Any spin $M^3$, exists a natural induced $text{Pin}^-$ structure on Poincare dual PD
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It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
given a spin structure on $M^3$, the submanifold $text{PD}(a)$ can be given a natural induced $text{Pin}^-$ structure.
$text{PD}(a)$ is a smooth, possibly non-orientable submanifold in $M^3$ representing Poincare dual to $ain H^1(M^3,mathbb{Z}_2)$ (it always exist in codimension 1 case).
Question 1: How do we digest this is always true?
My take is that:
(1) The normal bundle to the submanifold $text{PD}(a)equiv N^2subset M^3$ for oriented $M^3$ can be realized as
determinant line bundle
$det T{N^2}$, so that $TM^3|_{N^2}=TN^2oplus det TN^2$.(2) For a general vector bundle $V$, there is a natural bijection between Pin$^-$- structures on $V$ and Spin-structures on $Voplus det V$.
Question 2: How can one show that (2) is true?
algebraic-topology manifolds geometric-topology fiber-bundles spin-geometry
asked Nov 16 at 22:48
wonderich
2,06631229
add a comment |
up vote
2
down vote
favorite
It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
given a spin structure on $M^3$, the submanifold $text{PD}(a)$ can be given a natural induced $text{Pin}^-$ structure.
$text{PD}(a)$ is a smooth, possibly non-orientable submanifold in $M^3$ representing Poincare dual to $ain H^1(M^3,mathbb{Z}_2)$ (it always exist in codimension 1 case).
Question 1: How do we digest this is always true?
My take is that:
(1) The normal bundle to the submanifold $text{PD}(a)equiv N^2subset M^3$ for oriented $M^3$ can be realized as
determinant line bundle
$det T{N^2}$, so that $TM^3|_{N^2}=TN^2oplus det TN^2$.(2) For a general vector bundle $V$, there is a natural bijection between Pin$^-$- structures on $V$ and Spin-structures on $Voplus det V$.
Question 2: How can one show that (2) is true?
algebraic-topology manifolds geometric-topology fiber-bundles spin-geometry
asked Nov 16 at 22:48
wonderich
2,06631229
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
given a spin structure on $M^3$, the submanifold $text{PD}(a)$ can be given a natural induced $text{Pin}^-$ structure.
$text{PD}(a)$ is a smooth, possibly non-orientable submanifold in $M^3$ representing Poincare dual to $ain H^1(M^3,mathbb{Z}_2)$ (it always exist in codimension 1 case).
Question 1: How do we digest this is always true?
My take is that:
(1) The normal bundle to the submanifold $text{PD}(a)equiv N^2subset M^3$ for oriented $M^3$ can be realized as
determinant line bundle
$det T{N^2}$, so that $TM^3|_{N^2}=TN^2oplus det TN^2$.(2) For a general vector bundle $V$, there is a natural bijection between Pin$^-$- structures on $V$ and Spin-structures on $Voplus det V$.
Question 2: How can one show that (2) is true?
algebraic-topology manifolds geometric-topology fiber-bundles spin-geometry
asked Nov 16 at 22:48
wonderich
2,06631229
It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
given a spin structure on $M^3$, the submanifold $text{PD}(a)$ can be given a natural induced $text{Pin}^-$ structure.
$text{PD}(a)$ is a smooth, possibly non-orientable submanifold in $M^3$ representing Poincare dual to $ain H^1(M^3,mathbb{Z}_2)$ (it always exist in codimension 1 case).
Question 1: How do we digest this is always true?
My take is that:
(1) The normal bundle to the submanifold $text{PD}(a)equiv N^2subset M^3$ for oriented $M^3$ can be realized as
determinant line bundle
$det T{N^2}$, so that $TM^3|_{N^2}=TN^2oplus det TN^2$.(2) For a general vector bundle $V$, there is a natural bijection between Pin$^-$- structures on $V$ and Spin-structures on $Voplus det V$.
Question 2: How can one show that (2) is true?
algebraic-topology manifolds geometric-topology fiber-bundles spin-geometry
algebraic-topology manifolds geometric-topology fiber-bundles spin-geometry
asked Nov 16 at 22:48
wonderich
2,06631229
asked Nov 16 at 22:48
wonderich
2,06631229
asked Nov 16 at 22:48
wonderich
2,06631229
asked Nov 16 at 22:48
wonderich
2,06631229
asked Nov 16 at 22:48
wonderich
2,06631229
2,06631229
add a comment |
add a comment |
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