Piecewise-regular maps

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Math. Ann. (2018) 372:1545–1574 https://doi.org/10.1007/s00208-017-1607-2

Piecewise-regular maps

Wojciech Kucharz¹

Received: 27 April 2017 / Revised: 24 September 2017 / Published online: 20 October 2017

AbstractLetV,Wbe real algebraic varieties (that is, up to isomorphism, real alge- braic sets), andX Vsome subset. A map fromXintoWis said to beregularif it can be extended to a regular map defined on some Zariski locally closed subvariety ofVthat containsX. Furthermore, a continuous mapf X Wis said to bepiecewise- regularif there exists a stratificationSofVsuch that foreverystratumSSthe restriction offto each connected component ofX Sis a regular map.

ByastratificationofVwemeanafinitecollectionofpairwisedisjointZariskilocally closed subvarieties whose union is equal toV. Assuming that the subsetXofVis compact, we prove thateverycontinuous map fromXinto a Grassmann variety or a unit sphere can be approximated by piecewise-regular maps. As an application, we obtain a variant of the algebraization theorem for topological vector bundles. If the varietyVis compact and nonsingular, we prove that each continuous map fromVintoaunitsphereishomotopictoapiecewise-regularmapofclassC^k,wherekisan arbitrarynonnegative integer.

Mathematics Subject Classification14P05·14P99·57R22

Communicated by Ngaiming Mok.

BWojciech KucharzWojciech.Kucharz@im.uj.edu.pl

1Faculty of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Mathematische

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1 Introduction

Inthispaper,byarealalgebraicvarietywemeanalocallyringedspaceisomorphicto

analgebraicsubsetofR^m,forsomem,endowedwiththeZariskitopologyandthesheaf ofreal- valuedregularfunctions(suchanobjectiscalledanaffinerealalgebraicvariety

in[6]).Theclassofrealalgebraicvarietiesisidenticalwiththeclassofquasi-projective realalgebraicvarieties,cf.

[6,Proposition3.2.10,Theorem3.4.4].Morphismsofreal algebraic varieties are calledregularmaps. Each real algebraic variety carries also theEuclideantopology,inducedbythestandardmetriconR.Unlessexplicitlystated

otherwise,wealwaysassumethatrealalgebraicvarietiesandtheirsubsetsareendowed with the Euclideantopology.

Topologicalproperties of regular maps and their applications to algebraization of topologicalvectorbundleswereinvestigatedinnumerouspapers[2–14,16–25,27,32–

35,37,38,40–43,47–49,51,53,56,64,65,68,71,72,76–80,82,83]. In general, regular mapsaretoorigidtoreflectadequatelytopologicalphenomena.Itisthereforedesirable

tointroducemapswhichhavemanygoodfeaturesofregularmapsbutaremoreflexible.

We first generalize the definition of regular map.

Definition 1.1LetV,Wbe real algebraic varieties,X Vsome (nonempty) subset, andZthe Zariski closure ofXinV.

A mapf:X→Wis said to beregularif there exist a Zariski open neighborhood Z0⊆ZofXand a regular mapf˜:Z0→Wsuch thatf˜|X=f.

The next step requires the concept of stratification. By astratificationof a real algebraic varietyVwe mean a finite collection of pairwise disjoint Zariski locallyclosed subvarieties (some possibly empty) whose union is equal toV.

Definition 1.2LetV,Wbe real algebraic varieties,f X Wa continuous map defined on some subsetX V, andSa stratification ofV.

The mapfis said to beS-regularif foreverystratumSSthe restriction offtoX Sis a regular map. Also,fis said to bepiecewiseS-regularif foreverystratumSSthe restriction offto each connected component ofX Sis a regular map (whenX S∅ ).

Moreover,fis said to bestratified-regular(resp.piecewise-regular) if it isT- regular (resp. piecewiseT-regular) for some stratificationTofV.

Essentially,thesenotionsdonotdependontheambientvarietyV.Moreprecisely,

supposethatVisaZariskilocallyclosedsubvarietyofarealalgebraicvarietyV. The mapf X WisS-regular (resp. piecewiseS-regular) if and only if it isS- regular(resp.piecewiseS-regular),whereSi s thestratificationofVd e f i n e d b y S SV V V VwithVthe Zariski closure ofVinV.Conversely,given a stratificationPofV, the mapfisP-regular (resp. piecewiseP-regular) ifandonlyifitisP(V)-regular(resp.piecewiseP(V)-regular),whereP(V)isthe stratification ofVdefined byP(V)V P PP. Thus, in the definition of stratified- regularmap(resp.piecewise-regularmap)itdoesnotmatterwhetherXis

regardedasasubsetofVorasasubsetofV.

Evidently, each stratified-regular map is piecewise-regular, whereas the converse is not always true. General properties of these two classes of maps and relationships

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betweenthemarediscussedinSect.2.Stratified-regularmapsandfunctionsarethor- oughly investigated, in a more restrictive framework, in[4,29,30,44,45,50,52,54,

55,58,60–62,66,74,75,84],where they sometimes appear under different names (cf.

Remark2.5).IfXisasemialgebraicset,theneachpiecewise-regularmapdefinedonXis semialgebraic.

Theorem2.9provides a nontrivial characterization of piecewise- regularmapsamongsemialgebraicones.

Inthissectionweconcentrateontopologicalpropertiesofpiecewise-regularmaps.

WithnotationasinDefinition1.2,weletC(X,W)denotethespaceofallcontinuous

mapsfromXintoW,endowedwiththecompact-opentopology.Wesaythatthemapf can be approximated by piecewiseS-regularmapsifeveryneighborhood offinC(X,W)containsapiecewiseS-regularmap.Approximationoff b y mapsofother types (regular,S-regular, stratified-regular, piecewise-regular, etc.) is defined in the analogousway.

We pay special attention to maps with values in Grassmannians. We letFstand forR,CorH(the quaternions). When convenient,Fwill be identified withR^d(F), whered(F)dimRF. We will consider only leftF-vector spaces, which plays a role ifF Hsince the quaternions are noncommutative. For any integersrandn, with

0r n, we denote byGr(Fⁿ)the Grassmann space ofr-dimensionalF-vector subspaces ofFⁿ. As in [6, Sections 3.4 and 13.3],Gr(Fⁿ)will be regarded as a real algebraic variety. The disjoint union

is also a real algebraicvariety.

n

G(Fⁿ)= Gr(Fⁿ)

r=0

Theorem 1.3Let V be a real algebraic variety and let X V be a compact subset.Then, for each positive integer n, every continuous map from X intoG(Fⁿ)can beapproximated by piecewise-regular maps.

