The Conley index, gauge theory, and triangulations - UCLA ...

THE CONLEY INDEX, GAUGE THEORY, AND TRIANGULATIONS 17
for Y and −Y get matched with each other. We deduce that:
α(−Y ) = −γ(Y ), β(−Y ) = −β(Y ), γ(−Y ) = −α(Y ).
In conclusion, the middle invariant β satisfies the desired properties (a) and (b), and this
completes the proof of Theorem 4.7.
Example 4.9. Consider Σ(2, 3, 11) again. The three v-tails in the Pin(2)-equivariant Floer
homology (12) end in degrees (a, b, c) = (4, 1, 2). We get that (α, β, γ) = (2, 0, 0), all even,
in agreement with the fact that Σ(2, 3, 11) has Rokhlin invariant zero.
To review, the key reason why β worked better than the Frøyshov invariant for our
purposes was that the cohomology of B Pin(2) is 4-periodic, whereas the cohomology of
BS 1 is 2-periodic.
4.4. Failure of additivity. Frøyshov proved in [Frø10] that his h-invariant gives a homomorphism
from Θ H 3 to Z. Let us briefly explain the curious facts that h is a homomorphism
and β is not.
Frøyshov’s proof starts with the observation that his construction works also for disjoint
unions of homology spheres. One can define the group Θ H 3 as generated by all (possibly
disconnected) manifolds with H 1 (Y ; Z) = 0, modulo homology cobordism. The connected
sum of two manifolds is homology cobordant to their disjoint union, so we can use the
latter to define addition in Θ H 3 . Recall from Section 3.5.2 that the Floer stable homotopy
invariant of Y 0 ∐ Y 1 is obtained as the smash product of the invariants for the two factors.
This implies that the S 1 -equivariant Floer cochain complex for Y 0 ∐ Y 1 is obtained by
tensoring those for Y 0 and Y 1 over the ground ring H ∗ (BS 1 ; F) = F[U], where F is our
chosen field. Since F[U] is a principal ideal domain, every F[U]-chain complex is quasiisomorphic
to its homology. This implies that, in the derived category of F[U]-modules,
each SWFC S1
∗ (Y i ) can be decomposed as a direct sum of a U-tail and a finite piece, and
the two U-tails coming from Y 0 and Y 1 get tensored together to yield the tail for Y 0 ∐ Y 1 .
From here it is easy to deduce that h(Y 0 ∐ Y 1 ) = h(Y 0 ) + h(Y 1 ).
The argument above fails in the Pin(2) case, because H ∗ (B Pin(2); F 2 ) = F 2 [q, v]/(q 3 ) is
not a PID. In fact, one can give an explicit counterexample as follows. Let Y = Σ(2, 3, 11),
with Floer stable homotopy type X = SWF(Y ), the unreduced suspension of Pin(2). (Compare
Example 3.4.) Then SWF(Y ∐ Y ) = X ∧ X. The smash product of two unreduced
suspensions is the unreduced suspension of the join product. Moreover, the join of Pin(2)
with itself is a Pin(2)-bundle over the wedge sum S 2 ∨ S 2 ∨ S 1 . Starting from these observations
we can compute SWFH Pin(2)
∗ (Y ∐ Y ; F 2 ) to be
q
q
F 2 0 F 2 F 2 F 2 0 . . . . . . . . .
⊕
F 2 2
v
v
v
v
with the leftmost element in degree 2. This shows that the values of (α, β, γ) for Y ∐ Y are
(2, 2, 0). Therefore,
β(Y ∐ Y ) = 2 ≠ β(Y ) + β(Y ) = 0.
By considering disjoint unions of several copies of Σ(2, 3, 11) (with both possible orientations),
it can be shown that no non-trivial linear combination of α, β and γ is additive.