Vanishing and nearby boundary cycles of complex non-isolated singularities

Abstract The local topology of singular points in analytic spaces is known, by work of Milnor and others, to be determined by its link $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . In this work, we look at holomorphic map-germs $$(X,{\underline{0}}) \buildrel {f}\over {\rightarrow } ({{\mathbb {C}}},0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:munder> <mml:mn>0</mml:mn> <mml:mo>̲</mml:mo> </mml:munder> <mml:mo>)</mml:mo> </mml:mrow> <mml:mover> <mml:mo>→</mml:mo> <mml:mi>f</mml:mi> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where X is an arbitrary reduced and equidimensional complex analytic space in some $${\mathbb {C}}^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> </mml:math> . We compare the topology of the link $$L_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> with that of the boundary of a local non-critical level, i.e., with the boundary of its Milnor fiber $$F_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> . We study how the topology of the boundary $$\partial F_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> </mml:math> changes as the non-critical level degenerates to the critical one.