Non-orientable quotients of the product of two 2-spheres by Z/4Z

(Difference between revisions)
Jump to: navigation, search
m (Problem)
m (Problem)
Line 18: Line 18:
''Question'': Are $N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and $N \cup \rm{MCyl}(L(4, 1) \to L(8, 3))$ diffeomorphic?
''Question'': Are $N \cup \rm{MCyl}(L(4, 1) \to L(8, 1))$ and $N \cup \rm{MCyl}(L(4, 1) \to L(8, 3))$ diffeomorphic?
+
+
This question was posed by Jonathan Hillmann at the [[:Category:MATRIX 2019 Interactions|MATRIX meeting on Interactions between high and low dimensional topology.]]
</wikitex>
</wikitex>

Revision as of 06:00, 8 January 2019

1 Problem

Let \sigma be a generator of \mathbb{Z}/ 4\mathbb{Z} and consider the free action of \Z/4 on S^2 \times S^2 defined by

\displaystyle \sigma(x, y) = (y, -x),  \text{ where } (x,y)\in S^2 \times S^2.

Let M := S^2\times S^2/ \langle \sigma \rangle be the quotient of S^2 \times S^2 obtained from this free action.

To understand the structure of this quotient, first, notice that \sigma^2 restricted to the diagonal copy of S^2 \subset S^2 \times S^2 is the antipodal map.

So the diagonal projects down to the projective plane \mathbb{R}P^2 inside the quotient. Denote a normal disk bundle neighbourhood of this projective plane by N.

Off the diagonal, the structure of S^2 \times S^2/\langle \sigma \rangle is that of a mapping cylinder. Namely, the mapping cylinder of the double cover of the lens space L(8,1) by the lens space L(4, 1).

So
Tex syntax error
is a model for the quotient M = S^2\times S^2/ \langle \sigma \rangle. Modifying that mapping cylinder by taking the double covering L(4, 1) \to L(8, 3), it can be shown that
Tex syntax error
and
Tex syntax error
are homotopy equivalent.

In [Hambleton&Hillmann2017] it is shown that there are exactly four topological manifolds in this homotopy type, two of which are smoothable and two which have non-trivial Kirby-Siebenmann invariant.

Question: Are
Tex syntax error
and
Tex syntax error
diffeomorphic?

This question was posed by Jonathan Hillmann at the MATRIX meeting on Interactions between high and low dimensional topology.

2 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox