An Introduction to Algebraic Topology by Andrew H. Wallace

By Andrew H. Wallace

This self-contained therapy assumes just some wisdom of actual numbers and actual research. the 1st 3 chapters specialize in the fundamentals of point-set topology, and then the textual content proceeds to homology teams and non-stop mapping, barycentric subdivision, and simplicial complexes. routines shape a vital part of the textual content. 1961 version.

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The following definitions are essential in characterizing such groupoids. 36 Let G be a Lie groupoid. r G is proper if (s, t) : G1 → G0 × G0 is a proper map. Note that in a proper Lie groupoid G, every isotropy group is compact. r G is called a foliation groupoid if each isotropy group Gx is discrete. r G is e´ tale if s and t are local diffeomorphisms. If G is an e´ tale groupoid, we define its dimension dim G = dim G1 = dim G0 . Note that every e´ tale groupoid is a foliation groupoid. Let us try to understand the effects that these conditions have on a groupoid.

We can assume that U is a contractible open set in Rn with H acting linearly, and so p−1 (V ) B(H U) EH ×H U BH. As a result, p : BG → X is a map such that the inverse image of each point is rationally acyclic, because the reduced rational cohomology of BH always vanishes if H is finite. By the Vietoris–Begle Mapping Theorem (or the Leray spectral sequence), we conclude that p induces an isomorphism in rational homology: p∗ : H∗ (BG; Q) ∼ = H∗ (X; Q). 53 We now look more closely at the case of an orbifold X associated to a global quotient M/G.

A compactly supported orbifold n-form ω on Ui is by definition a compactly supported Gx -invariant n-form ω on Ux . We define ω= Ui 1 |Gx | ω. Ux Each arrow g : x → y in G1 induces a diffeomorphism g : Ux → Uy between components of the inverse image of Ui . It is not hard to show that 1 |Gy | ω= Uy 1 |Gx | g∗ω = Ux 1 |Gx | ω. 1 De Rham and singular cohomology of orbifolds 35 As a result, the value of the integral is independent of our choice of the component Ux . In general, let ω be a compactly supported G-invariant n-form.

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