Orientation reversal of manifolds - Universität Bonn

2.7 3-manifolds 21
n
θ n
≤ 6 0
7 Z/28
8 Z/2
9 (Z/2) 3
10 Z/6
11 Z/992
12 0
13 Z/3
14 Z/2
15 Z/8128 ⊕ Z/2
16 Z/2
17 (Z/2) 4
Table 2.1: The groups of homotopy spheres in dimensions ≤ 17 [KM], [Levine].
it must be topologically chiral. The most prominent example for a manifold
whose chirality is detected by the Casson invariant is the Poincaré homology
sphere. There are many descriptions for this 3-manifold. It can be obtained, e. g.,
by identifying opposite faces of a solid dodecahedron in the appropriate way or
by 1-surgery on the right-handed trefoil knot (see the section “Surgery” below).
The Casson invariant is normed so that the value of the Poincaré homology
sphere is −1 [Saveliev99, Ch. 17.5].
Another concept, which produces chiral 3-manifolds in abundance, is homology
bordism. We quote from [Saveliev99, Ch. 11.4]: Two oriented integral
homology 3-spheres are called homology cobordant “if there exists a smooth
compact oriented 4-manifold W with boundary ∂W = −Σ 0 ∪ Σ 1 such that the
inclusion induced homomorphisms H ∗ (Σ i ) → H ∗ (W) are isomorphisms.” The
homology cobordism group, denoted Θ 3 Z
, has the connected sum as group operation
and S 3 as the neutral element. The Poincaré homology sphere has infinite
order in Θ 3 Z . Even more, it is known that Θ3 Z
contains a free abelian group of
infinite rank. This provides us with a countable infinite number of homology
3-spheres, all of which are topologically chiral.
2.7.1
Knots and links
In this section, we review the connections between knot theory and the topology
of 3-mainfolds. There are two different constructions to obtain 3-manifolds
from a link in S 3 : branched coverings and surgery. Both constructions justify
the naming “amphicheiral” for manifolds with an orientation-reversing
self-map. In the following, we deal only with tame knots, i. e. topological