Under an additional assumption onX, we also have a stronger result.

Theorem 1.4Let V be a real algebraic variety and let X V be a compact locallycontractible subset. Then there exists a stratificationSof V such that, for each

positive integer n, every continuous map from X intoG(Fⁿ)can be approximated bypiecewiseS-regular maps.

Virtuallyalltopologicalspacesoneencountersinrealalgebraicgeometryarelocally contractible; for example, any semialgebraic set is locally contractible, cf. [6, Theo- rem9.3.6].

The proofs of Theorems1.3and1.4, given in Sect.4, are based on some fairly explicit constructions.

It is well-known that maps with values inG(Fⁿ)encode information on algebraic andtopologicalF-vectorbundles,cf.[6,39].Thisisalsothecaseforstratified-algebraicF-vector bundles introduced in [61] and further investigated in[57,59,63,66].Theo- rems1.3and1.4haveabearingonF-vectorbundlesaswell,whichiselaboratedupon

inSect.5.ThemainresultsofSect.5areTheorems5.10and5.11.

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Se

∼ Sⁿ=I(u0,...,un)∈Rⁿ⁺¹:u²+··· +u²=1.

Theorem 1.5Let V be arealalgebraic variety and let X V be a compact sub-set.

Then, for each positive integern,every continuous mapfromX intoSⁿcan beapproximatedbypiecewise-regularmaps.

Theorem1.5,whichisprovedinSect.6,impliesthateachcontinuousmapfromXintoSⁿis homotopic to a piecewise-regular map.However,the following homotopyresult requires a differentproof.

Theorem 1.6Let V be a compact nonsingularrealalgebraicvariety,n a positiveinteger,and f VSⁿa continuous map. Thenthereexists a stratificationSof Vsuch that, for each nonnegative integerk,the map f is homotopic to a piecewiseS- regular map g:V→ Sⁿof classC^k.

We prove Theorem1.6in Sect.7. It turns out that a suitable stratificationSofV is quite simple and consists of at most 3 strata.

Theorems1.3,1.5and1.6are optimal in the sense explained in the followingexample.

Example 1.7LetWbe a compact nonsingular real algebraic variety. A cohomology class inH^q(WZ/2)is said to bealgebraicif the homology class Poincaré dual to it can be represented by a Zariski closed subvariety ofWof codimensionq, cf. [3,6].

The setH^q(W; Z/2)of all algebraic cohomology classes inH^q(W; Z/2)forms a subgroup.alg viously,theuniquegeneratorsofH^q(Sq;Z/2)=Z/2isanalgebraic

Ob q

cohomology class.

TherealalgebraicvarietiesG1(F²)andS^d(F)arecanonicallybiregularlyisomorphic

andwillbeidentified.Foranypositiveintegerm,letT^mS1 S¹bethem- foldproduct ofS¹.Clearly,

T^m⊆ R^2m= R^2m× {0}⊆ R^2m+1. We will regardT^mas a subset ofR^2m+1.

Fixanintegerm≥d(F)+1,andlety0beapointinT^m−d(F).Letαbethehomology classinHd(F)

(T^m;Z/2)representedbytheC^∞submanifoldN:=T^d(F)×{y0}⊆T^m. G:= {u∈H^d(F)(T^m; Z/2):(u,α)=0},

where(u, α)stands for the Kronecker product. Letπ: T^m= T^d(F)×T^m−d(F)→ T^d(F)bethecanonicalprojectionandletτ:T^d(F)→S^d(F)beaC^∞mapoftopological

degree1.FortheC^∞mapg:=τ◦π:T^m→S^d(F),wehave g^∗(sd(F))∈/G.

Since the normal bundle toNinT^mis trivial andNis the boundary of a compactC^∞ manifold with boundary, it follows from [23, Proposition 2.5, Theorem 2.6] that there

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exist a nonsingular real algebraic varietyVand aC^∞diffeomorphismσV T^m with

The map is of classC^∞,and

H^d(F)(V; Z/2)⊆σ^∗(G).

h:=g◦σ:V→ G1(F²)= S^d(F) h^∗(sd(F))∈/H^d(F)(V;Z/2).

By[61,Propositions7.2and7.7],hisnothomotopictoanystratified-regularmap.In

particular,hcannot be approximated by stratified-regular maps, which is of interest in view of Theorems1.3and1.5. Also, by Proposition2.13,his not homotopic to any piecewise-regular map of classC^∞;thus thecasek cannot

beincludedinTheorem1.6.

Furthermore, according to [15, Proposition 1.2],Vas above can be assumed to be aZariskiclosedsubvarietyofR^2m+1thatisobtainedfromT^mviaanarbitrarilysmallC^∞isotopy.

From a different viewpoint, strengthening of Theorems1.3,1.5and1.6might be possible.Givena compact nonsingular real algebraic varietyVand two positive integersnandk,it remains an open question whethereverycontinuous map fromVintoG(Fⁿ)orS^ncanbeapproximatedbypiecewise-regularmapsofclassC^k.

Example 1.8Stratified-regular maps oftenhavebetter approximation and homotopy propertiesthanregularones.Forinstance,ifVisacompactnonsingularrealalgebraic variety of dimensionn1, theneverycontinuous map fromVi n t oSⁿcan be

approximated by stratified-regular maps, cf. [55, Corollary 1.3]. On the other hand, ifnis even, then each regular map fromTⁿintoSⁿis null homotopic, cf. [8] or [6, Theorem 13.5.1].

Piecewise-regular maps are not always more flexible than regular maps.

Example 1.9LetFrbe the Fermat curve of degreerin the real projective planeP²(R), Fr:= {(x:y:z)∈ P²(R):x^r+y^r=z^r}.

Clearly,F2canbeidentifiedwithS¹.Ifs>r2 , theneverypiecewise-regularmap fromFrintoFsisconstant.Thisclaimholdssince,byvirtueoftheHurwitz–Riemann

theorem[31,p.140],everyrationalmapfromFrintoFsisconstant.

ItwouldbeofinteresttodecidewhetherornotcounterpartsofTheorems1.3and1.5hold for maps with values in an arbitrary rational nonsingular real algebraicvariety.

Wehavealready indicatedhowthe present paper is organized. It should be added that Sect.3contains some preliminary technical results.

Henceforth, the following notation will be frequently used.

