How the Staircase Diagram changes when we pass to derived couple (Spectral Sequence)

Set A_{n,p}^1=H_n(X_p) and E_{n,p}^1=H_n(X_p,X_{p-1}). The diagram then has the following form:

When we pass to the derived couple, each group A_{n,p}^1 is replaced by a subgroup A_{n,p}^2=\text{Im}\,(i_1: A_{n,p-1}^1\to A_{n,p}^1). The differentials d_1=j_1k_1 go two units to the right, and we replace the term E_{n,p}^1 by the term E_{n,p}^2=\text{Ker}\, d_1/\text{Im}\,d_1, where the d_1‘s refer to the d_1‘s leaving and entering E_{n,p}^1 respectively.

The maps j_2 now go diagonally upward because of the formula j_2(i_1a)=[j_1a]. The maps i_2 and k_2 still go vertically and horizontally, i_2 being a restriction of i_1 and k_2 being induced by k_1.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s