114 A Appendix
A.3 Diffeomorphisms between products of lens
A.3 Diffeomorphisms between products of lens spaces 115 Theorem 99: [Metzler, Satz 5]
Lm(1,r) ×Ln(1, 1) is diffeomorphic to Lm(1, 1) ×Ln(1,r) for(r,mn) =1. Here, it is important to know if the diffeomorphism can be chosen to be ori-entation-preserving. Metzler does not consider this question in his dissertation but by tracing his construction it can be seen that he indeed gives an orienta-tion-preserving diffeomorphism. In fact, the diffeomorphism in Theorem 99 is given on the universal covering as a composition of five diffeomorphisms
f1, . . . ,f5∶S3×S3→S3×S3.
The diffeomorphisms f1 to f3 are all given by the scheme fi∶S3→S3, (z1,z2) ↦ (g(z1,z2) ⋅z1⋅h(z1,z2),z2)
or (z1,g(z1,z2) ⋅z2⋅h(z1,z2)) (3) for z1,z2∈S3⊂H, certain maps g,h∶S3×S3→S3 and multiplication under-stood in the quaternionsH. Hence, the induced map on H3(S3×S3) is given (with the obvious basis) by a triangular matrix (∗1 0
1) or (1∗
0 1), so f1 to f3 preserve the orientation.
The map f4 is the product of the identity in the first factor and a map of degree −1 in the second factor (compare [Metzler, eq. (37)]), hence it reverses the orientation. This is compensated by f5, which simply interchanges the two spheres. Altogether, Metzler constructs an orientation-preserving diffeomorph-ism.
Since the parameter r in Theorem 99 can be changed freely modulom in one product and modulo nin the other, we can vary the parameter modulo (m,n):
Lemma 100
The products Lm(1, 1) ×Ln(1,r1) and Lm(1, 1) ×Ln(1,r2) are oriented dif-feomorphic if r1−r2 is a multiple of(m,n).
Proof. Implicitly, we have (r1,n) =1. Let g ∶= (m,n). Choose x,y ∈Z such thatxn+ymg =1−r1. Then we have
(r1+xn,mg) =1
and (r1+xn,n) = (r1,n) =1 ⇒ (r1+xn,g) =1.
Both lines together imply(r1+xn,m) =1. Hence, we can replace r1 bys1∶=
r1+xnand have thusLn(1,r1) =Ln(1,s1),(s1,n) =1 and additionally(s1,m) = 1. Likewise, we replacer2bys2 such that(s2,m) = (s2,n) =1.
Leta,bbe integers such thats2−s1=an+bm. We have the following chain of orientation-preserving diffeomorphisms
Lm(1, 1) ×Ln(1,s1) =Lm(1, 1) ×Ln(1,s1+an) →Lm(1,s1+an) ×Ln(1, 1)
=Lm(1,s1+an+bm) ×Ln(1, 1) =Lm(1,s2) ×Ln(1, 1) →Lm(1, 1) ×Ln(1,s2)
116 A Appendix
The two arrows in this formula are given by Theorem 99. For the first arrow, we have to check that (s1+an,mn) =1 in order to apply Theorem 99. Since (s1,n) =1 and(s2,m) =1, we have
(s1+an,n) =1
and (s1+an,m) = (s2−bm,m) = (s2,m) =1
Both lines together imply (r+bn,mn) =1, so Theorem 99 applies. For the second arrow, we already ensured(s2,mn) =1.
This enables us to extend Corollary 98 to the case(m,n) =2.
Proposition 101
The product of 3-dimensional lens spaces Lm(1, 1) ×Ln(1,r) is smoothly amphicheiral if (m,n) ≤2.
Proof. By Lemma 100, the product Lm(1, 1) ×Ln(1,r)is clearly oriented diffeo-morphic toLm(1, 1) ×Ln(1,−r).
Theorem 102: [Metzler, Satz 7], restricted to two factors
The product Lm1n(1,r1) ×Lm2(1, 1)is diffeomorphic to Lm1(1, 1) ×Lm2n(1,r2) if(m1,n) = (m2,n) =1and
r1≡ { 1 mod m1
−1 mod n, r2≡ { 1 mod m2
−1 mod n.