Notation 1.10For any functionf:n→ Rdefined on some setn, we put Z(f):= {x∈n:f(x)=0}.

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2 General properties of piecewise-regularmaps We first deal with regular maps in the sense of Definition1.1.

Lemma 2.1Let V be arealalgebraicvariety,W R^pa Zariskiclosed subvariety,and

f=(f1,...,fp):X→W⊆ R^p

a map defined on some subset X⊆V . Then the following conditions are equivalent:

(a) The map f isregular.

(b) Each component function fj:X→ Risregular,j=1,..., p.ProofIt is clear that (a) implies(b).

Suppose that (b) holds, and letZbe the Zariski closure ofXinV. We can find a Zariski open neighborhoodZ0⊆ZofXand regular functionsf˜j:Z0→ Rsuch

thatf˜j|^X=f^jforj=1, . . . ,p.Theregularmapf˜=(f˜¹, . . . ,f˜p):Z0→R^p iscontinuousintheZariskitopologyandf˜(X)⊆W .Hencef˜(Z0)⊆W ,which

implies(a). nu

Regular functions can be characterized as follows.

Lemma 2.2Let V⊆ Rⁿbe a Zariski closed subvariety, and f:X→ Ra functiondefined on some subset X⊆V . Then the following conditions are equivalent:

(a) The function f isregular.

(b) Foreach point x∈X there exist a Zariski open neighborhood Vx⊆V of x andaregularfunctionFx:Vx→ Rsuch thatFx=f on X∩Vx.

(c) Thereexist tworegularfunctionsϕ,ψ:V→ Rsuch that X⊆V\Z(ψ)a n d

f=ϕ/ψonX. n

(d) Foreach point x∈X there exist a Zariski open neighborhood Ux⊆ Rofx and two polynomial functions Px,Qx: Rⁿ→ Rsuch that Ux⊆ Rⁿ\Z(Qx)and

f=Px/Qxon X∩ Ux. n n

(e) Thereexist two polynomial functionsP,Q:R and f=P/Q on X.

ProofIt readily follows that

→ Rsuch that X⊆ R \Z(Q)

(a)⇒(b)⇒(d),(e)⇒(c)⇒(a),and (e)⇒(d).

Supposethat(d)holds.Foreachpointx∈X,pickapolynomialfunctionSx: Rn→RwithZ(Sx)=Rⁿ\Ux. Since the Zariski topology onRⁿis Noetherian, we can find a finite subset{x1,...,xr} ⊆ Xsuch that

X⊆U:=Ux₁∪···∪Ux_r. SetPi=Px_i,Qi=Qx_i,Si=Sx_i, and

r r

P=S²Q

iPⁱ, Q=S²Q2.

i=1 i=1

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ThenU= Rⁿ\Z(Q)andS²Qif=S²PionX. Consequently,f=P/QonX,

hence(e)holds.Theproofiscomplete. nu

AfiltrationofarealalgebraicvarietyVisafinitesequenceF (V0,V1,...,Vm1) of Zariski closed subvarieties ofVsatisfying

V=V0⊇V1⊇···⊇Vm+1= ∅.

We allowVi=Vi+1for somei. Note thatF:= {Vi\Vi+1:0≤i≤m}is a stratifi-The following is a generalization of [61, Proposition 2.2].

Proposition 2.3Let V , W be real algebraic varieties, and f:X→W a map definedon some subset X⊆V . Then the following conditions are equivalent:

(a) There exists a stratificationSof V such that for every stratum S∈Stherestriction of f to X∩S is aregularmap.

(b) ThereexistsafiltrationFofVsuchthatforeverystratumT∈ Ftherestrictionof f to X∩T is aregularmap.

(c) Forevery Zariski closed subvariety Z⊆V ,thereexists a Zariski open densesubset Z⁰⊆Z such that the restriction of f to X∩Z⁰is aregularm a p . Inparticular,themapf i s stratified-regularifandonlyifitiscontinuousandsatisfiesthe equivalent conditions(a),(b),(c).

ProofIt is clear that (b) implies (a).

Suppose that (a) holds, and letZ Vbe an irreducible Zariski closedsubvariety.Wecan find a stratumSSsuch that the intersectionS Zis nonempty and Zariskiopen(henceZariskidense)subsetofZ.Thus(c)holdsforeachirreducibleZ. It immediately follows that (c) holds in the generalcase.

Nowsupposethat(c)issatisfied.SetV0=V.Makinguseof(c)withZ=V0,wefindaZariskiclo sednowheredensesubvarietyV1⊆V0suchthattherestrictionofftoX∩(V0\V1)is a regular map.

Note that dimV1<dimV0.WerepeatthisconstructionwithZ=V1togetV2⊆V1,andsoon.Thisprocesst erminatesafterfinitelymanystepswithVm+1=∅,whichproves(b).

nu LetV,Wbe real algebraic varieties,XVsome subset, andZthe Zariski closure ofXinV.Wesay that a mapf X Wisrationalif there exists a Zariski open dense subsetZ⁰Zsuch that the restriction offtoX Z⁰is a regular map (no condition on the restriction offtoX(XZ⁰)isimposed).

In view of Proposition2.3, each stratified-regular map is continuous rational. On the other hand, if the set Sing(V)of singular points ofV(that is, the complement inVof the locus of all nonsingular points ofV) is nonempty, then it can happen that a function fromVintoRis continuous rational but it is not stratified-regular, cf.

[45,Example 2]. However, the following holds.

Proposition 2.4Let V , W be real algebraic varieties, and f:U→W a map definedon an open subset U⊆V\Sing(V). Then the following conditions are equivalent:

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(a) The map f isstratified-regular.

(b) The map f is continuous andr a t i o n a l .

ProofWemay assume thatW⊆R^pis a Zariski closedsubvariety.Hence, by Lemma2.1, the proof is reduced to the caseW=R, which follows from [44, Propo-sition4.2]

(avariantof[45,Proposition8]).

nu

It is worthwhile to make a comment on the terminology used in different papers.

Remark 2.5In view of Proposition2.4, reading papers[50,52,54,55,58,60]one can substituteeverywherestratified-regularmapsforcontinuousrationalmaps.Itfollows

fromProposition2.3thatstratified-regularfunctionscoincidewithcontinuoushered-

itarilyrationalfunctionsstudiedin[44,45].Furthermore,asexplainedin[29,61,66],stratified -regularmapsdefinedonaconstructiblesubsetofarealalgebraicvarietyare identical withregulousmaps.

