Gromov norm and free $Z_2$-space
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The Gromov norm of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class. It is known, the following class of manifold has a non-zero Gromov norm:
$(1)$ Oriented closed connected Riemannian manifolds of negative sectional curvature.
$(2)$ Oriented closed connected hyperbolic manifolds.
$textbf{Q})$ I am interested in knowing a list of examples of orientable manifolds with non-zero Gromov norm that admit a free $mathbb{Z}_2$-action.
As one example, we can consider the genus $g>1$ torus $T_g$. Since $T_g$ is a hyperbolic manifold, its Gromov norm is non-zero by (2). Moreover, If we assume
this is embedded in $mathbb{R}^3$ in a standard way, with the “center” of the torus at the origin, then the antipodal map $xto -x$ define a free $mathbb{Z}_2$-action on it.
algebraic-topology group-actions
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
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add a comment |
$begingroup$
The Gromov norm of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class. It is known, the following class of manifold has a non-zero Gromov norm:
$(1)$ Oriented closed connected Riemannian manifolds of negative sectional curvature.
$(2)$ Oriented closed connected hyperbolic manifolds.
$textbf{Q})$ I am interested in knowing a list of examples of orientable manifolds with non-zero Gromov norm that admit a free $mathbb{Z}_2$-action.
As one example, we can consider the genus $g>1$ torus $T_g$. Since $T_g$ is a hyperbolic manifold, its Gromov norm is non-zero by (2). Moreover, If we assume
this is embedded in $mathbb{R}^3$ in a standard way, with the “center” of the torus at the origin, then the antipodal map $xto -x$ define a free $mathbb{Z}_2$-action on it.
algebraic-topology group-actions
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
$endgroup$
$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
1
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Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
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– 123...
Dec 15 '18 at 11:31
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(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54
add a comment |
$begingroup$
The Gromov norm of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class. It is known, the following class of manifold has a non-zero Gromov norm:
$(1)$ Oriented closed connected Riemannian manifolds of negative sectional curvature.
$(2)$ Oriented closed connected hyperbolic manifolds.
$textbf{Q})$ I am interested in knowing a list of examples of orientable manifolds with non-zero Gromov norm that admit a free $mathbb{Z}_2$-action.
As one example, we can consider the genus $g>1$ torus $T_g$. Since $T_g$ is a hyperbolic manifold, its Gromov norm is non-zero by (2). Moreover, If we assume
this is embedded in $mathbb{R}^3$ in a standard way, with the “center” of the torus at the origin, then the antipodal map $xto -x$ define a free $mathbb{Z}_2$-action on it.
algebraic-topology group-actions
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
$endgroup$
The Gromov norm of a compact oriented n-manifold is a norm on the homology (with real coefficients) given by minimizing the sum of the absolute values of the coefficients over all singular chains representing a cycle. The Gromov norm of the manifold is the Gromov norm of the fundamental class. It is known, the following class of manifold has a non-zero Gromov norm:
$(1)$ Oriented closed connected Riemannian manifolds of negative sectional curvature.
$(2)$ Oriented closed connected hyperbolic manifolds.
$textbf{Q})$ I am interested in knowing a list of examples of orientable manifolds with non-zero Gromov norm that admit a free $mathbb{Z}_2$-action.
As one example, we can consider the genus $g>1$ torus $T_g$. Since $T_g$ is a hyperbolic manifold, its Gromov norm is non-zero by (2). Moreover, If we assume
this is embedded in $mathbb{R}^3$ in a standard way, with the “center” of the torus at the origin, then the antipodal map $xto -x$ define a free $mathbb{Z}_2$-action on it.
algebraic-topology group-actions
algebraic-topology group-actions
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
edited Dec 15 '18 at 10:28
rtybase
11.3k21533
11.3k21533
asked Dec 15 '18 at 9:40
123...123...
426213
asked Dec 15 '18 at 9:40
123...123...
426213
asked Dec 15 '18 at 9:40
123...123...
426213
426213
$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
1
$begingroup$
Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
$endgroup$
– 123...
Dec 15 '18 at 11:31
$begingroup$
(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54
add a comment |
$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
1
$begingroup$
Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
$endgroup$
– 123...
Dec 15 '18 at 11:31
$begingroup$
(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54
$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
1
1
$begingroup$
Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
$endgroup$
– 123...
Dec 15 '18 at 11:31
$begingroup$
Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
$endgroup$
– 123...
Dec 15 '18 at 11:31
$begingroup$
(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54
$begingroup$
(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54
add a comment |
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$begingroup$
Of the examples you cite of manifolds with non-zero Gromov norm, do you know of any explicit calculations?
$endgroup$
– Tyrone
Dec 15 '18 at 11:18
1
$begingroup$
Indeed, the Gromov norm of $T_g$ is $2|chi(T_g)|$. Please see math.uchicago.edu/~cbutler/GromovNorm.pdf , page 3, for the calculation.
$endgroup$
– 123...
Dec 15 '18 at 11:31
$begingroup$
(Q) is essentially equivalent to asking for a list of manifolds with $H^1(M;Bbb Z/2) neq 0$ and non-zero Gromov norm. I do not see why this should be attainable. Certainly a great many hyperbolic manifolds have this property.
$endgroup$
– user98602
Dec 16 '18 at 0:54