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Submitted exclusively to the London Mathematical Society DOI: 10.1112/S000000 ON SMOOTH MAPS WITH FINITELY MANY CRITICAL POINTS DORIN ANDRICA and LOUIS FUNAR Abstract We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres. 1. Introduction If M,N are manifolds, possibly with boundary, consider maps f : M ? N with ∂M = f?1(∂N) such that f has no critical points on ∂M . Denote by ?(M,N) the minimal number of critical points of such maps. The reader may consult the survey [3] for an account of various features of this invariant (see also [24]). Most of the previously known results consist of sufficient conditions on M and N ensuring that ?(M,N) is infinite. The aim of this note is to find when non-trivial ?(Mm, Nn) can occur if the dimensions m and n of Mm and respectively Nn, satisfy m ≥ n ≥ 2. Non-trivial means here finite and non-zero. Our main result is the following: Theorem 1.1. Assume that Mm, Nn are compact orientable manifolds and ?(Mm, Nn) is finite, where 0 ≤ m? n ≤ 3 and (m,n) 6? {(2, 2), (4, 3), (4, 2), (5, 3), (5, 2

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London Okane ga nai Riemann surface

SubmittedexclusivelytotheLondonMathematicalSocietyDOI:10.1112/S0000000000000000ONSMOOTHMAPSWITHFINITELYMANYCRITICALPOINTSDORINANDRICAandLOUISFUNARAbstractWecomputetheminimumnumberofcriticalpointsofasmallcodimensionsmoothmapbetweentwomanifolds.Wegiveaswellsomepartialresultsforthecaseofhighercodimensionwhenthemanifoldsarespheres.1.IntroductionIfM,Naremanifolds,possiblywithboundary,considermapsf:M→Nwith∂M=f−1(∂N)suchthatfhasnocriticalpointson∂M.Denotebyϕ(M,N)theminimalnumberofcriticalpointsofsuchmaps.Thereadermayconsultthesurvey[3]foranaccountofvariousfeaturesofthisinvariant(seealso[24]).MostofthepreviouslyknownresultsconsistofsuﬃcientconditionsonMandNensuringthatϕ(M,N)isinﬁnite.Theaimofthisnoteistoﬁndwhennon-trivialϕ(Mm,Nn)canoccurifthedimensionsmandnofMmandrespectivelyNn,satisfym≥n≥2.Non-trivialmeanshereﬁniteandnon-zero.Ourmainresultisthefollowing:Theorem1.1.AssumethatMm,Nnarecompactorientablemanifoldsandϕ(Mm,Nn)isﬁnite,where0≤m−n≤3and(m,n)6∈{(2,2),(4,3),(4,2),(5,3),(5,2),(6,3),(8,5)}.Ifm−n=3wealsoassumethatthePoincare´conjectureindimension3holdstrue.Thenϕ(Mm,Nn)∈{0,1}andϕ(Mm,Nn)=1preciselywhenthefollowingtwoconditionsarefulﬁlled:(i)MmisdiﬀeomorphictotheconnectedsumNb]Σm,whereΣmisanexoticsphereandNbisam-manifoldwhichﬁbersoverNn.(ii)MmdoesnotﬁberoverNn.Proof.Thestatementisaconsequenceofpropositions3.1,4.1and5.1.Remark1.1.(i)Thesecondconditionisnecessary,ingeneral.Thereexistexamplesofcon-nectedsumsNb]ΣmwhichﬁberoverN,yettheyarenotdiﬀeomorphictonon-diﬀeomorphicﬁbrationsoverS4withﬁberS3.Nb.Infact,theexotic7-spheresconstructedbyMilnorin[20]arepairwise(ii)However,ifthecodimensionm−niszerowebelievethatthesecondcon-ditionisredundanti.e.ifMmisdiﬀeomorphictoNb]ΣmandMmisnot2000MathematicsSubjectClassiﬁcation57R45,58K05,57R60,57R70.PartiallysupportedbytheEuropeanResearchTrainingNetworkGeometricAnalysis.

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