Lecture s on Hilber t Cub e Manifold s

Th e goa l of thes e lecture s is to presen t an introductio n to the geometri c topolog y of the

Hilber t cub e Q an d separabl e metri c manifold s modele d on Q , whic h we :ai l Hilbert cube

manifolds or Q-manifo/ds. In the pas t te n year s ther e ha s bee n a grea t dea l of researc h on Q

an d Q-manifold s whic h is scattere d throughou t severa l paper s in the literature . We presen t

her e a self-containe d treatmen t of onl y a fe w of thes e result s in the hop e tha t it wil l stimulat e

furthe r interes t in this area . No ne w materia l is presente d her e an d no attemp t ha s bee n mad e

to be complete . Fo r exampl e we hav e omitte d the importan t theore m of Schori-Wes t statin g

tha t the hyperspac e of close d subset s of [0,1 ] is homeomorphi c to Q . In an appendi x (prepare d

independentl y by R. D. Anderson , D. W. Curtis , R. Schor i an d G . Kozlowski ) ther e is a list of

problem s whic h are of curren t interest . Thi s include s problem s on Q-manifold s as wel l as mani -

folds modele d on variou s linea r spaces . We refe r the reade r to this for a muc h broade r perspec -

tive of the field .

In som e vagu e sens e Q-manifol d theor y seem s to be a "stable " PL r-manifol d theory .

Thi s become s more precis e in ligh t of the Triangulatio n an d Classificatio n theorem s of Chapter s

X I an d XII . In particular , al l handle s ca n be straightene d an d consequentl y al l Q-manifold s

ca n be triangulated . Thu s ther e are delicat e finite-dimensiona l obstruction s whic h do no t

appea r in Q-manifol d theory . Thi s is perhap s wh y the proof s of the topologica l invarianc e of

Whitehea d torsio n (Chapte r XII ) an d the finitenes s of homotop y type s of compac t ANR s

(Chapte r XIV ) first surface d at the Q-manifol d level .

In the first fou r chapter s we presen t the basi c tool s whic h are neede d in al l of the remainin g

chapters . Beyon d this ther e see m to be a t leas t two possibl e course s of action . Th e reade r wh o

is intereste d onl y in the triangulatio n an d classificatio n of Q-manifold s shoul d rea d straigh t

throug h (avoidin g onl y Chapte r VI) . In particula r the topologica l invarianc e of Whitehea d torsio n

appear s in §38 . Th e reade r wh o is intereste d in R. D. Edwards ' recen t proo f tha t ever y AN R

is a Q-manifol d facto r shoul d rea d the first fou r chapter s an d the n (with the singl e exceptio n of

26.1 ) ski p ove r to Chapter s XII I an d XIV .

Thes e lecture s wer e delivere d in October , 1975 , a t Guilfor d Colleg e as par t of the Regiona l

Conferenc e Progra m sponsore d by the Conferenc e Boar d of the Mathematica l Science s wit h the

suppor t of the Nationa l Scienc e Foundation . I wis h to expres s my appreciatio n to the Conferenc e

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