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Diarmuid Crowley (HIM Bonn): Some new calculations for the Gromoll filtration (17 May 2011)


Let \Gamma_n denote the group of oriented homotopy n-spheres, n > 4. There is a decreasing filtration of \Gamma_n due to Gromoll:

\Gamma = \Gamma_{n, 2} \supset \Gamma_{n, 3} \supset ... \supset \Gamma_{n, n-3} = 0

The determination of the Gromoll filtration is a difficult open problem (it seems to be unknown for any n > 6, n \neq 12) which has many interesting applications in geometric topology.

In this talk I will survey approaches to finding elements deeper in the Gromoll filtration: plumbing and compostion and also obstructions to finding such elements from pseudo-isotopy theory. Then I will report on new computations for certain Gromoll groups. As time permits I will discuss applications and open problems.

Many of the new results are part of joint work with with Thomas Schick.

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