We now present a counterpart of Proposition2.3for piecewise-regular maps.

Proposition 2.6Let V , W be real algebraic varieties, and f:X→W a map definedon some subset X⊆V . Then the following conditions are equivalent:

(a) There exists a stratificationSof V such that for every stratum

S∈Stherestriction of f to each connected component of X∩S is aregularmap.

(b) ThereexistsafiltrationFofVsuchthatforeverystratumT∈ Ftherestrictionof f to each connected component of X∩T is aregularmap.

(c) Forevery Zariski closed subvariety Z⊆V there exists a Zariski open densesubset Z⁰⊆Z such that the restriction of f to each connected component ofX∩Z⁰isaregularmap.

Inparticular,themapfispiecewise-regularifandonlyifitiscontinuousandsatisfiesthe equivalent conditions(a), (b),(c).

ProofOnecanrepeattheproofofProposition2.3withonlyminormodifications.nuWealso havethe following characterization of piecewise-regular maps.

Proposition 2.7Let V , W be real algebraic varieties, and f:X→W a continuousmap defined on some subset X⊆V . Then the following conditions are equivalent:

(a) The map f ispiecewise-regular.

(b) There exists a stratificationSof V such that for every stratum S∈Stherestriction f|X∩S:X∩S→W is a piecewise-regular map.

ProofIt is clear that (a) implies (b).

Suppose that (b) holds for some stratificationSofV. For each stratumS∈Sthere exists a stratificationTSofVsuch that for every stratumT∈TSthe restriction offto each connected component ofX∩S∩Tis a regular map. Note that

P:={S∩T:S∈SandT∈TS}

is a stratification ofV, and the mapfis piecewiseP-regular. Thus (b) implies (a),asrequired.

nu

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Given a real algebraic varietyV, a subsetA⊆Vis said to be anonsingularalgebraic arcif its Zariski closureCinVis an algebraic curve (that is, dimC=1),A⊆C\Sing(C), andAis homeomorphic toR.

Proposition 2.8Let V , W berealalgebraic varieties, and f X W a continuousmap defined on a semialgebraic subset X V . Then the following conditionsareequivalent:

(b) There exists a stratificationSof V such that for every stratum SSand everynonsingular algebraicarcA in V , with A XS,the restriction fAis aregularmap.

ProofWemay assume thatWR^pis a Zariski closedsubvariety.Hence, by Lemma2.1, the proof is reduced to the caseWR.

It is clear that (a) implies (b).

Suppose that (b) holds for some stratificationSofV. In view of [44, Proposi- tion 3.5], foreverystratumSS, the restrictionfX Sis a piecewise-regular map.

Consequently, by Proposition2.7,fis a piecewise-regular map. Hence (b) implies(a),asrequired.

nu

Piecewise-regularmapscanbecharacterizedamongsemialgebraicmapsasfollows.

Theorem 2.9Let V , W berealalgebraic varieties, X V a semialgebraic subset,and f X W a continuous semialgebraic map. Then the following conditionsareequivalent:

(b) Forevery nonsingular algebraicarcA in V ,withA X,the restriction fAisapiecewise-regularmap.

(c) ForeverynonsingularalgebraicarcAinV,withA⊆X,thereexistsanonemptyopen subset A0⊆A such that the restriction f|A0is aregularmap.

ProofAs in the proof of Proposition2.8, we may assumethatW R.

Evidently, (a) implies (b), and (b) implies(c).

Suppose that (c) holds, and letZ Vbe a Zariski closedsubvariety.LetYbe the Zariski closure ofX ZinV. By Lemma2.11below (withXreplaced byX Z), there exists a Zariski open dense subsetY⁰Ysuch that the restriction offto each connected component of(XZ)Y⁰is a regular function. Note thatZ⁰Z(Y Y⁰)isaZariskiopendensesubsetofZ,and

(X∩Z)∩Y⁰=X∩Z⁰.

Hence,inviewofProposition2.6,condition(a)holds. nu

For background on Nash manifolds and Nash functions we refer to [6]. The following variant of [44, Propositon 2.5] will be useful in the proof of Lemma2.11.

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Lemma 2.10Let N⊆ Rⁿbe a connected Nash submanifold, and f:N→ Ra Nashfunction.AssumethatforeverynonsingularalgebraicarcAinRⁿ,withA⊆N,thereexists a nonempty open subset A0A such that the restriction fA₀is aregularfunction. Then f is a rationalf u n c t i o n .

ProofLetVbetheZariskiclosureofNinR^n.NotethatVisirreducible.Furthermore,

theZariskiclosureofthegraphoff i n V×Risalsoirreducible.Complexifyingthesedata,wecompletet heproofarguingasin[44,Proposition2.5].

nu

In the proof of Theorem2.9we used the following.

Lemma 2.11Let V be arealalgebraicvariety,X⊆V a semialgebraic set, andf:X→

Ra semialgebraic function. Let Y be the Zariski closure of X in V . AssumethatforeverynonsingularalgebraicarcAinV,withA⊆X,thereexistsanonemptyopen subset A0⊆A such that the restriction f|A0is aregularfunction. Thenthereexists a Zariski open dense subset Y⁰⊆Y such that the restriction of f to eachconnectedcomponentofX∩Y⁰isaregularfunction.

ProofWe may assume thatVRⁿis a Zariski closed subvariety. SetY0

YSing(Y), and letX^∗be the interior ofXY0inY0.ThenX X^∗is a semialgebraic subset ofYwhose Zariski closure is Zariski nowhere dense inY, cf. [6, Chapter 2].Clearly,eachconnectedcomponentofX^∗isaNashmanifold.Therefore,thereexists

a Zariski closed and Zariski nowhere dense subvarietyS⊆Ysuch that (X\X^∗)∪Sing(Y)⊆S

andtherestrictionoff to eachconnectedcomponentofXSisaNashfunction,cf.forexample[28, (2.4.1)].SinceXSisasemialgebraicset,ithasfinitelymanyconnectedcomponents.Hence,inviewofLe mma2.10,thereexistsaZariskiopendensesubsetY^0⊆Ywhichhastherequiredproperties.

nu We next deal with piecewise-regular maps of classC^∞. Initially, we

considerfunctions on nonsingular real algebraic arcs.

Lemma 2.12Let C be a real algebraic curve, A C Sing(C)a nonsingular realalgebraic arc, and f ARa piecewise-regular function of classC^∞. Then f is aregular function.

