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CONCLUSIONES Y RECOMENDACIONES
For example, sinceχ(RP2m) = 1 andχ(CPn) =n+ 1, this criterion shows that
RP2mandCP2mcannot be boundaries. Notice that we have proved that these are not boundaries of topological manifolds, let alone of smooth ones.
4. Poincar´e duality for manifolds with boundary
The index gives a more striking criterion: if a closed oriented 4k-manifoldM is the boundary of a (topological) manifold, thenI(M) = 0. To prove this, we must first obtain a relative form of the Poincar´e duality theorem applicable to manifolds with boundary.
We letM be ann-manifold with boundary,n >0, throughout this section, and we letRbe a given commutative ring. We say thatM isR-orientable (or orientable ifR=Z) if its interior ˚M =M−∂M isR-orientable; similarly, anR-orientation ofM is anR-orientation of its interior. To study these notions, we shall need the following result, which is intuitively clear but is somewhat technical to prove. In the case of smooth manifolds, it can be seen in terms of inward-pointing unit vectors of the normal line bundle of the embedding∂M −→M.
Theorem (Topological collaring). There is an open neighborhood V of ∂M
in M such that the identification ∂M = ∂M× {0} extends to a homeomorphism
V ∼=∂M×[0,1).
It follows that the inclusion ˚M −→ M is a homotopy equivalence and the inclusion ∂M −→M is a cofibration. We take homology with coefficients in Rin the next two results.
Proposition. AnR-orientation ofM determines an R-orientation of ∂M. Proof. Consider a coordinate chartU of a pointx∈∂M. If dimM =n, then
U is homeomorphic to an open half-disk inHn. Let V =∂U =U ∩∂M and let
y∈U˚=U−V. We have the following chain of isomorphisms:
Hn( ˚M ,M˚−U˚) ∼= Hn( ˚M ,M˚−y) ∼ = Hn(M, M−y) ∼ = Hn(M, M−U˚) ∂ −→ Hn−1(M−U , M˚ −U) ∼ = Hn−1(M−U ,˚ (M−U˚)−x) ∼ = Hn−1(∂M, ∂M−x) ∼ = Hn−1(∂M, ∂M−V).
The first and last isomorphisms are restrictions of the sort that enter into the definition of an R-orientation, and the third isomorphism is similar. We see by use of a small boundary collar that the inclusion ( ˚M ,M˚−y)−→ (M, M −y) is a homotopy equivalence, and that gives the second isomorphism. The connecting homomorphism is that of the triple (M, M−˚U , M−U) and is an isomorphism since
H∗(M, M −U)∼=H∗(M, M) = 0. The isomorphism that follows comes from the observation that the inclusion (M−U˚)−x−→M−U is a homotopy equivalence, and the next to last isomorphism is given by excision of ˚M−˚U. The conclusion is an easy consequence of these isomorphisms.
170 THE INDEX OF MANIFOLDS; MANIFOLDS WITH BOUNDARY
Proposition. If M is compact and R-oriented and z∂M ∈Hn−1(∂M) is the
fundamental class determined by the inducedR-orientation on ∂M, then there is a
unique elementz∈Hn(M, ∂M)such that∂z=z∂M;z is called theR-fundamental
class determined by theR-orientation ofM.
Proof. Since ˚M is a non-compact manifold without boundary and ˚M −→M is a homotopy equivalence,Hn(M)∼=Hn( ˚M) = 0 by the vanishing theorem. There- fore ∂ : Hn(M, ∂M) −→ Hn−1(∂M) is a monomorphism. Let V be a boundary collar and letN =M−V. ThenN is a closed subspace and a deformation retract of theR-oriented open manifold ˚M, and we have
Hn( ˚M ,M˚−N)∼=Hn(M, M−M˚) =Hn(M, ∂M).
SinceM is compact, N is a compact subspace of ˚M. Therefore theR-orientation of ˚M determines a fundamental class in Hn( ˚M ,M˚−N). Let z be its image in
Hn(M, ∂M). Then z restricts to a generator ofHn(M, M −y)∼=Hn( ˚M ,M˚−y) for every y ∈ M˚. Via naturality diagrams and the chain of isomorphisms in the previous proof, we see that∂zrestricts to a generator ofHn−1(∂M, ∂M−x) for all
x∈∂M and is the fundamental class determined by theR-orientation of∂M.
Theorem (Relative Poincar´e duality). Let M be a compact R-oriented n-
manifold with R-fundamental class z ∈ Hn(M, ∂M;R). Then, with coefficients
taken in anyR-moduleπ, capping withz specifies duality isomorphisms
D:Hp(M, ∂M)−→Hn−p(M) and D:Hp(M)−→Hn−p(M, ∂M).
Proof. The following diagram commutes by inspection of definitions:
Hp−1(∂M) // D Hp(M, ∂M) // D Hp(M) // D Hp(∂M) D Hn−p(∂M) //Hn−p(M) //Hn−p(M, ∂M) //Hn−p−1(∂M).
HereDfor∂M is obtained by capping with∂zand is an isomorphism. By the five lemma, it suffices to prove thatD :Hp(M)−→Hn−p(M, ∂M) is an isomorphism. To this end, letN =M ∪∂M M be the “double” ofM and let M1 andM2 be the two copies of M inN. ClearlyN is a compact manifold without boundary, and it is easy to see thatN inherits anR-orientation from the orientation onM1and the negative of the orientation onM2. Of course,∂M =M1∩M2. IfU is the union of
M1and a boundary collar inM2and V is the union ofM2 and a boundary collar in M1, then we have a Mayer-Vietoris sequence for the triad (N;U, V). Using the evident equivalences ofU withM1,V withM2, andU∩V with∂M, this gives the exact sequence in the top row of the following commutative diagram. The bottom row is the exact sequence of the pair (N, ∂M), and the isomorphism results from the homeomorphism N/∂M ∼= (M1/∂M)∨(M2/∂M); we abbreviateN1 = (M1, ∂M)
5. THE INDEX OF MANIFOLDS THAT ARE BOUNDARIES 171