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Hi,All:

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and

  consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$

  through the induced map on homology .Why is this action a surjection onto

 $Sp^2(2g,\mathbb Z)$? ; I cannot see why every m in $Sp^2(2g,\mathbb Z)$ necessarily comes from

  some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Sp^2 is the

  kernel of the mod2 reduction map from $H_1(Sg,\mathbb Z)$ to $H_1(Sg.\mathbb Z_2)$ , which I

  know is natural, but I cannot if the naturality helps.

  A similar second one: Let SPg be the spin-mapping class group , and consider this time

  the action on $H_1(Sg,\mathbb Z_2)$, sending g in SPg to m' in Og (orthogonal group,

  preserving the Rokhlin form, but I'm pretty sure this generalizes to all orthogonal groups),

  via the induced map on homology again. I cannot see either why this second action is

  surjective onto Og, tho I suspect that since Og is defined over $Z_2$ (more precisely

  over $H_1(Sg,\mathbb Z_2)$, this makes it a symplectic group, so that this case may reduce

  to the one above with $Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from some f in Mg. My group theory is a bit weak, so I may be missing some basic results.

Any Ideas?

Thanks.

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$ through the induced map on homology .Why is this action a surjection onto  $Sp^2(2g,\mathbb Z)$? ; I cannot see why every m in $Sp^2(2g,\mathbb Z)$ necessarily comes from some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Sp^2 is the kernel of the mod2 reduction map from $H_1(Sg,\mathbb Z)$ to $H_1(Sg.\mathbb Z_2)$ , which I know is natural, but I cannot if the naturality helps. A similar second one: Let SPg be the spin-mapping class group , and consider this time the action on $H_1(Sg,\mathbb Z_2)$, sending g in SPg to m' in Og (orthogonal group, preserving the Rokhlin form, but I'm pretty sure this generalizes to all orthogonal groups), via the induced map on homology again. I cannot see either why this second action is surjective onto Og, tho I suspect that since Og is defined over $Z_2$ (more precisely over $H_1(Sg,\mathbb Z_2)$, this makes it a symplectic group, so that this case may reduce to the one above with $Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from some f in Mg. My group theory is a bit weak, so I may be missing some basic results.

Any Ideas?

Hi,All:

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and

consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$

through the induced map on homology .Why is this action a surjection onto

$Sp^2(2g,\mathbb Z)$? ; I cannot see why every m in $Sp^2(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Sp^2 is the

kernel of the mod2 reduction map from $H_1(Sg,\mathbb Z)$ to $H_1(Sg.\mathbb Z_2)$ , which I

know is natural, but I cannot if the naturality helps.

  A similar second one: Let SPg be the spin-mapping class group , and consider this time

 the action on $H_1(Sg,\mathbb Z_2)$, sending g in SPg to m' in Og (orthogonal group,

preserving the Rokhlin form, but I'm pretty sure this generalizes to all orthogonal groups),

 via the induced map on homology again. I cannot see either why this second action is

 surjective onto Og, tho I suspect that since Og is defined over $Z_2$ (more precisely


over $H_1(Sg,\mathbb Z_2)$, this makes it a

  symplectic group, so that this case may reduce to the one above with Sp(2g)  

 $Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results.

Any Ideas?

Thanks.

Hi,All:

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and

consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$

through the induced map on homology .Why is this action a surjection onto

$Sp^2(2g,\mathbb Z)$? ; I cannot see why every m in $Sp^2(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Sp^2 is the

kernel of the mod2 reduction map from $H_1(Sg,\mathbb Z)$ to $H_1(Sg.\mathbb Z_2)$ , which I

know is natural, but I cannot if the naturality helps.

A similar second one: Let SPg be the spin-mapping class group , and consider this time

the action on $H_1(Sg,\mathbb Z_2)$, sending g in SPg to m' in Og (orthogonal group,

preserving the Rokhlin form, but I'm pretty sure this generalizes to all orthogonal groups),

via the induced map on homology again. I cannot see either why this second action is

surjective onto Og, tho I suspect that since Og is defined over $Z_2$ (more precisely

over $H_1(Sg,\mathbb Z_2)$, this makes it a symplectic group, so that this case may reduce

to the one above with $Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from some f in Mg. My group theory is a bit weak, so I may be missing some basic results.

Any Ideas?

Thanks.

Hi,All:

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and

consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp(2g,\mathbb Z)$

through the induced map on homology .Why is this action a surjection onto

$Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Any Ideas?

Thanks.

Hi,All:

Let Mg be the Mapping Class Group for $S_g$, the genus-g orientable surface, and

consider the action of Mg on $H_1(S_g,\mathbb Z)$ sending f in Mg to m in $Sp^2(2g,\mathbb Z)$

through the induced map on homology .Why is this action a surjection onto

$Sp^2(2g,\mathbb Z)$? ; I cannot see why every m in $Sp^2(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results. Sp^2 is the

kernel of the mod2 reduction map from $H_1(Sg,\mathbb Z)$ to $H_1(Sg.\mathbb Z_2)$ , which I

know is natural, but I cannot if the naturality helps.

  A similar second one: Let SPg be the spin-mapping class group , and consider this time

 the action on $H_1(Sg,\mathbb Z_2)$, sending g in SPg to m' in Og (orthogonal group,

preserving the Rokhlin form, but I'm pretty sure this generalizes to all orthogonal groups),

 via the induced map on homology again. I cannot see either why this second action is

 surjective onto Og, tho I suspect that since Og is defined over $Z_2$ (more precisely


over $H_1(Sg,\mathbb Z_2)$, this makes it a

  symplectic group, so that this case may reduce to the one above with Sp(2g)  

 $Sp(2g,\mathbb Z)$? ; I cannot see why every m in $Sp(2g,\mathbb Z)$ necessarily comes from

some f in Mg. My group theory is a bit weak, so I may be missing some basic results.

Any Ideas?

Thanks.