ProofT h e functionf i s analytic,beingsemialgebraicandofclassC^∞,cf.[6,Propo- sition8.1.8].

Bydefinitionofpiecewise- regularmap,wecanfindaZariskiopendensesub-setC⁰⊆C\Sing(C), a regular functionϕ:C⁰→ Rand a nonemptyo p e n subsetU⊆A∩C⁰such thatf|U=ϕ|U. RegardingRas a subset ofP¹(R), we getareg-ularmapψ:C\Sing(C)→P¹(R)withψ|

C0=ϕ.Hencef=ψ|

Abytheidentityprincipleforanalyticmaps.Consequently,fisaregularfunction.

nu LetV,Wbe real algebraic varieties and letA⊆V,B⊆Wbe arbitrary subsets.We say that a mapg:A→Bis ofclassC^∞if for some algebraic embeddings

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˜: → ˜ =

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ϕVRⁿ,ψWR^pthere exist an open neighborhoodURⁿofϕ(A)and aC^∞mapg UR^psuch thatg(ϕ(x)) ψ(g(x))for allx A. If this holds, then the same condition holds for any choice of the algebraic embeddingsϕandψ.

Lemma2.12can be generalized as follows.

Proposition2.13L e t V,Wberealalgebraicvarieties,andf:U→ Wamapdefinedonanopensubset U⊆V\Sing(V).Thenthefollowingconditionsareequivalent:

(a) The map f is piecewise-regular and of classC^∞.

(b) The restriction of f to each connected component of U is aregularm a p . ProofAs in the proof of Proposition2.8, we may assume thatWR.

Suppose that (a) holds and letU0be a connected component ofU. For each non- singular algebraic arcAinV, withA⊆U0, the restrictionf|Ais a regular function by Lemma2.12. Hence, in view of [44, Theorem 2.4],f|U0is a rational function.

Consequently,f|U0is a regular function according to [50, Proposition 2.1].

Itisclearthat(b)implies(a). nu

We also have the following variant of Proposition2.13.

Proposition2.14L e t V,Wberealalgebraicvarieties,andf:U→ Wamapdefinedonanopensubset U⊆V\Sing(V).Thenthefollowingconditionsareequivalent:

(a) The map f ispiecewise-regular,and for every nonsingular algebraicarcA in V ,withA U,the restriction fAis of classC^∞.

(b) TherestrictionofftoeachconnectedcomponentofUisastratified-regularmap.

ProofAs in the proof of Proposition2.8, we may assume thatWR. It suffices to consider the case whereUis connected.

Suppose that (a) holds. For each nonsingular algebraic arcAinV, withA U, the restrictionfAis a regular function by Proposition2.13. Hence, in view of [44, Theorem 2.4],fis a rational function. Consequently,fis a stratified-regular function according to Proposition2.4.

Itisclearthat(b)implies(a). nu

3 Functionson asimplex

Thissectionisofatechnicalnature.OurmaingoalisLemma3.7,whichisneededinSects.4and6.For thesakeofclarity,webeginwithsomepreliminaryfacts.

Lemma 3.1Let V be a real algebraic variety, W⊆V a Zariski closed subvariety,and f:W→ Ra regular function. Then there exists a regular function F:V→ Rsuch that F|W=f.

ProofBy Lemma2.1, there exist regular functionsϕ, ψVRsuch thatW

V Z(ψ)andfϕ/ψonW. Pick a regular functionαVRwithZ(α)W. Then the functionF^ϕψhas the required properties.

α²+ψ²

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Kucharz

=

∈

⊆

∈

⊆

|

: →

⊆

: ∪ → | |

ForanyrealalgebraicvarietyV,weletR(V)denotetheringofreal-valuedregular functions onV.

IfW⊆Vis a Zariski closedsubvariety,then theideal IV(W)={f∈R(V):f|W=0}

ofR(V)is called theideal of W in V.

Lemma 3.2Let V be a real algebraic variety, and W1, W2Zariski closed subvarietiesof V for which

IV(W1∩W2)=IV(W1)+IV(W2)inR(V).

Let f W1W2Rbe a function such that the restrictions fW1and fW2areregularfunctions. Then f is aregularfunction.

ProofBy Lemma3.1, there exists a regular functiongi:V→ RwithgSinceg1i−g|W_i=f|2∈IWV(W_ifori=1,2.1∩W2), wehaveg1−g2=h1−h2,wherehi∈IV(Wi).Henceg1−h1=g2−h2is a regular function onVwhoserestrictiontoW1∪W²isequaltof.Consequently,fisaregularfunction.

nu LetVbeanonsingularrealalgebraicvariety,andDVaZariskiclosedsubvariety.Wesay thatDis asimple normal crossing hypersurfaceif for each pointp Dthere exist a Zariski open

neighborhoodU Vofpand local

coordinatesx1,...,xnonU(aregularsystemofparametersatpV )suchthattheintersectionofeachirreduci ble component ofDwithUisgivenby the equationxi0 for a suitablei. In particular, ifDisasimplenormalcrossinghypersurface,theneachirreduciblecomponentofDis nonsingular of codimension1.

Lemma 3.3Let V be a nonsingular real algebraic variety, D V a simple normalcrossing hypersurface, and f DRa function whose restriction to each irreduciblecomponent of D is a regular function. Then f is a regular function.

ProofWeuse induction on the numberkof irreducible components ofD.The casek=1 is obvious. Suppose thatk≥2. LetD1be an irreducible component ofD,and letDbe the union of the remaining irreducible components. The restrictionf|D₁is a regular function by assumption, whereas the restrictionfDis a regular function by the inductionhypothesis.

Pick a pointp∈D1∩D. It suffices to find a Zariski open neighborhoodU⊆Vofpsuch thatf|D∩Uis a regular function. IfUis small enough, there exist local coordinatesx1,...,xnonUsuch that the idealIU(D1∩U)is generated byx1,and

theidealIU(D∩U)isgeneratedbytheproductx2··· x^lforsomelwith2≤l≤n.NotethattheidealI

U(D1∩D∩U)isgeneratedbyx¹andx²··· x^l;inotherwords,

IU(D1∩D∩U)=IU(D1∩U)+IU(D∩U)inR(U).

Hencef|D∩UisaregularfunctioninviewofLemma3.2. nu

We give next the following variant of Lemma3.1.

