DOI: 10.24193/subbmath.2020.4.11
The size of some vanishing and critical sets
Cornel Pintea
Abstract. We prove that the vanishing sets of all top forms on a non-orientable manifold are at least 1-dimensional in the general case and at most 1-codimen- sional in the compact case. We apply these facts to show that the critical sets of some differentiable maps are at least 1-dimensional in the general case and at most 1-codimensional when the source manifold is compact.
Mathematics Subject Classification (2010):57R70, 57R35, 57M10.
Keywords:Critical and vanishing sets.
1. Introduction
It is well-known that the orientability of a manifold is characterized by the existence of a top differential form which never vanishes. Therefore it is natural to investigate the size of the vanishing setsV(θ) :={p∈M :θp = 0} of the top forms θ∈Ωm(M) towards a measure of thedeviation from orientabilityof the involved non- orientable manifold M. Indeed, the complement of every vanishing set of a top form is orientable and the smallest such vanishing sets are good candidates to measure this deviation. In this paper we show that the top forms of non-orientable manifolds cannot have arbitrarily small vanishing sets and apply this fact to show that some maps cannot have arbitrarily small critical sets. For instance the zero dimensional subsets of the non-orientable manifolds are neither vanishing sets of the top differentiable forms, nor critical sets of any differentiable function with orientable regular set, for the orientable option of the target manifold. Similar lower bounds for the size of the branch locus arise due to Church and Timourian [5, 6] in the codimension cases 0,
−1 and−2. On the other hand, the critical set of a zero codimensional differentiable map was treated before in [17], where the critical set is realised as the vanishing set of the pull-back of a volume form on the oriented target manifold.
Note that the other extreme is well represented in the recent years, as quite some effort oriented towards the maps with finite critical sets has been done, not only for one dimensional, but also for higher dimensional target manifolds [1, 2, 3, 8, 9, 10].
The paper is organized as follows: In the second and third sections we quickly review the tools and emphasize the preparatory results needed to prove the main results of the paper, which are also stated here. In the fourth section we prove the main results of the paper, the first of which concerns the surjectivity of the group homomorphism induced, at the level of fundamental groups, by the inclusionM\A ,→ M, where Mm (m ≥2) is a manifold andA ⊂M is a closed zero dimensional set.
As a consequence we observe that the dimension of the critical set of a zero or lower codimensional map, whose target manifold is orientable and the source manifold is non-orientable, is at least 1-dimensional. Relying, all over this paper, on the inductive definition of the ’dimension’ [7, 13], we prove that the dimension of the critical set of a zero or lower codimensional map, whose target manifold is compact orientable and the source manifoldMnis compact non-orientable, is at least (n−1)-dimensional. Recall however that the small and large inductive dimensions are equal to each other and both are equal with the covering dimension whenever the evaluated space is separable [7, p. 65]. Since differential manifolds are metrizable metric spaces, it follows that the inductive dimensions of a certain subset of a given manifold are equal to each other and both are equal with the covering dimension of that subset.
2. Main results
In order to achieve such results we rely on the characterization of orientability of a connected differential manifoldM by means of theorientation character, i.e. the group homomorphismwM :π1(M)−→C2:={−1,1}defined by
wM([γ]) =
1 if ˜γ(1) = ˜x1
−1 if ˜γ(1) = ˜x−1,
where ˜γ: [0,1]−→Mˆ is the lift of the loop γ: [0,1]−→M,γ(0) =γ(1) =x, with
˜
γ(0) = ˜x1,p: ˆM −→M is the orientable double cover ofM andp−1(x) ={x˜1,x˜−1}.
Indeed,M is orientable if and only if the orientation character is trivial. Equivalently, M is non-orientable if and only ifwM is onto. Taking into account that the orientation double cover ofOisp
p−1(O):p−1(O)−→O, we deduce that the orientation character of a connected open setO⊆M can be decomposed as
ωO =wM◦π1(iO), whereπ1(iO) :π1(O)−→π1(M)
is the group homomorphism induced by the inclusion mapiO :O ,→M. Consequently the open connected subsetO of a non-orientable manifoldM remains non-orientable wheneverπ1(iO) is surjective. Note that the orientation characterωM ofM coincides with w1(M)◦ρ, where ρ:π1(M)−→H1(M,Z) stands for the Hurewicz homomor- phism andw1(M) for the first Stiefel-Whitney class regarded as a homomorphism via the homomorphism of the universal coefficient Theorem
H1(M;Z2)−→Hom(H1(M;Z),Z2) andC2is identified with Z2.