Again, it is important to know whether the constructed diffeomorphism preserves the orientation. Metzler proves the theorem with a composite of two maps on the universal covering f2○ f1∶S3×S3 →S3×S3. Both maps were designed according to the scheme (3), so they preserve the orientation. In the first half of the proof, he also states the covering transformations onS3×S3(on both ends of the map) carefully, thereby identifying the fundamental groups Z/(m1n) ⊕Z/m2 and Z/m1⊕Z/(m2n). By checking these actions one sees that Metzler really generates the space Lm
1(1, 1) ×Lm
2n(1,r2) with its canonical orientation on the right hand side and not, e. g. the unoriented diffeomorphic space Lm1(1,−1) ×Lm2n(1,r2).
This shows that Theorem 102 can in fact be read with an orientation-pre-serving diffeomorphism understood, which gives us some amphicheiral pro-ducts of lens spaces that were not yet covered by Theorems 97 and 99. The other theorems in [Metzler] do not add to the present results.
Proposition 103
Let a,b,c be pairwise coprime integers and let r≡1 mod a. Then the product L∶=Lab(1,r) ×Lac(1, 1) is smoothly amphicheiral.
A.3 Diffeomorphisms between products of lens spaces 117 Proof. By Lemma 100,Lis oriented diffeomorphic toL2∶=Lab(1,r2) ×Lac(1, 1) with
r2≡ { 1 mod a
−1 mod b.
Thus, we can apply Theorem 102 withm1=a, n=b and m2=ac. This gives a diffeomorphism toLa(1, 1) ×Labc(1,r3) with
r3≡
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1 mod a
−1 mod b 1 mod c.
Since a,band c are pairwise coprime, we can apply Lemma 100 again and obtain a diffeomorphism toLa(1, 1) ×Labc(1,r4) with
r4≡
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1 mod a 1 mod b
−1 mod c.
Theorem 102, applied this time with m1=a,n=c and m2=ab, gives L5∶=
Lac(1,r5) ×Lab(1, 1)with
r5≡ { 1 mod a
−1 mod b.
By Lemma 100, L2 is oriented diffeomorphic toLab(1, 1) ×Lac(1, 1), and L5 is oriented diffeomorphic to Lac(1, 1) ×Lab(1, 1). Since these are products of odd-dimensional manifolds with their factors interchanged, this proves that L is amphicheiral.
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Summary
We study the phenomenon of orientation reversal of manifolds. An orientable manifold is calledamphicheiral if it admits an orientation-reversing self-map and chiral if it does not. Many familiar manifolds like spheres or orientable surfaces are amphicheiral: they can be embedded mirror-symmetrically into Rn, as the following figure illustrates.
Reflect at the equator:
Similarly and
On the other hand, examples of chiral manifolds have been known for many decades, e. g. the complex projective spacesCP2kor some lens spaces in dimen-sions congruent 3 mod 4. However, this phenomenon has not been analysed systematically.
Chiral manifolds can be studied in various categories by restricting the orientation-reversing map to homotopy equivalences, homeomorphisms or dif-feomorphisms. The various notions of chirality do not coincide, and we extend the definition of chiral and amphicheiral manifolds by attributes, e. g. “topolo-gically chiral” or “smoothly amphicheiral” that express the various restrictions on the orientation-reversing map.
We start with a survey of known results and examples of chiral manifolds, observing the basic facts that the point in dimension 0 is chiral and every closed, orientable 1- and 2-dimensional manifold is amphicheiral. A funda-mental question is whether there are chiral manifolds in every dimension≥3, and we prove this as the first main result. Our general aim is to produce manifolds which are chiral in the strongest sense, so we construct manifolds in every dimension≥3 which do not admit a self-map of degree −1.
The obstruction to orientation reversal in the constructed manifolds lies in the fundamental group since, e. g., the odd-dimensional examples are Eilenberg-MacLane spaces, and the proof of chirality uses as a substantial ingredient
125