Homotopy type of connected sum depends on choice of gluing map
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Statement
It is possible to find an example of compact connected orientable manifolds and
such that the homotopy type of the connected sum
is not well defined, i.e., we can get connected sums of different homotopy types depending on the choice of the gluing map.
Facts used
Proof
To construct an example, we need to find a case where both and
are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both
and
as isomorphic to the complex projective plane
.
There are two possible connected sums:
- Connected sum of two complex projective planes with same orientation: This has cohomology ring isomorphic to
, where
are additive generators of the free abelian group
and
is the additive generator for
.
- Connected sum of two complex projective planes with opposite orientation: This has cohomology ring isomorphic to
, where
are additive generators of the free abelian group
and
is the additive generator for
.