Hill and Mike Hopkins. I will outline our proof that the Kervaire invariant elements θj in the stable homotopy groups of spheres do NOT exist for j ≥ 7. Our methods include complex cobordism theory, equivariant stable homotopy theory and a new computational tool, the slice spectral sequence. The gap theorem This is joint work with Mike Hill and Mike Hopkins. I will discuss the proof of the Gap Theorem, which says that our periodic spectrum Ω has no homotopy in dimension -2. Assuming the Slice Theorem is true, the proof of the Gap Theorem boils down to some surprisingly easy algebra.
Sadofsky, Hal
The Detection Theorem
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The Detection Theorem asserts the following. Given a candidate (in the Adams-Novikov E2 term) for an element of Kervaire invariant one in dimension at least 126, it is detected in the homotopy of a certain cohomology theory, ΩO. (The non-existence of certain elements of Kervaire invariant one will then follow by showing that ΩO has no homotopy in the relevant dimensions, but this is not part of the Detection Theorem.) We prove the Detection Theorem by showing that it follows from an algebraic result about maps on Ext groups, and proving the algebraic result. In particular we study the map Ext2, 2j+1BP*BP(BP *, BP*) → H2(C8, ΩO*) → H2(C8 , R*) The first map is a map of Adams spectral sequences and the second map is induced by a map on coefficients. The Detection Theorem is proved by showing that candidates for elements detecting Kervaire invariant one have non-zero images under the composite map (for j ≤ 6).
Schwede, Stefan
Ultra-commutative ring spectra
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I will discuss a specific setting for "global" equivariant stable homotopy theory where an object encodes one equivariant homotopy type for every finite group, compatible with restriction and transfers. I will explain that the strictly commutative ring spectra in this context are `more commutative' than E-infinity ring spectra and how to exploit the extra structure that they come with.

Participants

Name	Institution
Adamaszek, Michal	University of Warwick
Adams-Florou, Spiros	University of Edinburgh
Akhmetev, Peter	IZMIRAN
Atiyah, Sir Michael	University of Edinburgh
Baker, Andrew	University of Glasgow
Barnes, David	The University of Sheffield
Barthel, Tobias	University of Oxford
Bauer, Tilman	VU University Amsterdam
Brito, Pedro	University of Aberdeen
Bruner, Robert	Wayne State University
Cazanave, Christophe	University of Nice
Chacholski, Wojciech	KTH
Cheung, Pokman	University of Sheffield
Collins, Julia	University of Edinburgh
Cranch, James	University of Leicester
Crowley, Diarmuid	Hausdorff Research Institute for Mathematics
Garcia Pulido, Ana Lucia	University of Warwick
Garkusha, Grigory	Swansea University
Gasparim, Elizabeth	University of Edinburgh
Gepner, David	University of Regensburg
Grant, Mark	The University of Edinburgh
Gray, Brayton	Univerity of Illinois at Chicago
Greenlees, John	University of Sheffield
Henn, Hans-Werner	IRMA, Université de Strasbourg
Hill, Michael	University of Virginia
Hopkins, Mike	Harvard University
Hubbuck, John	University of Aberdeen
Husemoller, Dale	Max Planck Institute for Mathematics
Joachim, Michael	Westfälische Wilhelms-Universität
Jones, John	University of Warwick
Kedziorek, Magdalena	University of Sheffield
Lam, Kee Yuen	University of British Columbia
Lam, S.P.	University of Cambridge
Lannes, Jean	Ecole Polytechnique
Lawson, Tyler	University of Minnesota
Lesh, Kathryn	Union College
Levikov, Filipp	University of Aberdeen
Mahowald, Mark	Northwestern University
Malhotra, Arjun	University of Sheffield
McTague, Carl	University of Southampton
Meier, Lennart	Universität Bonn
Morava, Jack	The Johns Hopkins University
Noel, Justin	Max Planck Institute for Mathematics
Nogami, Jumpei	Universität Duisburg-Essen
Oliver, Bob	[permanent address] Universite Paris 13 ; [currently address] Copenhagen University
Orson, Patrick	University of Edinburgh
Patchkoria, Irakli	University of Bonn
Pedersen, Erik Kjær	University of Copenhagen
Powell, Geoffrey	CNRS and Université Paris 13
Randal-Williams, Oscar	Copenhagen University
Randall, Duane	Loyola University New Orleans
Ranicki, Andrew	University of Edinburgh
Ravenel, Douglas	University of Rochester
Ray, Nigel	University of Manchester
Sadofsky, Hal	University of Oregon
Schwede, Stefan	Universität Bonn
Strickland, Neil	University of Sheffield
Swarbrick, Rupert	University of Warwick
Szymik, Markus	Heinrich-Heine-Universität Düsseldorf
Tangora, Martin	University of Illinois at Chicago
Theriault, Stephen	University of Aberdeen
Ullman, Harry	University of Sheffield
Zare, Hadi	The University of Manchester