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homotopy groups of a Lie groupoid

Context

Homotopy theory

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The geometric homotopy groups of a Lie groupoid XX are those of its geometric realization |X||X| when regarded as a simplicial manifold. Equivalently, regarding XX as an object in the (∞,1)-topos ∞LieGrpd, its homotopy groups are those of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos Π(X)\Pi(X) \in ∞Grpd.

Definition

For X=(X 1X 0)X = (X_1 \stackrel{\to}{\to} X_0) a Lie groupoid and x:*Xx : * \to X a point, let

X =(X 1× X 0X 1× X 0X 1X 1× X 0X 1X 0) X_\bullet = \left( \cdots X_1 \times_{X_0} X_1 \times_{X_0} X_1 \stackrel{\to}{\stackrel{\to}{\to}}X_1 \times_{X_0} X_1 \stackrel{\to}{\to} X_0 \right)

be its nerve regarded as a simplicial manifold.

Remark

When regarding each manifold X nX_n as a diffeological space, hence a sheaf on the site CartSp then X inPSh(CartSp) Δ op[CartSp op,sSet]X_\bullet in PSh(CartSp)^{\Delta^{op}} \simeq [CartSp^{op},sSet] is the simplicial presheaf on CartSp that presents XX as an object in the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids.

Definition

Regard X X_\bullet as a simplicial topological space by forgetting the smooth structure. Write |X ||X_\bullet| \in Top for its geometric realization as a simplicial topological space.

The geometric homotopy groups of XX are defined to be the ordinary homotopy groups of the topological space |X ||X_\bullet|:

π n(X,x):=π n(|X |,x). \pi_n(X,x) := \pi_n(|X_\bullet|,x) \,.

In this form the definition originates in (Segal).

Properties

Regard XX as an ∞-Lie groupoid under the natural embedding LieGrpdLieGrpdLieGrpd \hookrightarrow \infty LieGrpd. By the discussion at ∞LieGrpd this is a locally ∞-connected (∞,1)-topos, which means that its terminal geometric morphism comes with a further left adjoint Π\Pi

(ΠΔΓ):LieGrpdGrpd. (\Pi \dashv \Delta \dashv \Gamma) : \infty LieGrpd \to \infty Grpd \,.

We say that Π(X)GrpdTop\Pi(X) \in \infty Grpd \simeq Top is the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of XX.

Observation

The geometric homotopy groups of XX are those of Π(X)Top\Pi(X) \in Top.

Proof

By the discussion at ∞-Lie groupoid we have precisely that Π(X)\Pi(X) is presented by the geometric realization of the simplicial topological space underlying the nerve of XX.

References

The definition of the homotopy groups of a Lie groupoid as those of its geometric realization appearently goes back to

  • Graeme Segal, Classifying spaces and spectral sequences , IHES Publ. Math. 34 (1968) 105–112.

An equivalent definition is in

  • A. Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97

reproduced in section 3 of

  • Graeme Segal, Classifying spaces related to foliations , Topology 17 (1978), 367-382.

Revised on January 3, 2011 10:30:41 by Anonymous Coward (81.153.251.158)