Full Record

Author | Stapleton, Nathaniel J. |

Title | Transchromatic generalized character maps |

URL | http://hdl.handle.net/2142/26269 |

Publication Date | 2011 |

Date Accessioned | 2011-08-25 22:21:33 |

Degree | PhD |

Discipline/Department | 0439 |

Degree Level | doctoral |

University/Publisher | University of Illinois – Urbana-Champaign |

Abstract | In "Generalized Group Characters and Complex Oriented Cohomology Theories", Hopkins, Kuhn, and Ravenel discovered a generalized character theory that proved useful in studying cohomology rings of the form E^*(BG). In this paper we use the geometry of p-divisible groups to describe a sequence of "intermediate" character theories that retain more information about the cohomology theory E and yield the related result of Hopkins, Kuhn, and Ravenel as a special case. |

Subjects/Keywords | Algebraic Topology; Stable Homotopy Theory; Generalized Cohomology Theory; p-Divisible Group; Barsotti-Tate Group; Morava E-theory |

Contributors | Rezk, Charles (advisor); Ando, Matthew (Committee Chair); Rezk, Charles (committee member); McCarthy, Randy (committee member); Schenck, Henry K. (committee member) |

Language | en |

Rights | Copyright 2011 Nathaniel Stapleton |

Country of Publication | us |

Record ID | handle:2142/26269 |

Repository | uiuc |

Date Retrieved | 2020-03-09 |

Date Indexed | 2020-03-09 |

Grantor | University of Illinois at Urbana-Champaign |

Issued Date | 2011-08-25 22:21:33 |

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…*generalized* *cohomology* *theory* on the category of finite spaces (spaces
equivalent to a finite CW-complex), Topf . That is a functor
(Topf )op −→ AB*
from finite spaces to graded abelian groups that satisfies all of the Eilenberg-Steenrod…

…Hopkins, Kuhn, and Ravenel build, for each Morava E-*theory*, an equivariant *cohomology* *theory* that
mimics the properties of Cl(G, L) and is the receptacle for a map from Borel equivariant En . They begin by
constructing a ring L(En )…

…p-divisible group. Ct is flat over En0 and can
X im α then for
, G) and Fix(X) =
be used to make a height t *cohomology* *theory*. Let Gp = hom(Zn−t
p
α∈Gp
all finite G there is a map of equivariant theories
En∗ (EG ×G X)…

…character maps induce an isomorphism
when the source is tensored up to Ct .
5
Chapter 2
Preliminaries
2.1
Conventions
Within this paper all rings are commutative with unit and all graded rings are graded commutative.
By a *cohomology* *theory* we mean a…

…axioms except for
the dimension axiom. We choose finite spaces as our source category because it allows for flat extension of
*cohomology* theories. By an equivariant *cohomology* *theory* we will always mean a Borel-equivariant *theory*.
For G an abelian group…

…K(t)-*theory* for 0 ≤ t < n: LK(t) En . En is an even periodic height n *theory* and LK(t) En is an even periodic
height t *theory*. Basic properties of these *cohomology* theories can be found in ([15], [6…

…Let K be complex K-*theory* and let R(G) be the complex representation ring of a finite group G.
Consider a complex representation of G as a G-vector bundle over a point. Then there is a natural map
R(G) → K 0 (BG). This…

…virtual bundles of dimension 0. [1]
Let L be the smallest characteristic zero field containing all roots of unity and let Cl(G; L) be the ring
of class functions on G taking values in L. A classical result in representation *theory*…