Remark 2.1. LetMmis a connected non-orientable manifold.
1. If the 1-skeletonM1of a certainCW-decomposition ofM is a strong deformation retract of some of its open neighbourhoodU, then the complementM\U cannot be the vanishing set of any top form onM, as the group homomorphism
π1(iM\U) :π1(M \U)−→π1(M) is onto [11, p. 39].
2. Them−2 and lower dimensional submanifolds of Mmcannot be the vanishing sets of any top forms onM, as the group homomorphism
π1(iM\X) :π1(M\X)−→π1(M)
is an isomorphism for m ≥ 3 and an epimorphism for m = 2, whenever X is such a submanifold ofM. In particular the discrete subsets ofM cannot be the vanishing sets of any top forms on M [16, Proposition 2.3]. By using the same type of arguments one can actually show that no countable subset of M can be the vanishing sets of any top form on M. In other words the vanishing set of every top form onM is uncountable. In fact the zero dimensional subsets of M cannot be the vanishing sets of any top forms onM, as we shall see in the Theorem 2.1 and Corollary 2.2.
Theorem 2.1. IfMmis a smooth connected manifold (m≥2) andA⊆M is a closed zero dimensional set, thenM\A is also connected and the group homomorphism
π1(i) :π1(M\A)−→π1(M),
induced by the inclusioni:M\A ,→M, is onto, i.e. π1(M, M\A) = 0.
Corollary 2.2. If Mm is a non-orientable manifold, then dimV(ω) ≥ 1 for every differentiable formω∈Ωm(M).
Proof. Assume that dimV(ω) = 0 for some differentiable formω∈Ωm(M). Accord- ing to Theorem 2.1, the complementM \V(θ) of the vanishing set is also connected and the group homomorphism
π1(i) :π1(M\V(θ))−→π1(M)
is onto. The non-orientability of M shows that the orientation characterwM is onto.
Consequently the orientaion characterωM\V(θ) =wM◦π1(iM\V(θ)), ofM\V(θ), is also onto, due to Theorem 2.1.
On the other hand the restriction θ|M\V(θ) is a nowhere vanishing top form of M \V(θ), which shows that M \V(θ) is an orientable open submanifold of M. In other words, the orientation character ωM\V(θ) = wM ◦π1(iM\V(θ)) is trivial, which implies that either the orientation character wM is not onto or the induced group homomorphismπ1(iM\V(θ)) :π1(M\V(θ))−→π1(M) is not onto, which is absurd.
In the compact non-orientable case we can provide, by using some different type of arguments, a much larger lower bound for the vanishing sets of all top forms.
Theorem 2.3. If Mm is a compact connected non-orientable manifold, then dimV(ω)≥m−1 for every differentiable form ω∈Ωm(M).
Remark 2.2. The estimate, provided by Corollary 2.2 is sometimes sharp. Indeed, by removing a suitable circle out of Klein bottle we obtain a cylinder, which is orientable.
Also by removing a suitable copy of the (2n−1)-dimensional real projective space, out of the 2n-dimensional real projective space we obtain a 2n-disc, which is also orientable. In both casses the removed submanifolds have, due to Corollary 2.2 and Theorem 2.3, the smallest possible dimension in order to get orientability on their complements.
3. Preliminary results
3.1. Vanishing sets of differentiable forms
If ω is a k-differential on M, recall that the vanishing set V(ω) of ω is the collection of pointsz∈U at whichω vanishes, i.e.
V(ω) :={z∈M :ωz(v1, . . . , vk) = 0 for allvi ∈Tz(M)}.
We shall only use in this paper the vanishing sets of the top differential forms ofM. In this subsection we investigate the size of critical sets of maps between two manifolds with the same dimension via the vanishing set of the pull-back form of a volume form on the target manifold.
Remark 3.1. Iff :Mn→Nnis a local diffeomorphism andθ∈Ωk(N), thenV(f∗θ) = f−1(V(θ)). If f is additionally surjective, then this equality can be rewritten as f(V(f∗θ)) =V(θ), which shows, by means of Hodel [12],
dim (V(f∗θ)) = dimV(θ) (3.1)
wheneverV(f∗θ) is compact.
Theorem 3.1. ([17]) If Mm, Nn, m≥n are differential manifolds withN orientable and f : M −→ N is a differential map, then C(f) = V f∗volN
, where volN is a volume form onN.
Corollary 3.2. Let Mn, Nn be differential manifolds. If N is orientable and M is non-orientable thendimC(f)≥1 for every differentiable functionf :M →N. Proof. Let volN be a volume form onN. Combining Theorem 3.1 with Corollary 2.2
we deduce that dimC(f) = dimV(f∗volN)≥1.
