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What is $H_3Spin(3)$, and how is this related with the twist of framing on a 3-manifold?
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From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=frac14piint_MmathrmTr(omegawedge domega+frac23omegawedgeomegawedgeomega)$$
i.e. it changes under a twist of framing on $M$ by $I(g)rightarrow I(g)+2pi s$ with $sinmathbbZ$, is related with the group $H_3Spin(3)=mathbbZ$.
What is this group $H_3Spin(3)$?
Why is it isomorphic to $mathbbZ$?
How exactly is it related with the change of Pontryagin class under a change of framing on $M$?
They also talked about $Omega_3^fr=mathbbZ_24$.
- What exactly is this $Omega_3^fr$?
(I also posted my question at https://www.physicsoverflow.org/41425 hoping to receive answers from physicists)
differential-geometry quantum-field-theory tqft
asked Jul 26 at 22:46
New Student
1434
migrated from mathoverflow.net Jul 27 at 3:41
This question came from our site for professional mathematicians.
add a comment |Â
up vote
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From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=frac14piint_MmathrmTr(omegawedge domega+frac23omegawedgeomegawedgeomega)$$
i.e. it changes under a twist of framing on $M$ by $I(g)rightarrow I(g)+2pi s$ with $sinmathbbZ$, is related with the group $H_3Spin(3)=mathbbZ$.
What is this group $H_3Spin(3)$?
Why is it isomorphic to $mathbbZ$?
How exactly is it related with the change of Pontryagin class under a change of framing on $M$?
They also talked about $Omega_3^fr=mathbbZ_24$.
- What exactly is this $Omega_3^fr$?
(I also posted my question at https://www.physicsoverflow.org/41425 hoping to receive answers from physicists)
differential-geometry quantum-field-theory tqft
asked Jul 26 at 22:46
New Student
1434
migrated from mathoverflow.net Jul 27 at 3:41
This question came from our site for professional mathematicians.
3
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
1
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=frac14piint_MmathrmTr(omegawedge domega+frac23omegawedgeomegawedgeomega)$$
i.e. it changes under a twist of framing on $M$ by $I(g)rightarrow I(g)+2pi s$ with $sinmathbbZ$, is related with the group $H_3Spin(3)=mathbbZ$.
What is this group $H_3Spin(3)$?
Why is it isomorphic to $mathbbZ$?
How exactly is it related with the change of Pontryagin class under a change of framing on $M$?
They also talked about $Omega_3^fr=mathbbZ_24$.
- What exactly is this $Omega_3^fr$?
(I also posted my question at https://www.physicsoverflow.org/41425 hoping to receive answers from physicists)
differential-geometry quantum-field-theory tqft
asked Jul 26 at 22:46
New Student
1434
From the question https://www.physicsoverflow.org/32208, Mr Ryan Thorngren said in the answer that the the framing anomaly of the gravitational Chern-Simons action
$$I(g)=frac14piint_MmathrmTr(omegawedge domega+frac23omegawedgeomegawedgeomega)$$
i.e. it changes under a twist of framing on $M$ by $I(g)rightarrow I(g)+2pi s$ with $sinmathbbZ$, is related with the group $H_3Spin(3)=mathbbZ$.
What is this group $H_3Spin(3)$?
Why is it isomorphic to $mathbbZ$?
How exactly is it related with the change of Pontryagin class under a change of framing on $M$?
They also talked about $Omega_3^fr=mathbbZ_24$.
- What exactly is this $Omega_3^fr$?
(I also posted my question at https://www.physicsoverflow.org/41425 hoping to receive answers from physicists)
differential-geometry quantum-field-theory tqft
asked Jul 26 at 22:46
New Student
1434
asked Jul 26 at 22:46
New Student
1434
asked Jul 26 at 22:46
New Student
1434
asked Jul 26 at 22:46
New Student
1434
1434
migrated from mathoverflow.net Jul 27 at 3:41
This question came from our site for professional mathematicians.
migrated from mathoverflow.net Jul 27 at 3:41
This question came from our site for professional mathematicians.
3
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
1
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13
add a comment |Â
3
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
1
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13
3
3
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
1
1
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13
add a comment |Â
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3
That's homology, a very basic invariant of topological spaces studied in algebraic topology. The group in the last question may be framed cobordism. That said, your questions would be better suited for other forums, like the one you alude or math stack exchange.
– Fernando Muro
Jul 26 at 23:40
1
Both the computation of H_3(Spin(3)) and the third framed cobordism group can be found in many classical algebraic topology texts
– Tobias Shin
Jul 27 at 1:19
@Tobias Shin How is $H_3(Spin(3))$ related with the twist of framing? Why does the gravitational Chern-Simons change by $2pimathbbZ$ under the twist of framing?
– New Student
Jul 27 at 5:13