Talk:S-duality I (Ex)
Exercise 0.1 (first part). Any finite CW complex has an -dual, for sufficiently large.
Proof. (tentative: see remarks below) Let be a finite pointed CW complex with basepoint . Embed with regular neighbourhood , so that
with a homotopy-equivalence and an -dimensional manifold-with-boundary embedded in .
Now let
be the collapse map,
be the map induced by
and let be the composite
where is a chosen homotopy-inverse for .
The map is degree-1 by construction, so .
For any , we have the identity
so by Poincare-Lefschetz duality,
is an isomorphism for all .
In general for any and ,
So in our case we have
which is an isomorphism by above and the fact that is a homotopy-equivalence. This witnesses that is an -dual of .
Remark 0.2. This uses reduced homology instead of unreduced homology in the definition of -duality, which is correct (?) as we are using smash products rather than direct products.
Remark 0.3. It doesn't seem to be immediately clear why the map defined above is the correct geometric `diagonal' map to use so that the claimed identity holds. It seems more natural to use the map
but this would witness that is an -dual of , rather than