Preface

The tim e evolutio n o f some processes , such a s the movement o f th e planets , is very

accurately modele d b y differentia l equations . Suc h accurat e modelin g requires th e identifica -

tion o f a small number o f measurable quantities , like the position s an d velocities o f th e

planets, whose behavior ove r the tim e span considere d is almost independent o f th e neglecte d

factors. Ther e ar e othe r importan t processes , such as the fluctuation o f anima l populations ,

for whic h the identificatio n o f such quantitie s i s not possibl e bu t wher e a rough relation be -

tween th e mor e obviou s variables an d their "rate s o f change " may be evident. Ther e is

some questio n a s to th e valu e o f a differential equatio n mode l fo r suc h processes; in man y

cases it is clear tha t th e predictio n derive d fro m th e equatio n wa s in fact know n beforehan d

and used to writ e dow n th e equatio n itself . O f course thi s is no failing i f on e ca n the n us e

the equatio n t o mak e less obvious predictions .

In an y case , if suc h rough equations ar e to b e o f use it i s necessary t o stud y the m in

rough terms, and tha t i s the ai m o f thes e notes. Th e ideas ar e familia r fro m th e inde x the -

ory o f singular point s o f a vector fiel d o n th e plane . Give n a bounded region, no boundary

point o f whic h is a singular point , an integer ca n b e define d (an d sometime s computed )

which must b e zero if ther e ar e no singula r points inside th e region . Th e integer is an alge-

braic topological invarian t (th e windin g number) whic h depend s onl y o n the behavio r o f th e

vector fiel d o n th e boundary . I n fac t th e boundar y dat a itself nee d no t b e precise. Fo r

example, if th e vecto r fiel d i s deforme d t o a new on e in such a way that n o boundar y poin t

is ever a singular poin t the n th e intege r correspondin g t o th e ne w field i s the sam e a s that o f

the old . Th e integer i s therefore relevan t eve n if th e equation s ar e onl y roughl y known . I n

the approac h here, other algebrai c topologica l invariant s ar e associated t o specia l sets o f solu-

tions o f differentia l equation s an d thes e invariants ar e shown t o have similar "stability "

properties wit h respec t t o change s in th e equations .

In the firs t chapte r th e ideas are describe d heuristicall y an d a t length. Thei r simplicit y

and naturalness ar e supposed t o mak e u p fo r th e absenc e o f definition s an d proofs . Mos t of

the latter ar e supplied i n th e remainin g three chapter s alon g with som e further examples .

In particular , th e omitte d proof s (o f an y length) are wel l represented b y one s that ar e given .

The previou s statement s appl y onl y t o th e mai n subjec t matte r though . I n orde r t o achiev e

a kind o f impressionistic completeness , a n outlin e containin g some o f th e basi c definition s

and theorem s o f algebrai c topolog y i s provided; but n o rea l understanding o f th e language

can be gleane d fro m it .

The note s ar e base d o n lecture s give n i n Boulder , Colorado , i n June , 1976, a t

a Regiona l Researc h Conferenc e supporte d b y th e Nationa l Scienc e Foundation . Th e

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http://dx.doi.org/10.1090/cbms/038/01