Técnicas e instrumentos de recolección de datos

Step 5. _{The result holds for any open subsetU of M.}

We may as well takeM =U. By Step 3, we may apply Zorn’s lemma to conclude that there is a maximal open subsetV ofM for which the conclusion holds. IfV is not all ofM, sayx6∈V, we may choose a coordinate chartU such thatx∈U. By Steps 2 and 4, the result holds forU ∪V, contradicting the maximality ofV. This completes the proof of the Poincar´e duality theorem.

6. The orientation cover

There is an orientation cover of a manifold that helps illuminate the notion of orientability. For the moment, we relax the requirement that the total space of a cover be connected. Here we take homology with integer coefficients.

Proposition. _{Let M be a connectedn-manifold. Then there is a2-fold cover}

p: ˜M −→M such that M˜ is connected if and only ifM is not orientable.

Proof. _{Define ˜M to be the set of pairs (x, α), wherex∈M and whereα∈}

Hn(M, M −x) ∼= Z is a generator. Define p(x, α) = x. If U ⊂ M is open and

β∈Hn(M, M−U) is a fundamental class ofM atU, define hU, βi={(x, α)|x∈U and β maps toα}.

The setshU, βiform a base for a topology on ˜M. In fact, if (x, α)∈ hU, βi ∩ hV, γi, we can choose a coordinate neighborhood W ⊂U∩V such that x∈W. There is a unique classα′_{∈Hn(M, M−W) that maps toα, and bothβ andγmap toα}′_. Therefore

hW, α′_{i ⊂ hU, βi ∩ hV, γi.} ClearlypmapshU, βihomeomorphically ontoU and

p−1_{(U) =hU, βi ∪ hU,−βi.}

Therefore ˜M is an n-manifold and pis a 2-fold cover. Moreover, ˜M is oriented. Indeed, ifU is a coordinate chart and (x, α)∈ hU, βi, then the following maps all induce isomorphisms on passage to homology:

( ˜M ,M˜ − hU, βi) (M, M −U) ( ˜M ,M˜ −(x, α)) (M, M−x) (hU, βi,hU, βi −(x, α)) O O p ∼ = //(U, U−x). O O

Via the diagram,β ∈Hn(M, M−U) specifies an element ˜β ∈Hn( ˜M ,M˜ − hU, βi), and ˜βis independent of the choice of (x, α). These classes are easily seen to specify an orientation of ˜M. Essentially by definition, an orientation of M is a cross sections:M −→M˜: ifs(U) =hU, βi, then theseβ specify an orientation. Given one section s, changing the signs of the β gives a second section −s such that

˜

M = im(s)∐im(−s), showing that ˜M is not connected if M is oriented. The theory of covering spaces gives the following consequence.

164 THE POINCAR ´E DUALITY THEOREM

Corollary. _{If M is simply connected, or if π1(M)contains no subgroup of}

index 2, then M is orientable. If M is orientable, then M admits exactly two

orientations.

Proof. _{If M is not orientable, then p∗(π1( ˜M)) is a subgroup of π1(M) of} index 2. This implies the first statement, and the second statement is clear. We can use homology with coefficients in a commutative ring R to construct an analogous R-orientation cover. It depends on the units of R. For example, if R = Z2, then the R-orientation cover is the identity map of M since there is a unique unit in R. This reproves the obvious fact that any manifold is Z2- oriented. The evident ring homomorphism Z−→ R induces a natural homomor- phism H∗(X;Z)−→H∗(X;R), and we see immediately that an orientation of M

induces anR-orientation ofM for anyR. PROBLEMS

(1) Prove: there is no homotopy equivalencef :CP2n_−→CP2n_{that reverses} orientation (induces multiplication by−1 onH4n(CP2n_)).

In the problems below, M is assumed to be a compact connectedn-manifold (without boundary), wheren≥2.

2. Prove that ifM is a Lie group, then M is orientable.

3. Prove that ifM is orientable, thenHn−1(M;Z) is a free Abelian group. 4. Prove that ifM is not orientable, then the torsion subgroup ofHn−1(M;Z)

is cyclic of order 2 and Hn(M;Zq) is zero if q is odd and is cyclic of order 2 if q is even. (Hint: use universal coefficients and the transfer homomorphism of the orientation cover.)

5. LetM be oriented with fundamental classz. Letf :Sn _{−→M be a map} such thatf∗(in) =qz, wherein∈Hn(Sn;Z) is the fundamental class and

q6= 0.

(a) Show thatf∗: H∗(Sn;Zp)−→H∗(M;Zp) is an isomorphism ifpis a prime that does not divideq.

(b) Show that multiplication byqannihilatesHi(M;Z) if 1≤i≤n−1. 6. (a) LetM be a compactn-manifold. Suppose thatM is homotopy equiv- alent to ΣY for some connected based spaceY. Deduce thatM has the same integral homology groups as Sn_{. (Hint: use the vanishing} of cup products on ˜H∗_{(ΣY) and Poincar´e duality, treating the cases}

M orientable andM non-orientable separately.)

(b) Deduce thatM is homotopy equivalent toSn_{. Does it follow thatY} is homotopy equivalent toSn−1_?

7. * Essay: The singular cohomologyH∗_{(M;R) is isomorphic to the de Rham} cohomology ofM. Why is this plausible? Sketch proof?

CHAPTER 21