Onwards and Upwards Mark Behrens Shameless Plug ND

Onwards and Upwards Mark Behrens

Shameless Plug: ND undergrad Geom/Top workshop • Annual, starting summer 2017 • First part “basic notions in geometey/topology” • Second part “undergrad research conference” ND graduate Ph. D. program • 6 geometry faculty • 6 topology faculty • 10 grad students in topology • Several topology/geometry seminars • RTG grant in geometry and topology

Pontryagin-Thom Construction

Pontryagin-Thom Construction

Pontryagin-Thom Construction

Pontryagin-Thom Construction

Homotopy groups of spheres in low dimensions

<k(Sk) >k(Sk)

Infinite subgroups completely understood

Values stabilize along diagonals: n+k(Sk) = n+k+1(Sk+1) for k >> 0

Stable homotopy groups: ns : = lim n+k(Sk) k Primary decomposition: ns = ( ns)(p) p prime e. g. : 3 s = Z 24 = z 8 + Z 3

Computation: Mahowald-Tangora-Kochman Picture: A. Hatcher 12

J-homomorphism •

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Computation: Nakamura -Tangora Picture: A. Hatcher 15

( ns)(5) Computation: D. Ravenel Picture: A. Hatcher n 16

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• Trivial for n even • Cyclic for n odd – Generated by boundary of an explicit parallelizable manifold given by plumbing construction

24 KO Hurewicz homomorphism

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( �S)(p) : p = 5 Computation: Ravenel Picture: Hatcher

( �S)(p) : p = 5 Computation: Ravenel Picture: Hatcher

( �S)(p) : p = 5 Computation: Ravenel Picture: Hatcher

Hurewicz image of TMF (p = 3) 29

Hurewicz image of TMF (p = 2) 30