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A tutorial on constructions of finite complexes with specified cohomology (after Steve Mitchell and Jeff Smith)

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HHH - Homotopy harnessing higher structures

Central to the study of modern homotopy theory is the Periodicity Theorem of Mike Hopkins and Jeff Smith, which says that any type n finite complex admits a v_n self map. Their theorem follows from the Devanitz-Hopkins-Smith Nilpotence Theorem once one has constructed at least one example of v_n self map of a type n complex. The construction of such an ur-example uses a construction due to Jeff Smith making use of the modular representation theory of the symmetric groups. This followed the first construction of a type n complex for all n by Steve Mitchell, which used the modular representation theory of the general linear groups over Z/p. The fine points of the Smith construction are not in the only published source: Ravenel's write-up in his book on the Nilpotence Theorems. I'll discuss some of this, and illustrate the ideas with a construction of a spectrum whose mod 2 cohomology is free on one generator as a module over A(3), the 1024 dimensional subalgebra of the Steenrod algebra generated by Sq1, Sq2, Sq4, and Sq8.

This talk is part of the Isaac Newton Institute Seminar Series series.

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