In addition to the usefulness of the vanishing sets of differentiable forms in evaluating the size of the critical sets, they are also useful in evaluating the size of the tangency sets [4].
3.2. Zero dimensional subsets of manifolds
Lemma 3.3. If C is a closed subset of a smooth manifold Mn, then there exists a smooth nonnegative function f :M −→Rsuch thatf−1(0) =C.
Proof. We first consider an embedding j : M ,→ R2n+1, whose existence is ensured by Whitney’s embedding theorem.
If K ⊆ R2n+1 is a closed subset such that j(C) = K∩j(M), i.e. j−1(K) = C, then the required function is f =g◦j, whereg :R2n+1 −→R is a smooth positive function such thatg−1(0) =K, whose existence is ensured by the Whitney theorem
([18, Th´eor`eme 1, p. 17]).
Proposition 3.4. If Ais a closed zero dimensional subset of a smooth manifold Mn, then for eachx∈M and every neighbourhoodU ofx, there exists an open neighbour- hoodV ofxsuch that V ⊆U, ∂V ∩A=∅ and∂V is smooth.
Proof. Ifx6∈A, then the existence of V is immediate. Assume now thata∈A and consider an open and relatively compact neighbourhoodV0 ofasuch thatV0⊆U and
∂V0∩A=∅. We may assume thatV0 is actually connected, as otherwise we reduce V0 to its connected component containinga. Ifϕ:M −→Ris a smooth nonnegative function such that ϕ−1(0) =A, whose existence is ensured by Lemma 3.3, observe that m:= min{ϕ(x)|x∈∂V0} >0, since the compact set A∩cl(V0) =A∩V0 has no common points with the compact boundary∂V0. Ify∈(0, m) is a regular value of ϕ|V0 :V0 →R, then (ϕ|V0)−1(y) is a compact hypersurface in V0, as (ϕ|V0)−1(y) = ϕ−1(y)∩cl(V0). Indeed, the inclusion (ϕ|V0)−1(y) ⊆ ϕ−1(y)∩cl(V0) is obvious. If x∈ϕ−1(y)∩cl(V0), thenϕ(x) =y and x∈cl(V0) =V0∪∂V0. But sincey >0, it follows that x6∈∂V0, which shows that x∈V0 andx∈(ϕ|V0)−1(y) as well. Because y < m, it follows that (ϕ|V0)−1(y)∩A=∅.
Finally, we consider a regular value y ∈ (0, m) of ϕ|V0 : V0 → R and observe that the inverse image ϕ|V0
−1
(−∞, y)⊆V0 is an open neighbourhood ofAand
∂h
ϕ|V0−1
(−∞, y)i
= ϕ|V0−1
(y),
which shows that∂h
ϕ|V0−1
(−∞, y)i
∩A=∅. IfV is the connected component of the inverse image ϕ|V0−1
(−∞, y) containinga, then its boundary is a collection of connected components of ϕ|V0
−1
(y) and therefore∂V ∩A=∅.
Remark 3.2. If A is a closed zero dimensional subset of a smooth surface Σ, then for each x ∈Σ and every neighbourhood U of x, there exists an open disk D such that x ∈D ⊆U, ∂D∩A =∅ and ∂D is a smooth circle. Indeed, we consider, via Proposition 3.4, a local chart (W, ψ) of Σ atxas well as a connected neighbourhood V of x with smooth boundary such that x ∈ V, cl(V) ⊆ W ⊆ U, ψ(W) = D2 and ∂V ∩A = ∅. Note that the boundary of ψ(V) is a union of pairwise disjoint circles, as the circle is the only compact boundaryless one dimensional manifold. One of these circles, say C, is the boundary of the unbounded component of R2\ψ(V).
The bounded component ofR2\Cis completely contained inD2, containsψ(V) and we may choose its inverse image throughψto play the role ofD.
3.3. Deformations of punctured manifolds
Since the deformations of the punctured Euclidean space and the punctured manifolds [16] will be repeatedly used in what follows, we shall review them shortly.
Forr >0 andn∈N∗ denote byDnr and Srn−1the open disk and the sphere respec- tively, both of them having the center at the origin of the spaceRn and radiusr.D1n andS1n−1will be simply denoted byDn andSn−1respectively. Forx∈Dn, consider the map hx:Rn\ {x} −→Rn\ {x}defined to be the identity outside the open disc Dn andhx(y) =Sn−1∩]xy for everyy∈Dn\ {x}, where ]xy stands for the half line {(1−s)x+sy : s >0}. In particularhx(y) =y, ∀y∈Sn−1.
