Acknowledgements ................................................ 4
Introduction .................................................... 7
1 Preliminaries ............................................... 10
1.1 Stable maps and their friends .......................... 10
1.2 Differential topology .................................. 18
2 E-tame retractions .......................................... 20
2.1 Tame retractions and topological stability ............. 21
2.2 Properties of E-tame retractions ....................... 23
2.3 Combining retractions .................................. 27
2.4 Weighted homogeneous maps and E-tame retractions ....... 34
3 Multigerm equivalences ...................................... 41
3.1 Groups of multigerm equivalences and their maximal
compact subgroups ...................................... 42
3.2 The quotient  ƒ/MC(ƒ) is contractible ................ 47
3.3 Factorization of MC(ƒ) for multigerms ................ 59
3.4 Computing maximal compact subgroups .................... 62
4 Construction of E-tame retractions in examples .............. 69
4.1 E-singularities ........................................ 69
4.2 Z-singularities ........................................ 99
4.3 E-tame retractions for the non-weighted homogeneous
cases ................................................. 111
Conclusion and future work .................................... 111
Index ......................................................... 113
References .................................................... 116
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