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 {{citeD|Wall1999|Chapter 6}}, and / or {{citeD|Ranicki2002|Chapter 12}}) | + | 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 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? | ||
# and of adding a new element which is a linear of combination of the others? | # and of adding a new element which is a linear of combination of the others? | ||
</wikitex> | </wikitex> | ||
| − | == References == | + | <!-- == References == |
| − | {{#RefList:}} | + | {{#RefList:}} --> |
[[Category:Exercises]] | [[Category:Exercises]] | ||
| + | [[Category:Exercises without solution]] | ||
Revision as of 14:47, 1 April 2012
Exercise 1:
Let 
and 
be two sets of ![\mathbb{Z}[\pi_1(X)]](/images/math/6/0/0/6000cfbde265d4e4a836a3979b050aa4.png)
-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?
