- #1

Bacle

- 662

- 1

Hi, All:

I'm a bit confused about this: the 1st homology of the torus T(over Z) is Z(+)Z,

so that elements of the form (a,b) =/ (0,0) are non-trivial , meaning these are

cycles (closed curves) that do not bound subsurfaces of the torus, and every cycle

that does bound is

equivalent/homologous to (0,0). Still, I don't see why, e.g., the cycle (1,1) of a meridian

and then a parallel, is non-bounding: if we went once around meridionally and once

longitudinally, we would disconnect the torus. Isn't (1,1) then, a cycle that bounds?

Additionally, the genus of the torus is 1 , meaning that we can remove at most one

SCC (simple-closed curve) from the torus without disconnecting it, so it would seem

like any pair (a,b) would consist of 2 curves ( one going a times meridionally and b

times longitudinally ), so that, since the genus is 1, (a,b) would disconnect the torus,

right? But then (a,b) would be a cycle that bounds, so that it would be homologous

to (0,0), which it is not. What am I missing?

EDIT: I realize the obvious fact that (a,b) is not a SCCurve. Is this the issue, i.e.,

must homology classes (at least for the torus) be represented by SCC's?

I'm a bit confused about this: the 1st homology of the torus T(over Z) is Z(+)Z,

so that elements of the form (a,b) =/ (0,0) are non-trivial , meaning these are

cycles (closed curves) that do not bound subsurfaces of the torus, and every cycle

that does bound is

equivalent/homologous to (0,0). Still, I don't see why, e.g., the cycle (1,1) of a meridian

and then a parallel, is non-bounding: if we went once around meridionally and once

longitudinally, we would disconnect the torus. Isn't (1,1) then, a cycle that bounds?

Additionally, the genus of the torus is 1 , meaning that we can remove at most one

SCC (simple-closed curve) from the torus without disconnecting it, so it would seem

like any pair (a,b) would consist of 2 curves ( one going a times meridionally and b

times longitudinally ), so that, since the genus is 1, (a,b) would disconnect the torus,

right? But then (a,b) would be a cycle that bounds, so that it would be homologous

to (0,0), which it is not. What am I missing?

EDIT: I realize the obvious fact that (a,b) is not a SCCurve. Is this the issue, i.e.,

must homology classes (at least for the torus) be represented by SCC's?

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