Applications of algebra to a problem in topology

Tuesday, April 21, 2009

Joint work with

Mike Hill

Doug Ravenel

Pontryagin (1930’s)

cobordism group of stably framed k-manifolds

Pontryagin (1930s) k=2

Pontryagin (1930s) This defines a fuction

If genus , and so

You can always lower the genus with surgery

Pontryagin (?)

it’s quadratic and refines the intersection pairing

is not linear

Pontryagin (?)

Kervaire (1960)(framed)

defined

showed

quadratic refinement of the intersection pairing

produced a piecewise linear

Kervaire (1960)

has no smooth structurehence

represented bythere exists

Browder (1969)

Barratt-Jones-Mahowald (1969, 1984)

so the first open dimension is 126

The elements exist

dimensions

The Kervaire invariant problem

In which dimensions can

be non-zero?

Doomsday Theorem (Hill, H., Ravenel)

Adams SpectralSequence

Adams-NovikovSpectral Sequence

Something easierto compute

supports afornon-zero differential

Rochlin’s Theorem

periodicity Theorem

K-theory and reality (Atiyah, 1966)

space with a action

vector bundles with compatible conjugate-

linear action

Assemble K-theory from the equivariant

chains on

slice filtration(Dugger, Hu-Kriz, H.-Morel, Voevodsky)

slice filtration

periodicity

level 5 topological modular forms

Like with instead of

the period2 below the period

Assemble tmf(5) from the equivariant chains

slice filtration

the 4 dimensional real regular representation of

gap + periodicity

differentials on the

The actual proof

Show that all the choices of are distinguished

Step 2:

Prove a gap theorem (easy)Step 3:

Use and an appropriateStep 1:cohomology theory

Prove a periodicity theorem (of period 256)Step 4:

Relation to Geometry/Physics?

4 dimensional field theory?

generalization of Clifford algebras with periodicity of

(maybe twice that)

Question

Applications of algebra to a problem in topology - by Michael Hopkins

Transcript of Applications of algebra to a problem in topology - by Michael Hopkins

Applications of algebra to a problem in topology - by Michael Hopkins

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