Preface t o Lectur e 1 The singularit y theor y approac h t o topologica l classificatio n o f object s o f an y nature follows a standard strategy going back to Poincare. On e considers the infinite- dimensional spac e o f object s unde r consideration , includin g generi c object s a s wel l as degenerat e objects . Degenerat e object s for m a hypersurfac e ( a codimensio n on e subvariety) i n thi s infinite-dimensional space . Thi s discriminant hypersurface divide s the infinite-dimensional spac e F int o parts. Eac h connected componen t o f the com - plement t o th e discriminant hypersurfac e Z consists o f the nondegenerate object s o f the sam e topologica l typ e (havin g th e sam e discret e invariants) . The invariants, which form the ring H°{F \L) , the zero-dimensional cohomolog y classes o f th e complement , ar e locall y constan t function s o n th e complement . T o study th e cohomology o f the complement t o the discriminant hypersurfac e on e may use the infinite-dimensional version s of Alexander duality (provide d that the infinite - dimensional spac e F itsel f i s homologicall y sufficientl y simple) . Namely , on e ma y use the natura l stratificatio n o f th e discriminan t hypersurfac e accordin g t o differen t singularities o f analyze d object s t o obtai n informatio n abou t th e (co)homolog y o f the discriminan t hypersurface , an d the n b y dualit y t o transfor m i t into informatio n about th e (co)homolog y o f th e complement . This general approach was used with great succes s in recent work of Vassiliev on knot invariants. However , the potential applications of this method to many problems in th e topolog y o f space s o f mapping s an d varietie s i n rea l an d comple x geometr y are stil l waiting courageou s researchers . 3 http://dx.doi.org/10.1090/ulect/005/01

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