Let N be an n-dimensional manifold and c = (U, ϕ) be a local chart of N such that cl(Dn) ⊆ϕ(U). Denote by Dϕ and Sϕ the sets ϕ−1(Dn) and ϕ−1(Sn−1) respectively. Forx∈Dϕwe define the continuous maphc,x:N\ {x} −→N\ {x}by
hc,x(y) =
y ify∈M\Dϕ ϕ−1 hϕ(x)(ϕ(y))
ify∈U\ {x}.
Note thathc,x(Dϕ\{x}) =Sϕ andhc,x(y) =y, ∀y∈Sϕ.
Remark 3.3. 1. hx(Dn\{x}) =Sn−1 andhx'HxidRn\{x}(relRn\Dn), where Hx:Rn\{x} ×[0,1]→Rn\{x}, Hx(y, t) = (1−t)y+thx(y).
2. hc,x'Hc,xidM\{x}, whereHc,x: (M\{x})×[0,1]→M\ {x},
Hc,x(y, t) =
y ify∈M\Dϕ
ϕ−1 Hϕ(x)(ϕ(y), t)
ify∈U\{x}.
If P is a given manifold and f :P −→ M is a continuous map whose image avoids the pointx, thenf 'hc,x◦f and a homotopy betweenf andhc,x◦f isHc,x(·, t)◦f. We shall refere to each hc,x◦f andHc,x(·, t)◦f as thepunctured deformationof f fromxontoSϕ.
4. The proofs of theorems 2.1 and 2.3
Proof of Theorem 2.1. Consider a homotopy class of curves in π1(M, M \A) repre- sented by a continuous curve α : [0,1] −→ M, α(0), α(1) ∈ M \A and deform α rel{0,1} to some differentiable curveβ with non vanishing tangent vector field. The immersionβmight actually be chosen to be a geodesic fromα(0) toα(1) with respect to some Riemannian metric on M (see e.g. [14, Theorem 1.4.6, p. 24]). Obviously dim (A∩Im(β))≤dim(A) = 0 and dim Im(β) = 1.
From this point we continue the proof by induction with respect to the dimension mof the manifoldM. First assume thatm= 2 and observe that for eacht∈β−1(A) there exists, via Remark 3.2, a two dimensional discDt⊆M with circular boundary, neighbourhood of β(t), such that its circular boundary Ct has no common points with A. Since β is locally an embedding, Dt might be chosen inside the domain Ut of a coordinate chart ct = (Ut, ϕt) in such a way that Dt = Dϕt, Ct = Sϕt, α(0) = β(0), α(1) = β(1) ∈ M \cl(Dt), Jt := β−1(Dt) is an open interval and ϕt(Dt∩Im(β|Jt)) =ϕ(Dt)∩R. Since{Dt |t∈β−1(A)} is an open covering of the compact set Im(β)∩A, we may extract a finite open cover, sayDt1, . . . , Dts. We may
assume that Dti \cl(Dtj)6=∅ whenever i6=j. Since dim(Im(β) = 1 it follows that Im(β) cannot fill the open set
Dti\
r
[
j=1
j6=i
cl(Dtj).
For eachi∈ {1, . . . , r}, consider a point
xi∈Dti\
Im(β)∪
r
[
j=1
j6=i
cl(Dtj)
and the mapsgcti,xi,:M\ {x1, . . . , xs} −→M\ {x1, . . . , xs}given by gcti,xi(y) =hcti,xi(y).
Note that idM\{x1,...,xs}'gct
1,x1◦· · ·◦gcts,xs(relM \(Dt1∪ · · · ∪Dts)) as each of the mapsgct1,x1, . . . , gcts,xs is homotopic to idM\{x1,...,xs}relative toM\(Dt1∪ · · · ∪Dts).
Thus
β'(gct1,x1◦ · · · ◦gcts,xs◦β)(rel{0,1}) and Im(gct
1,x1◦ · · · ◦gcts,xs◦β)⊆M\A, as (gct
1,x1◦ · · · ◦gcts,xs) ((Dt1∪ · · · ∪Dts)\ {x1, . . . , xs})⊆Ct1∪ · · · ∪Cts⊆M \A andβ−1(M\(Dt1∪ · · · ∪Dts))⊆β−1(M\A).
