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Bockstein homomorphim and obstruction of spin-c structure
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Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $beta$ be the
$$
H^2(mathbbZ_2,M) to H^3(mathbbZ,M),
$$
such that $beta(w_2)$ is the integral cohomology class.
Is this true that for certain dimensions of $M=M^d$, say $d=5$, the existence of such a nontrivial $beta(w_2)$ indicates the obstruction of the spin$^c$ structure of $M$? How do we show this?
general-topology differential-topology homology-cohomology geometric-topology characteristic-classes
asked Jul 19 at 15:53
annie heart
549616
 |Â
show 4 more comments
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Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $beta$ be the
$$
H^2(mathbbZ_2,M) to H^3(mathbbZ,M),
$$
such that $beta(w_2)$ is the integral cohomology class.
Is this true that for certain dimensions of $M=M^d$, say $d=5$, the existence of such a nontrivial $beta(w_2)$ indicates the obstruction of the spin$^c$ structure of $M$? How do we show this?
general-topology differential-topology homology-cohomology geometric-topology characteristic-classes
asked Jul 19 at 15:53
annie heart
549616
1
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
1
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
2
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21
 |Â
show 4 more comments
up vote
0
down vote
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2
up vote
0
down vote
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2
Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $beta$ be the
$$
H^2(mathbbZ_2,M) to H^3(mathbbZ,M),
$$
such that $beta(w_2)$ is the integral cohomology class.
Is this true that for certain dimensions of $M=M^d$, say $d=5$, the existence of such a nontrivial $beta(w_2)$ indicates the obstruction of the spin$^c$ structure of $M$? How do we show this?
general-topology differential-topology homology-cohomology geometric-topology characteristic-classes
asked Jul 19 at 15:53
annie heart
549616
Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $beta$ be the
$$
H^2(mathbbZ_2,M) to H^3(mathbbZ,M),
$$
such that $beta(w_2)$ is the integral cohomology class.
Is this true that for certain dimensions of $M=M^d$, say $d=5$, the existence of such a nontrivial $beta(w_2)$ indicates the obstruction of the spin$^c$ structure of $M$? How do we show this?
general-topology differential-topology homology-cohomology geometric-topology characteristic-classes
asked Jul 19 at 15:53
annie heart
549616
asked Jul 19 at 15:53
annie heart
549616
asked Jul 19 at 15:53
annie heart
549616
asked Jul 19 at 15:53
annie heart
549616
549616
1
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
1
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
2
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21
 |Â
show 4 more comments
1
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
1
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
2
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21
1
1
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
1
1
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
2
2
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21
 |Â
show 4 more comments
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1
$SU(3)/SO(3)$ (called the Wu manifold) is a 5-dimensional example of this.
– Mike Miller
Jul 19 at 16:17
Thanks, can you be more explicit? You mean it is an obstruction, only in 5d?
– annie heart
Jul 19 at 16:19
1
An oriented vector bundle $E$ has a $textspin^c$ structure iff $beta w_2(E) = 0$. You can prove this via obstruction theory, as one possible approach. This is just a fact for all vector bundles over (paracompact) spaces. Now, if $E = TM$, this automatically vanishes for dimensions up to 4 - for easy reasons when the dimension is at most 3. See here for dimension 4. The Wu manifold is a simple 5-manifold that does not carry a spin^c structure, showing these arguments don't generalize.
– Mike Miller
Jul 19 at 16:29
Wu x torus is also not spin^c, so you would get examples in any higher dimension.
– Mike Miller
Jul 19 at 16:32
2
I believe that is spin when N is even, and is neither spin nor spin-c when N is odd. This seems to be a difficult calculation. If you have some specific reason to care about whether that manifold is spin-c I can write up the calculation as an answer to a separate question.
– Mike Miller
Jul 22 at 11:21