Generator sets and the kernel formation (Ex)

Revision as of 00:23, 27 March 2012

Exercise 1: Let $<wikitex>; '''Exercise 1:''' Let $\{x_1, x_2, \dots, x_k\}$ and $\{y_1, y_2, \dots, y_l\}$ be two sets of $\mathbb{Z}[\pi_1(X)]$-module generators for the kernel module $K_n(M)$. These are related by a sequence of elementaray operations. (See {{citeD|Wall1999|Chapter 6}}, and / or {{citeD|Ranicki2002|Chapter 12}}) # What is the effect on the kernel formation of $(f, b)$ of adjoining or deleting a zero? # What is the effect of permuting elements? # and of adding a new element which is a linear of combination of the others? </wikitex> == References == {{#RefList:}} [[Category:Exercises]]\{x_1, x_2, \dots, x_k\}$ and $\{y_1, y_2, \dots, y_l\}$ be two sets of $\mathbb{Z}[\pi_1(X)]$-module generators for the kernel module $K_n(M)$. These are related by a sequence of elementaray operations. (See [Wall1999, Chapter 6], and / or [Ranicki2002, Chapter 12])

1. What is the effect on the kernel formation of $(f, b)$ of adjoining or deleting a zero?
2. What is the effect of permuting elements?
3. and of adding a new element which is a linear of combination of the others?

References