ORIENTABILITY

MATH 60440 LAURENCE R. TAYLOR SPRING, 2014

Contents 1. The edge-path groupoid1

1. The edge-path

Recall the map ψK : Hn(M,M − K) → × Hn(M,M − p) which is p∈M injective and the definition Definition 1.1. Let M be a manifold. An orientation for M is a choice of generator up ∈ Hn(M,M − p) for each p ∈ M such that the collection “varies continuously”. “Varies continuously” means the following. For any compact subset K ⊂ M, let uK ∈ ×p∈K Hn(M,M − p) be the product of the up. The collection {up} “varies continuously” provided for each p ∈ M there exists a compact neighborhood, Kp, and a class µKp such that ψK(µKp) = uKp. Instead of thinking directly about orientability, let us try a different approach. Pick generators in Hn(M,M − p) arbitrarily and cover M by charts which are open balls. To each path λ: [0, 1] → M assign a +1 or a −1 as follows. Pick a finite set of charts so that [0, 1] is divided into finitely many intervals [ti, ti+1] with each contained in the image of a chart. Use the chart and the orientations at λ(ti) and λ(ti+1) to assign ±1 depending on whether the orientations at the two points agree or disagree using the chart to relate the two. Assign ±1 to λ depending on the sign assigned to λ(1). Lemma 1.2. The sign assigned to λ is independent of the choice of cover. Lemma 1.3. The sign assigned to λ is constant as λ varies by a relative to the end points. 2 The set of paths modulo homotopy rel endpoints is called the edge-path groupoid of the space, denoted E(X). There is a function s: E(X) → X × X defined by e(λ) = λ(0), λ(1). The choice of generators of Hn(M,M − p) defines a function u: E(X) → {±1}. There is a partial composition on the edge-path groupoid. If λ1(1) = λ2(0) define ( 1 λ1(2t) 0 6 t 6 2 λ2 ◦ λ1(t) = 1 λ2(2t − 1) 2 6 t 6 1 Of course you have to check that this composition is well-defined and associative and the constant path at a point is a unit whenever composition with it is defined. The composition of λ(t) with λ(1 − t) is homotopic rel either endpoint to the constant path. The formulas are just like those for the fundamental . −1 In fact e (x) is just π1(X, x). Lemma 1.4. The function u is multiplicative. A not so good theorem is

Theorem 1.5. If M is orientable, generators for Hn(M,M − p) can be chosen so that u is always +1. A better theorem is Theorem 1.6. M is orientable if and only if u factors through e. In other words u(λ) only depends on the starting point and the ending point. There exists a homomorphism

w1 : π1(X, x) → Z/2Z defined by w1(λ) = u(λ) for any path representing the element in π1.

Corollary 1.7. M is orientable if and only if w1 is trivial. Theorem 1.8. Simply-connected manifolds are orientable. 3 Last update: March 20, 2015 at 16:00