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| | ⊆ ||

− ||

−

: → | = |

=

:= ∩ := ∪···∪

Lemma 3.4Let V be arealalgebraicvariety,Y⊆X some subsets of V , and f:Y→

Raregularfunction. Assume that Y=X∩W,where W is the Zariski closure ofY in V . Thenthereexists aregularfunction F:V0→ R, defined on a Zariski openneighborhoodV0of X in V , such that F|Y=f.

ProofBy Lemma2.2, there exist regular functionsϕ, ψ:V→ Rsuch that Y⊆V\Z(ψ)andf=ϕ/ψonY. Pick a regular functionα:V→ RwithZ(α)=W.

SetV0:=V\Z(α²+ψ²)andF:=^ϕψonV0. ThenFhas the required properties.

α²+ψ²

nu Notation 3.5By a simplex inR^mwe always mean a closed geometric simplex.Forany finite (geometric) simplicial complexKinR^m, we writeKfor the union of all simplices inK; thusKR^mis a compact polyhedron.Wedenote byK⁽ⁿ⁾then-skeleton ofK.If�⊆R^misad-

simplex,then�˙s ta n d s forthesimplicialcomplexwhichconsistsofallfacesof�ofdimensi

onatmostd 1.Clearly,�˙istheunionofall(d 1)-

dimensional faces of�. The Zariski closure of�inR^m, denoted byH�, is an affine subspace of dimensiond.

Lemma3.6Let�⊆R^mbead-simplexandletf:|�˙|

→Rbeafunctionsuchthattherestrictionf|r:r→Risaregularfunctionforevery(d−1)- simplexr∈�˙.ThenthereexistsaregularfunctionF:�→RwithF|_|�˙|=f .

ProofLet�0,...,�dbeallthe(d−1)-

dimensionalfacesof�.WesetH:=H�andHi:=H�ifori=0,..., d.Obviously,Hi⊆Hand

�i=�∩Hi. By Lemma3.4,there exists a Zariski open neighborhoodn⊆Hof�and a regularfunction

Gi:n→RwithGi|�i=f |^�ⁱfori=0, . . . ,d.Inparticular,Gⁱ=m

�i∩�jfor alli,j. SinceHi∩Hjis the Zariski closure of�i∩�jinR Gi=Gjonn∩Hi∩Hj.

Gjon , we get

SetDinHiandDD0 Dd. ThenDis a

simplen o r m a l crossinghypersurface inn. Define a

functionϕDRbyϕD_iGiD_ifori0,. . . , d.By Lemma3.3,ϕis a regular function. In view of Lemma3.1, there exists a regularfunction<P:n→Rwith<P|

D=ϕ.ThefunctionF:=<P|�hastherequiredproperties.

nu

We need the following approximation result for functions defined on a simplex.

Lemma 3.7Let�⊆ R^mbe a d-simplex and let f:�→ Rbe a continuousfunctionsuchthattherestrictionf|r:r→Risaregularfunctionforevery(d−1)- simplexr∈�˙.Then,foreveryε>0,thereexistsaregularfunctiong:�→Rsatisfying

|f(x)−g(x)|<εfor allx∈�

andf|_|�˙|=g|^|�˙|.

ProofAccording to Lemma3.6, there exists a regular functionh:�→ Rwith h|_|�˙|=f||�˙|.Byreplacingfwithf−h,theproofisreducedtothecasef|_|�˙|=0.

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∪

4

⊆

:=

: → | = ≤

≤

:= = = := ∪···∪

| = :={∈ :| |

}

of�.WesetHH�andHiH�ifori0,. . . , d.The unionDH0

Hdisasimplenormalcrossinghypersurface inH. The function on�Dwhich is equal tofon�andidenticallyequal to 0 onDis continuous. Hence, byTietze’sextension theorem, there exists a continuous functionϕ:H→ Rwithϕ|�=fandϕ|D=0.

Fixε >0. Note that there exists aC^∞functionψ:H→ Rsatisfying ε

|ϕ(x)−ψ(x)|<

2for allx∈H

andψ|D=0. Indeed, by the Whitney approximation theorem [67, Theorem 10.16], one can find aC^∞functionλ:H→ Rfor which

ε

|ϕ(x)−λ(x)|<

4for allx∈H.

SinceϕD0, the setU x Hλ(x) <^εis an open neighborhood ofDinH. IfαHRis aC^∞function withαD1, 0α1 onHand support contained inU, then the functionψ (1α)λhas the required properties.

DenotebyC^∞(H)theringofC∞real-valuedfunctionsonH.Onereadilyseesthat theidealIC^∞(H)ofallC^∞functionsvanishingonDisgeneratedbypolynomial

functions,say,q1,...,qr(alternatively, one caninvokea much more general result [81,p.52,Proposition1]).Consequently,ψcanbewrittenintheform

ψ=ψ1q1+···+ψrqr, where theψkareC^∞functions onH. Let

M:=sup{|qk(x)|:x∈ ,� k=1,...,r}.

By the Weierstrass approximation theorem, there exists a polynomial function pk:H→ Rsatisfying

ε

|ψk(x)−pk(x)|<

2rMfor allx∈ .� Forp:=p1q1+···+prqr, wehave

ε ε

|ϕ(x)−p(x)|≤ |ϕ(x)−ψ(x)|+ |ψ(x)−p(x)|<

2+

2=εfor allx∈�

andp|D=0.Wecompletetheproofsettingg:=p|�. nu

4 Piecewise-regular maps intoGrassmannians

The role of Sects.4.1and4.2is to review some notation and terminology.

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∈

: →

X X

0

n, r

4.1 Inner product and matrices

As in Sect.1, we letFdenoteR,CorH. TheF-vector spaceFⁿis endowed with the standard inner product

givenby

(−,−) : Fⁿ× Fⁿ→ F

n

((x1,...,xn),(y1,...,yn))= xiyⁱ,

i=1

whereyistands for the conjugate ofyiinF.

Let Matm,n(F), or simply Matn(F)ifm=n, denote the set of allm-by-nmatrices with entries inF. For any matrixA= [aij]∈Matm,n(F), the correspondingF-linear transformationLA: Fⁿ→ F^mis given by

n

(x1,...,xn)i→(y1,...,ym),whereyi= xjaijfori=1 ,...,m

j=1

(recall that we always consider leftF-vector spaces). We will identifyAwithLAand write

A(v)=LA(v)forv∈ Fⁿ.

IfB= [bjk]∈Matn,r(F), then we define the productAB= [cik]by

n

cik= bjkaij._j=1 This convention implies thatLAB=LA◦LB.

