Applications Of Algebra To A Problem In Topology - By Michael Hopkins

Applications of algebra to a problem in topology

Tuesday, April 21, 2009

Joint work with

Mike Hill and

Doug Ravenel

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Pontryagin (1930’s)

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Pontryagin (1930’s)

cobordism group of stably framed k-manifolds

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Pontryagin (1930’s) k=0

k=1

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Pontryagin (1930s)

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k=2

Pontryagin (1930s)

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k=2

Pontryagin (1930s)

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k=2

Pontryagin (1930s)

If genus

This defines a fuction

,

and so

You can always lower the genus with surgery

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Pontryagin (?)

is not linear it’s quadratic and refines the intersection pairing

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Pontryagin (?)

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Kervaire (1960) (framed) defined quadratic refinement of the intersection pairing

showed

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Kervaire (1960) produced a piecewise linear with

hence

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has no smooth structure

Browder (1969)

there exists represented by

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Barratt-Jones-Mahowald (1969, 1984) The elements

exist

dimensions

for

so the first open dimension is 126

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The Kervaire invariant problem

In which dimensions can

be non-zero?

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Doomsday Theorem (Hill, H., Ravenel)

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Adams-Novikov Spectral Sequence Something easier to compute Adams Spectral Sequence

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Adams-Novikov Spectral Sequence Something easier to compute Adams Spectral Sequence

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for

supports a

non-zero differential

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Adams-Novikov Spectral Sequence Something easier to compute Adams Spectral Sequence

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periodicity Theorem

Rochlin’s Theorem

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K-theory and reality (Atiyah, 1966) space with a

action

vector bundles with compatible conjugatelinear action

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

Assemble K-theory from the equivariant chains on

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slice filtration

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periodicity

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periodicity

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level 5 topological modular forms Like

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with

instead of

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32 Tuesday, April 21, 2009

32 Tuesday, April 21, 2009

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2 below the period

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the period

2 below the period

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the period

slice filtration

Assemble tmf(5) from the equivariant chains on the 4 dimensional real regular representation of

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-2

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0

2 below the period

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the period

gap + periodicity

differentials on the

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The actual proof Step 1: Use

and an appropriate cohomology theory

Step 2: Show that all the choices of are distinguished Step 3: Prove a gap theorem (easy) Step 4:

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Prove a periodicity theorem (of period 256)

Relation to Geometry/Physics? 4 dimensional field theory? generalization of Clifford algebras with periodicity of (maybe twice that)

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Question

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