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Sullivan conjecture
Contents
Context
Homotopy theory
homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
The Sullivan conjecture (due to Dennis Sullivan , now a theorem due to Miller 84 ) states that – under certain conditions and after suitable p-adic completion – the canonical map for a G-space from its ordinary fixed points to its homotopy fixed points is a weak homotopy equivalence .

A proof was given in Carlsson 91 , using the Segal-Carlsson theorem .

References
References
Haynes Miller , The Sullivan conjecture on maps from classifying spaces , Annals of Mathematics Second Series, Vol. 120, No. 1 (Jul., 1984), pp. 39-87 (jstor:2007071 )

Gunnar Carlsson , Equivariant stable homotopy and Sullivan’s conjecture . Invent. Math. 103: 497–525, 1991 (dml:143867 , pdf )

See also

Last revised on June 3, 2021 at 10:31:15.
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