We regard Matn,r(F)as a real algebraic variety. If 1≤r≤n, then the subset

n,r(F)⊆Matn,r(F)

of all matrices with linearly independent columns is Zariski open. Furthermore, the map

0(F)→Gr(Fⁿ), Ai→A(F^r)

is a regular map, as is immediately seen by using the standard charts onGr(Fⁿ).

4.2 Vectorbundles

For any topologicalF-vector bundleξon a topological spaceX, we denote byE(ξ)its total space and byp(ξ)E(ξ)Xthe bundle projection. The fiber ofξovera pointx XisE(ξ)xp(ξ)⁻¹(x).

Given a nonnegative integern, we letεⁿ(F)denote the standard productF-vector

X

bundle onXwith total spaceX×Fⁿ. Any ϕ:ε^r(F)→εⁿ(F)of topological Ma

Ma

morphism

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Kucharz

: →

×

V

V V

G(F)

εⁿ(F), hence

: → ⊆

∈

⊆ : →

: →

X X

X

F-vector bundles is of the form

ϕ(x, v)=(x,A_ϕ(x)(v))for all(x, v)∈X× F^r,

whereAϕX Matn,r(F)is a uniquely determined map, called thematrixrepre- sentationofϕ.Obviously,Aϕis a continuousmap.

Ifξis a topologicalF-vector subbundle ofεⁿ(F), thenεⁿ(F)=ξ⊕ξ^⊥, whereξ^⊥ is the orthogonal complement ofξwith respect to the s^X tandard inner product onF^X ⁿ. The orthogonal projectionρξ:εⁿ(F)→εⁿ(F)ontoξis a topological morphism of

X X

F-vector bundles.

We will also consider algebraic vector bundles on a real algebraic varietyV. The productVFⁿwill be regarded as a real algebraic variety. By analgebraicF- vectorbundle on Vwe mean an algebraicF-vector subbundle ofεⁿ(F)for somen(cf. [6, Chapters 12 and 13] and [37,38] for various characterizations of algebraicF-vector bundles).

Ifϕ:ε^r(F)→εⁿ(F)is an algebraic morphism, then the matrix representation

V V

Aϕ:V→Matn,r(F) ofϕis a regular map.

IfξisanalgebraicF-vectorsubbundleofεⁿ⁽F),thenitsorthogonalcomplementξ^⊥

is also an algebraicF-vector subbundle, and the orthogonal projectionρV ξ :εⁿ(F)→

εⁿ(F)ontoξis an algebraic morphism.The tautologicalF-vector bundleγ (Fⁿ)onG(Fⁿ)is an algebraicF-vector sub- bundle ofεⁿ

r

r r

(Fⁿ).

Lemma 4.1Let V be arealalgebraicvariety,and f XGr(Fⁿ)a continuousmap defined on some subset X V . Then the mapPfX Matn(F),wherePf(x)FⁿFⁿis the orthogonal projection onto f(x)Fⁿfor all xX,iscontinuous. Furthermore, if the map f isregular,then so is the mapPf.

ProofWe regard the pullbackξ:=f^∗γr(Fⁿ)as a topologicalF-vector subbundle of

X

E(ξ)x={x} ×f(x)for allx∈X.

It follows thatPfis the matrix representation of the orthogonal projection ρξ:εⁿ(F)→εⁿ(F)ontoξ. Consequently,Pfis a continuous map.

Now,suppose thatfis a regular map. It suffices to consider the case whereXis a Zariski locally closed subvariety ofV. Thenξis an algebraicF-vector subbundle ofεⁿ⁽F),hencetheargumentaboveshowsthatPfisaregularmap.

nu

4.3 Maps intoGrassmannians

We can now prove the following variant of Lemma3.7.

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n

� �

n, r

⊆ : | | →

Lemma 4.2Let�⊆ R^mbe a d-simplex and let f:�→Gr(Fⁿ)be acontinuousmapsuchthattherestrictionf|_r:r→G^r(Fⁿ)isaregularmapforevery(d−1)- simplexr∈�˙.Then,foreachneighborhoodU⊆C(�,Gr(Fⁿ))off,thereexistsaregular mapg:�→Gr(Fⁿ)suchthatg∈Uandg|_|�˙|=f|^|�˙|.

ProofConsider the mapP=Pf:�→Matn(F), whereP(x):Fⁿ→ Fⁿis the orthogonalprojectionontof(x)⊆Fⁿforallx∈�.ByLemma4.1,Pisacontinuousmapandtherestrictio nP|_r:r→Matn(F)isaregularmapforevery(d−1)-simplex

r∈�˙.

Weregard the pullbackξ:=f ^∗γr(Fⁿ)as a topologicalF-vector subbundle of ε_�(F). Sinceξis topologically trivial, there exists an injective topological morphism ϕ:ε^r(F)→εⁿ(F)whoseimageisequaltoξ.LetA=A_ϕ:�→Mat^n,r(F)bethe

matrix representation ofϕ. ThenAis a continuous map and P(x)A(x)=A(x)f o r a l l x∈ .�

By the Weierstrass approximation theorem, there exists a regular map B:�→Matn,r(F)arbitrarily close toA. DefineC:�→Matn,r(F)by

C(x)=P(x)B(x)for allx∈ .� ThenCisacontinuousmap,closetoA,suchthattherestrictionC|

r:r→Matn,r(F)isaregularmapforevery(d−1)-

simplexr∈�˙.Hence,accordingtoLemma3.7, there exists a regular map<P:�→Matn,r(F), arbitrarily close toC,with

<P|_|�˙|=C|^|�˙|.Inparticular,theF-lineartransformation<P(x):F^r→Fⁿisinjective for allx∈�. In other words,<P( )� ⊆Mat⁰(F). Thus

g:�→ Gr(Fⁿ),g(x)=<P(x)(F^r)

is a well-defined regular map, close tof. We may assume thatg∈U. Furthermore,

g|_|�˙|=f|^|�˙|since<P|^|�˙|=C|^|�˙|. un

An important consequence of Lemma4.2is the following.

Proposition 4.3Let K be a finite simplicial complex inR^mand let f:|K| → G(Fⁿ)be a continuous map. Then, for each open neighborhoodU⊆C(|K|,G(Fⁿ))of f ,thereexists a continuous mapϕ:|K| → G(Fⁿ)such thatϕ∈Uand the restrictionϕ|

�:�→G(Fⁿ)is aregularmap for every simplex�∈K.

ProofWeuse induction ond=dimK. The cased=0 is obvious. Supposenowthatd≥1.

