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Four Cohomological Methods from 1893--1957 in Topology

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Seth Arnold, SLU

What
  • PhD Oral Exam
When Mon, Apr 23, 2018
from 10:00 AM to 10:50 AM
Where 204 Ritter Hall
Contact Name
Attendees PhD Candidacy Committee:
B Clair, K Druschel, J Kalliongis, Q Khan (chair), A Srivastava
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Explanation of four 6000-level graduate exercises:

  • Represent the isomorphism classes of group extensions 1 ⟶ Zn ⟶ Γ ⟶ C2 ⟶ 1.  First step in this algebra problem is a paper of I Reiner (1957).  Second step is calculation of the group cohomology H2(C2;Zn), whose origin is O Hölder's factorset (1893).  Third step is construction of models.
  • Compute the rational cohomology ring of the Eilenberg–Maclane space K(Z,n).  First done by J P Serre (1951) with spectral sequences.  Applied to show the famous result:  the stable homotopy groups of spheres are finite.
  • Compute the homotopy set [X,Sn] and for n odd [X,RPn] in terms of basic invariants, where X is an n-dimensional CW complex.  Application of 1940s and 1950s obstruction theory, respectively.
  • Define the obstructions to lifting a map X ⟶ B to X ⟶ E, where X is a CW complex and E ⟶ B is a fibration with simple fiber F.  Standard extension of 1950s obstruction theory with the use of twisted coefficients.

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