Let $\mathcal{P}$ be a finite presentation of some group. When we apply some Nielsen transformations on $\mathcal{P}$, will the homotope type of the presentation complex $K\_{\mathcal{P}}$ of $\mathcal{P}$ always be preserved?



Do Nielsen transformations on a presentation preserve the homotopy type of the corresponding presentation complex?
Yes. In fact, slightly more is true: the **simple** homotopy type of the presentation 2-complex is preserved under Nielsen transformations. For a proof of this fact, see
Micheal N. Dyer and Allan J. Sieradski, *Trees of homotopy types of two-dimensional CW-complexes. I.*, Comment. Math. Helv. **48** (1973), 31–44. [MR0377905](http://www.ams.org/mathscinet-getitem?mr=MR0377905)