By the induction hypothesis, there exists a continuous mapψ:|K^(d−1)|

→G(Fⁿ),arbitrarilyclosetof|_|K(d−1)|,suchthattherestrictionψ|r:r→G(Fⁿ)isaregular map for every simplexr∈K^(d−1).

Weclaimthatψhasa continuousextensionψ˜ K G(Fⁿ)thatbelongsto U. For the proof, we may assume thatG(Fⁿ)R^Nis a nonsingular Zariski closed subvariety. Regardingfandψas maps with values inR^N, we have

1f(x)−ψ(x)1<εfor allx∈ |K^(d−1)|,

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Kucharz

=

⊆

whereε>0 is small and1−1stands for the Euclidean norm onR^N. According toTietze’sextension theorem, there exists a continuous mapσ:|K| →R^Nwithσ_|K^(d−1)_|

=ψ.If

U:= {x∈ |K|: 1σ (x)−f(x)1<ε}

andα:|K| → Risacontinuousfunctionwithα|_|K(d−1)|=1,0≤α≤1andsupportcontained inU, then settingh:=f+α(σ−f), weget

1f(x)−h(x)1= |α(x)|1σ (x)−f(x)1<εfor allx∈ |K|.

Choose a tubular neighborhoodρ:T→G(Fⁿ)ofG(Fⁿ)inR^N. Ifεis sufficientlysmall,thenhhasvaluesinT,andthecompositeψ˜

ρhhasthepropertiesrequired in the claim.

By Lemma4.2, for everyd-simplex�∈K, there exists a regular map g�:�→G(Fⁿ)closetoψ˜|�andsuchthatg�|_|�˙|=ψ˜|_|�˙|.Thenthemap ϕ: |K| → G(Fⁿ), defined byϕ|_|K(d−1)|=ψandϕ|�=g�for everyd-simplex

�∈K,hasalltherequiredproperties. nu

In what follows, we will use stratifications constructed in a fairly simpleway.Any finite simplicial complexKinR^mgives rise to a filtration

FK=(Z0(K),Z1(K), . . . ,Zm+1(K))

ofR^m, whereZ0(K)= R^m,Zm+1(K)= ∅, andZd(K)is the union of theH�for all simplices�∈Kof dimension at mostm−dwithd=1,...,m. Here, as in Notation3.5,H�⊆ R^mstands for the Zariski closure of�. Setting

SK:={Zi(K)\Zi+1(K):i=0, ..., m},

we obtain a stratification ofR^m. More generally, ifVR^mis a Zariski closed subvariety, then the collection

SK(V):= {V∩S:S∈SK}

is a stratification ofV, which is said to beinduced by K. Obviously, SK(R^m)=SK.

Lemma 4.4Let K be a finite simplicial complex inR^m,V⊆ R^ma Zariski closedsubvariety,andX⊆|K|∩Vanarbitrarysubset.Then,foreverystratumT∈ SK(V),each connected component of X∩T is contained in some simplex�∈K.

ProofIt suffices to consider the caseV= R^m. IfS∈SK(R^m)=SK, then each connected component ofX∩Sis contained in a connected component of|K|

∩ S,whichinturniscontainedinsomesimplex�∈KbyconstructionofSK.

nu

We are now ready to prove the first two theorems announced in Sect.1.

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⊆

: →

⊆

Proof of Theorem1.3Letnbe a positive integer. We may assume thatV⊆ R^mand G(Fⁿ)⊆ R^Nare Zariski closed subvarieties.

Consider a continuous mapf:X→ G(Fⁿ)⊆ R^N. By Tietze’s extension theorem, there exists a continuous mapF: R^m→ R^NwithF|X=f. Letρ:T→ G(Fⁿ)be a tubular neighborhood ofG(Fⁿ)inR^N. ThenU:=F⁻¹(T)⊆ R^mis a neighborhood

ofX,andf˜:U→G(Fⁿ),givenbyf˜(x)= ρ(F(x))forx∈U,isacontinuous extension off. SinceXis a compact subset ofU, we getX B U, where

Bis the union of a finite collection of simplices inR^m. Then there exists a finite simplicial complexKinR^mwith|K| = B.In view of Proposition4.3, there exists a continuousmapϕ:|K| → G(Fⁿ),arbitrarilyclosetof˜||K|,suchthattherestrictionϕ|

�:�→G(Fⁿ)isaregularmapforeverysimplex�∈K.AccordingtoLemma4.4,the

restrictiong:=ϕ|X:X→G(Fⁿ)is a piecewiseSK(V)-regular map. The proofiscompletesincegisclosetof.

nu

Proofof Theorem1.4It can be assumed thatV⊆ R^mis a Zariski closedsubvariety,henceX⊆R^m.ByBorsuk’stheorem[26,p.537],Xisaretractofsomeneighborhood U⊆R^m.WecanfindafinitesimplicialcomplexKinR^mwithX⊆|K| ⊆ U.Thus

thereexistsaretractionr:|K| → X.

Weclaim that the induced stratificationS:=SK(V)ofVhas all the required properties. Indeed, letnbe a positive integer and letf:X→G(Fⁿ)be a continuous map.

Thenf◦r:|K| → G(Fⁿ)is a continuous extension off. By Proposition4.3, there exists a continuous mapϕ:|K| → G(Fⁿ), arbitrarily close tof◦r, such that the restrictionϕ|

�:�→G(Fⁿ)is a regular map foreverysimplex�∈K. Inviewof Lemma4.4, the

restrictiong:=ϕ|X:X→G(Fⁿ)is a piecewiseS-regular

map.Thiscompletestheproofsincegisclosetof.

nu

We also have the following variant of Theorem1.4.

Theorem 4.5LetX0R^mbe a compact subset, UR^ma neighborhood ofX0,βU U a homeomorphism, and K is a finite simplicial complex inR^m.AssumethatX0is aretractof Ua n d

X:=β(X0)⊆|K| ⊆ U.

Then, for each positive integer n, every continuous map from X intoG(Fⁿ)can beapproximated by piecewiseSK-regular maps.

ProofLetr0:U→X0be a retraction. ThenrX:U→X, given by rX(x)=β(r0(β⁻¹(x)))for allx∈U, is a well-defined retraction.

Letnbe a positive integer and letf:X→G(Fⁿ)be a continuous map.

Sincef◦rX:U→G(Fⁿ)is a continuous extension off, we complete the proof as in thecaseofTheorem1.4.

nu

Theorem4.5can be illustrated by revisiting Example1.7.