Smooth structures on noncompact 6-manifolds
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Does a noncompact, simply connected 6-manifold have a unique smooth structure? Maybe with more assumptions like having torsion-free homology and being spin?
Note that in the compact case C.T.C. Wall showed that this is true. Can counterexamples be constructed with exotic $mathbb{R}^4$s, e.g. $Xtimes S^2$ with $X$ an exotic $mathbb{R}^4$?
Any related references would be appreciated.
differential-geometry algebraic-topology differential-topology
asked Nov 15 at 2:59
srp
1827
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show 2 more comments
up vote
4
down vote
favorite
Does a noncompact, simply connected 6-manifold have a unique smooth structure? Maybe with more assumptions like having torsion-free homology and being spin?
Note that in the compact case C.T.C. Wall showed that this is true. Can counterexamples be constructed with exotic $mathbb{R}^4$s, e.g. $Xtimes S^2$ with $X$ an exotic $mathbb{R}^4$?
Any related references would be appreciated.
differential-geometry algebraic-topology differential-topology
asked Nov 15 at 2:59
srp
1827
For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
1
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31
|
show 2 more comments
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Does a noncompact, simply connected 6-manifold have a unique smooth structure? Maybe with more assumptions like having torsion-free homology and being spin?
Note that in the compact case C.T.C. Wall showed that this is true. Can counterexamples be constructed with exotic $mathbb{R}^4$s, e.g. $Xtimes S^2$ with $X$ an exotic $mathbb{R}^4$?
Any related references would be appreciated.
differential-geometry algebraic-topology differential-topology
asked Nov 15 at 2:59
srp
1827
Does a noncompact, simply connected 6-manifold have a unique smooth structure? Maybe with more assumptions like having torsion-free homology and being spin?
Note that in the compact case C.T.C. Wall showed that this is true. Can counterexamples be constructed with exotic $mathbb{R}^4$s, e.g. $Xtimes S^2$ with $X$ an exotic $mathbb{R}^4$?
Any related references would be appreciated.
differential-geometry algebraic-topology differential-topology
differential-geometry algebraic-topology differential-topology
asked Nov 15 at 2:59
srp
1827
asked Nov 15 at 2:59
srp
1827
edited Nov 15 at 3:10
asked Nov 15 at 2:59
srp
1827
asked Nov 15 at 2:59
srp
1827
asked Nov 15 at 2:59
srp
1827
1827
For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
1
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31
|
show 2 more comments
For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
1
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31
For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
1
1
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31
|
show 2 more comments
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For compact part Wall proved that if torsion vanish then it is true
– Anubhav Mukherjee
Nov 15 at 16:54
1
He also assumed the manifold is spin, as I stated above.
– srp
Nov 15 at 18:10
Yeah and $w_2=0$.
– Anubhav Mukherjee
Nov 15 at 18:18
@AnubhavMukherjee That is what spin means.
– Mike Miller
Nov 15 at 18:30
@MikeMiller In my first comment I forgot to mention $w_2$...and I know that it is the meaning of spin :P
– Anubhav Mukherjee
Nov 15 at 18:31