Orientation reversal of manifolds - Universität Bonn

2.2 The cup product and the intersection form 11
Reflect at the equator:
Similarly
and
Figure 2.1: Mirror-symmetric embeddings of oriented surfaces.
The sign depends on the orientation of CP 2n but not on the choice of t since
t is raised to an even power. With the preferred orientation on CP 2n , which
is induced by the complex structure, the value is in fact +1 [MS, Thm. 14.1,
Thm. 14.10, p. 170].
Suppose an orientation-preserving homotopy equivalence f ∶ CP 2n → −CP 2n
would exist. Since f ∗ (t) is again a generator, we have f ∗ (t) = ±t and by the
naturality of the Kronecker product
⟨t 2n , [CP 2n ]⟩ = ⟨( f ∗ t) 2n , [CP 2n ]⟩ = ⟨t 2n , [−CP 2n ]⟩ = −⟨t 2n , [CP 2n ]⟩,
which is impossible. Thus, the complex projective spaces CP 2n in dimensions
4k are homotopically chiral. On the other hand, CP 2n+1 is smoothly amphicheiral.
The orientation is reversed by the map that conjugates the homogeneous
coordinates.
Another point of view is the intersection form on middle cohomology. The
element t n ∈ H 2n (CP 2n ) intersects with itself with intersection number 1 since
⟨t n ∪ t n , [CP 2n ]⟩ = 1. In the following, we recall the definition and the main
properties of the intersection form.
Denote the torsion subgroup of an abelian group A by Tor A ∶= Tor(A, Q/Z).
Moreover, let A free ∶= A/Tor A. (We use this only for finitely generated abelian
groups so that A free is indeed a free abelian group, not only torsion-free.)
Theorem 2
The intersection form on a closed, oriented 2k-dimensional manifold
Q ∶ H k (M) × H k (M) → Z,
Q(a, b) ∶= ⟨a ∪ b, [M]⟩
is a (−1) k -symmetric bilinear form (i. e. it is symmetric if k is even and
antisymmetric if k is odd). Since every homomorphism from a torsion group
to Z is trivial, the intersection form is well-defined on the free quotient
H k (M) free . This bilinear form is unimodular.