We next assume that the statement holds for (m−1)-dimensional manifolds and we shall prove it for the m-dimensional manifold M. In this respect we consider a partition 0 =t0< t1<· · ·< tr= 1 of the interval [0,1] with small enough norm such that:
1. β([t0, t1])∩A=β([tr−1, tr])∩A=∅andβ(t1), . . . , β(tr−1)∈M\A.
2. there are small enough open discsD1 =Dϕ1, . . . , Dr−2 =Dϕr−2 with spherical boundariesS1=Sϕ1, . . . , Sr−2=Sϕr−2, for some charts c1= (U1, ϕ1), . . . , cr= (Ur−2, ϕr−2), with the following properties:
(a) β−1(Di) is the open interval (ti, ti+1) and the restriction (ti, ti+1)−→Di, t7→β(t) is an embedding, for everyi= 1, r−2;
(b) cl(Di)∩Im(β) = Si∩Im(β) ={β(ti), β(ti+1)} and Di∩Di+1 =∅ while cl(Di)∩cl(Di+1) =Si∩Si+1={β(ti+1)}, for everyi= 1, r−3.
Note that Im(β)∩A⊂D1∪ · · · ∪Dr−2. For everyi∈ {1, . . . , r−2}, considerxi∈Di\ Im(β) and observe thatβ|[xi,xi+1]'hcti,xi◦β|[xi,xi+1](rel({xi, xi+1})). By applying the inductive hypothesis to the punctured deformationhcti,xi◦β|[xi,xi+1](rel({xi, xi+1})) of β|[xi,xi+1] from xi onto Si, whose image is contained in the (m−1)-dimensional sphere Si, one can conclude that hcti,xi◦β|[xi,xi+1] is homotopic rel({xi, xi+1}) to some continuous curve γi : [xi, xi+1] −→ Si whose image avoids the set A, i.e.
γi([xi, xi+1])⊆Si\A. Thushcti,xi◦βis homotopic rel({0,1}) to the continuous curve γ : [0,1]−→M \A defined by γ|[x0,x1] =β|[x0,x1], γ|[xi,xi+1] =γi for 1≤i≤r−2
andγ|[xr−1,xr]=β|[xr−1,xr].
Proof of Theorem 2.3. We first observe that every top-formω ∈Ωm(M) is exact, as the top de Rham cohomology group HdRm(M) is trivial [15, Th. 15.21, p. 405], i.e.
ω =dθfor someθ∈Ωm−1(M). If p: ˜M −→M is the orientable double cover, then p∗ω=p∗(dθ) =d(p∗θ), which shows that
Z
M˜
p∗ω= 0 and that dim (V(p∗ω))≥m−1 therefore. Thus
dim (V(ω)) = dim (p(V(p∗ω))) = dim (V(p∗ω))≥m−1,
asp(V(p∗ω)) =V(ω).
Corollary 4.1. Let Mn, Nn be differential manifolds. If N is orientable and M is compact and non-orientable then dimC(f)≥n−1 for every differentiable function f :M →N.
Proof. Let volN be a volume form onN. Combining Theorem 3.1 with Theorem 2.3 we deduce that dimC(f) = dimV(f∗volN)≥n−1, for every differentiable function
f :M −→N.
A proof of Corollary 4.1 of similar flavor appears in [17, Theorem 2.4.(b)].
Remark 4.1. Corollaries 3.2 and 4.1 rely on the orientability of the regular set R(f) = M \C(f) in the 0 = dim(N)−dim(M) codimension case which is a pri- ori ensured by the nowhere vanishing restricted top formf∗volN
R(f)onR(f). In the lower codimension case dim(M)>dim(N), the lack of orientability of the regular set is obvious, even for the orientable option of the target manifoldN. We stress this by the example of the projection of a productM =N×X on the first factor, whenN is orientable and X is non-orientable. The critical set of this projection is obviously empty, but its regular set is the whole non-orientable product M =N×X.
However, the orientability of the regular set R(f) ensure similar lower bounds even in the lower codimensional context. More precisely, ifNn is orientable andMm (m > n) is connected non-orientable and f : M −→ N is a differentiable function with orientable regular set R(f), then dim(C(f))≥ 1. The proof of this statement works along the same lines with the one of Corollary 2.2, the role of the vanishing set V(θ) is played here by the critical setC(f).
Acknowledgment. The author is grateful to the anonymous referee for his (or her) useful comments, which have helped him to improve the presentation.
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Cornel Pintea
Babe¸s-Bolyai University,
Faculty of Mathematics and Computer Sciences, 1, Kog˘alniceanu Street,
400084 Cluj-Napoca, Romania e-mail:[email protected]
