Orientation reversal of manifolds - Universität Bonn

2.4 Lens spaces 15
factor 4 or a prime congruent 3 mod 4 is chiral in the strongest sense. We can
thus add the linking form to our list of obstructions to orientation reversal.
2.4
Lens spaces
Lens spaces form a very important class of manifolds for this work. They
appear in many different situations, both in proofs and as illustrations of various
aspects of chirality. Since the conventions about the parameters in lens spaces
differ between sources, they are defined here.
Choose integers n ≥ 2 and t ≥ 1 and parameters k 1 , . . . , k n ∈ (Z/t) × . The lens
space L t (k 1 , . . . , k n ) is defined as the quotient of the unit sphere S 2n−1 ⊂ C n
under the free action of {γ ∣ γ t = 1} ≅ Z/t by γ(c 1 , . . . , c n ) = (ξ k 1
c 1 , . . . , ξ kn c n ).
Here, ξ denotes the t-th root of unity e 2πi/t .
This lens space is a (2n − 1)-dimensional closed, orientable, smooth, connected
manifold with fundamental group Z/t. It has a preferred orientation
induced from the canonical orientation on C n and the outer normal vector
field of S 2n−1 . Furthermore, its fundamental group has a preferred generator γ
(if the fundamental group is nontrivial, i. e. if t > 1). The preferred generator
of the fundamental group is given by the covering translation γ; alternatively,
it can be described by any path from a basepoint x 0 to γ(x 0 ) in S 2n−1 . The
choice of x 0 is irrelevant since the fundamental group is abelian.
The notation L t (k 1 , . . . , k n ) implies that the parameters k i are relatively prime
to t. This will be implicitly assumed in all statements in this work.
In some definitions, instead of the k i their multiplicative inverses modulo t
are used, e. g. [Milnor66, §12]. The classification theorems below are literally
the same for both conventions, but the notation matters of course if individual
lens spaces are identified.
The orientation of a lens space can be reversed by multiplying one of its
parameters by −1; this corresponds to complex conjugation in the respective
coordinate of C n and preserves the preferred generator of the fundamental
group. More precisely, write L ∶= L t (k 1 , . . . , k n ) and L ′ ∶= L t (l 1 , . . . , l n ) and
let l i = −k i for exactly one i, otherwise l i = k i . Then there is an orientationreversing
diffeomorphism L → L ′ which maps the preferred generator of π 1 (L)
to the preferred generator of π 1 (L ′ ).
Lens spaces are classified (besides other concepts like simple homotopy type)
up to oriented homotopy equivalence, homeomorphism and diffeomorphism.
Theorem 9: homotopy classification [Milnor66, 12.1], [Lück, Thm. 2.31]
The lens spaces L t (k 1 , . . . , k n ) and L t ′(l 1 , . . . , l n ) are orientation-preserving
homotopy equivalent if and only if t = t ′ and there is e ∈ (Z/t) × such that
∏ n i=1 k i = e n ⋅ ∏ n i=1 l i in (Z/t) × . The same conditions apply for a map of
degree 1 between the two lens spaces.