高等数学(16)

934A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUEL

WEINBERGER

ThisgivesacoverORofcXbyopensetsofdiameter 5RwithLebesguenumber 1R.Considerthenerveofthiscover,restrictingattentionforthemomenttotwoconsecutivelevelsasinthepictureabove.Wewillcalltheopencovercorrespondingtopointsinthe(k+1)R-levelUandthecovercorrespondingtopointsatthekR-levelV.

ForU∈UandV∈V,wewillletU andV denotetheintersectionsofUandVwiththepartofcXbetweentheconepointandthekR-level.

Lemma7.3.IfU1∩...∩Uk∩V1∩...∩V = ,thenU1 ∩...∩Uk ∩V1 ∩...∩V = .

Proof.Thisisclear,sinceifapointxabovethekR-levelisinthe rstintersection,thenfollowingtheconelinesbacktothekR-levelgivesapointinthesecondintersection.

For{ k}small,thereisamapφ:U→Vsothat

( )U φ(U) forallU∈U

whichisde nedbychoosingφ(U)tobeanelementV∈VsothatU∩kR V∩kR.De neamapρ:N(U∪V)→N(V)byρ(V)=Vandρ(U)=φ(U).

Proposition7.4.

retraction.Themapρde nesasimplicialstrongdeformation

Proof.Toseethatρissimplicial,supposethatwehaveasimplex

U1,...,Un,V1,...,Vk ∈N(U∪V).

Byde nition,

U1∩...∩Un∩V1∩...∩Vk= ,

so

U1 ∩...∩Un ∩V1 ∩...∩Vk = ;

whence( )guaranteesthat

φ(U1)∩...∩φ(Un)∩V1∩...∩Vk= ,

soρde nesasimplicialmap.

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