Generator sets and the kernel formation (Ex)

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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 [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?
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