Talk:K-groups of oriented surface groups (Ex)
First we look at the Atiyah-Hirzebruch spectral sequence. Recall that a closed orientable surface is either or aspherical. Hence if is not the trivial group, then the surface is itself a model for . Let us fix the genus of the surface. Then the integral homology looks like
Moreover since the coefficients of are only or we see that all -terms of the spectral sequence are free abelian groups. By purely formal reasons there cannot be any non-trivial differential (the only possible ones have target the -axis of the spectral sequence, but since the coefficients in the AHSS always split off, these maps must have trivial image). Hence also the -terms of the spectral sequence are torsion free. In particular there are no extension problems to solve and we can read off the abutment to be
We want to remark that also for other homology theories there cannot occur non-trivial extensions when we want to compute the abutment from the -page. This follows from the following observation. The surface is given by a pushout
where the attaching map is of the form where the -cells are indexed over . In particular we see that suspending this map once, it becomes null-homotopic (higher homotopy groups are abelian). Hence, stably, the top cell of splits off and we can use the Mayer-Vietoris sequence to compute the homology of for any homology theory in terms of the coefficients.
We also want to remark that for the non-orientable surfaces this is not true anymore, and indeed one can show that there is a non-trivial extension in the calculation
Moreover since the coefficients of are only or we see that all -terms of the spectral sequence are free abelian groups. By purely formal reasons there cannot be any non-trivial differential (the only possible ones have target the -axis of the spectral sequence, but since the coefficients in the AHSS always split off, these maps must have trivial image). Hence also the -terms of the spectral sequence are torsion free. In particular there are no extension problems to solve and we can read off the abutment to be
We want to remark that also for other homology theories there cannot occur non-trivial extensions when we want to compute the abutment from the -page. This follows from the following observation. The surface is given by a pushout
where the attaching map is of the form where the -cells are indexed over . In particular we see that suspending this map once, it becomes null-homotopic (higher homotopy groups are abelian). Hence, stably, the top cell of splits off and we can use the Mayer-Vietoris sequence to compute the homology of for any homology theory in terms of the coefficients.
We also want to remark that for the non-orientable surfaces this is not true anymore, and indeed one can show that there is a non-trivial extension in the calculation
Moreover since the coefficients of are only or we see that all -terms of the spectral sequence are free abelian groups. By purely formal reasons there cannot be any non-trivial differential (the only possible ones have target the -axis of the spectral sequence, but since the coefficients in the AHSS always split off, these maps must have trivial image). Hence also the -terms of the spectral sequence are torsion free. In particular there are no extension problems to solve and we can read off the abutment to be
We want to remark that also for other homology theories there cannot occur non-trivial extensions when we want to compute the abutment from the -page. This follows from the following observation. The surface is given by a pushout
where the attaching map is of the form where the -cells are indexed over . In particular we see that suspending this map once, it becomes null-homotopic (higher homotopy groups are abelian). Hence, stably, the top cell of splits off and we can use the Mayer-Vietoris sequence to compute the homology of for any homology theory in terms of the coefficients.
We also want to remark that for the non-orientable surfaces this is not true anymore, and indeed one can show that there is a non-trivial extension in the calculation