Generator sets and the kernel formation (Ex)
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'''Exercise 1:''' | '''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 \cite{ | + | 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 \cite{Wall1999|Chapter 6}, and/or \cite{Ranicki2002|Chapter 12}) |
# What is the effect on the kernel formation of $(f, b)$ of adjoining or deleting a zero? | # What is the effect on the kernel formation of $(f, b)$ of adjoining or deleting a zero? | ||
# What is the effect of permuting elements? | # What is the effect of permuting elements? |
Latest revision as of 22:57, 25 August 2013
Exercise 1: Let and be two sets of -module generators for the kernel module . These are related by a sequence of elementaray operations. (See [Wall1999, Chapter 6], and/or [Ranicki2002, Chapter 12])
- What is the effect on the kernel formation of 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?