Smooth structures on pseudomanifolds with isolated conical singularities

0.29 MB·English

by  Hong Van Le

SMOOTH STRUCTURES ON PSEUDOMANIFOLDS
WITH ISOLATED CONICAL SINGULARITIES
1
1 HOˆNGVAˆNLEˆ, PETR SOMBERGAND JIRˇ´I VANZˇURA
0
2
r Abstract. In thisnotewe introducethenotion of a smooth structure
p onaconicalpseudomanifoldM intermsofC∞-ringsofsmoothfunctions
A on M. For a finitely generated smooth structure C∞(M) we introduce
the notion of the Nash tangent bundle, the Zariski tangent bundle, the
2
tangent bundleof M,andthenotion ofcharacteristic classes of M. We
1
provethevanishingofaNashvectorfieldatasingularpointforaspecial

] class ofEuclidean smooth structureson M. Weintroducethenotion of
G aconicalsymplecticformonM andshowthatitissmoothwithrespect
D toaEuclideansmoothstructureonM. Ifaconicalsymplecticstructure
is also smooth with respect to a compatible Poisson smooth structure
h. C∞(M), we show that its Brylinski-Poisson homology groups coincide
t withthedeRhamhomologygroupsofM. Weshownontrivialexamples
a
m of these smooth conical symplectic-Poisson pseudomanifolds.
[

5
v
Contents
7
0
1. Introduction 1
7
5 2. Pseudomanifolds w.i.c.s. and their smooth structures 3
. 3. Tangent bundles and vector fields on a pseudomanifold w.i.c.s.
6
0 with a finitely generated smooth structure 12
0 4. Symplectic pseudomanifolds w.i.c.s. 17
1
5. Concluding remarks 24
:
v References 25
i
X
r
a
AMSC: 51H25, 53D05, 53D17
Key words: C∞-ring, conical pseudomanifold, symplectic form, Poisson
structure
1. Introduction
Since the second half of the last century the theory of smooth manifolds
has been extended from various points of view to a large class of topological
spaces admitting singularities, see e.g. [8], [11], [12], [23], [24], [26], [27].
Roughly speaking, a Ck-structure, 1 k , on a topological space M
is defined by a choice of a subalgebra ≤Ck(M≤)∞of the R-algebra C0(X) of all
Date: April13, 2011.
1
2 HOˆNGVAˆNLEˆ,PETRSOMBERGANDJIRˇ´IVANZˇURA
continuous R-valued functions on M, which satisfies certain axioms varying
in different approaches. Most of efforts have been spent on construction of
a convenient category of smooth spaces, which should satisfy good formal
properties, see [1] for a survey. Notably, the theory of de Rham cohomology
has been extended to a large class of singular spaces, see [24], [27].
Inthisnotewedevelop thetheoryofsmoothstructuresonsingularspaces
in a different direction. We pick a class of topological spaces and ask, if we
can provide these spaces with a family of reasonable smooth structures and
what is the best smooth structure on a singular space. This question is
motivated by the question of finding the best compactification of an open
smooth manifold. We are looking not only for an extension of classical
theorems on smooth manifolds, but we are also looking for new phenomena
on these manifolds, which are caused by presence of nontrivial singularities.
We study in this note pseudomanifolds with isolated conical singularities.
Our choice is motivated by the following reasons. Firstly, isolated conical
singularities are geometrically the simplest possible, but they already serve
to illustrate new phenomena that are typical for the more general situation.
Secondly, the theory of smooth structures on singular spaces should include
investigationsrelatedtodifferentgeometricstructurescompatiblewiththese
smooth structures. A closely related field of research has been developed
since Cheeger wrote the seminal paper on spectral geometry of Riemannian
spaces with isolated conical singularities [5]. We would like to emphasize
that Cheeger and other people working on spectral geometry and index
theory on singular spaces, e.g. [6], [7], [16], deal with the analysis on the
open regular strata Mreg of a compact singular space M. Although Mreg
is open, for a large class of spaces the compactness of M forces the most
fundamentalfeaturesofthetheoryoncompactmanifoldstocontinuetohold
for Mreg. They did not consider M as a smooth space.
The plan of our note is as follows. In section 2 we introduce the notion
of a pseudomanifold M with isolated conical singularities, which we will ab-
breviate as a pseudomanifold w.i.c.s., their cotangent bundles, the notion of
smooth functions, smooth differential forms on these spaces and the notion
of smooth mappings between these spaces. Known and new examples are
given, see Example 2.5, some important properties of these smooth struc-
tures are proved, see Lemma 2.4, Proposition 2.12, Corollary 2.13, which
are important in later sections. Our approach is close to the approach by
Mostow in [24], which is formalized in the theory of C∞-rings as in [23].
Roughly speaking, a smooth structure on a pseudomanifold w.i.c.s. is speci-
fied by the canonical smooth structure on its regular stratum and a smooth
structure around its singular points, which dictates the way to “compact-
ify” the smooth structure around the singular point, see Definition 2.3. In
section3weconsiderfinitelygeneratedsmoothstructures. Weintroducedif-
ferentnotions oftangent bundlesofasmoothpseudomanifoldw.i.c.s., which
leads to the notion of characteristic classes of a finitely generated smooth
pseudomanifold w.i.c.s., see Remark 3.3. We investigate some properties of
SMOOTH STRUCTURES ON CONICAL PSEUDOMANIFOLDS 3
related smooth vector fields, see Proposition 3.6, Lemma 3.7.2. In section 4
we introduce the notion of a conical symplectic form on a pseudomanifold
w.i.c.s. M. We show that this symplectic form is smooth with respect to
a Euclidean smooth structure on M, see Corollary 4.6, and it possesses a
unique up to homotopy compatible C1-smooth conical Riemmanian metric,
seeLemma4.7. We alsoshow that if this conical symplectic formis compat-
ible with a Poisson smooth structure on M, the symplectic homology of M
is well-defined, see Remark 4.8, Lemma 4.9. If the symplectic form is also
smooth with respect to the compatible Poisson smooth structure, we prove
thatthesymplectic homology coincides withthedeRhamcohomology of M
with reverse grading, see Corollary 4.13. In Remark 4.8 we show non-trivial
examples of smooth Poisson structures which are compatible with a conical
symplectic structure on M. In section 5 we summarize our main results,
and pose some questions for further investigations.
2. Pseudomanifolds w.i.c.s. and their smooth structures
In this section we introduce the notion of a pseudomanifold w.i.c.s. M,
their cotangent bundles, the notion of a smooth structure, smooth differ-
ential forms on these spaces and the notion of smooth mappings between
thesespaces. Weprovideknownandnewexamples,representingthealgebra
of smooth functions in terms of generators and relations, see Example 2.5,
Lemma 2.14.1, Remark 2.16. We compare our concepts with some existing
concepts. We prove some important properties of these smooth structures,
see Lemma 2.4, Proposition 2.12, Corollary 2.13, Lemma 2.14.2, and we
characterize the removability of a singular point s M in terms of the local
∈
algebra of smooth functions on a neighborhood of s, see Lemma 2.21.
s
N
If L is a smooth manifold, the cone over L is the topological space
cL := L [0, )/L 0 .
× ∞ ×{ }
The image of L 0 is the singular point of cone cL. Let [z,t] denote
× { }
the image of (z,t) in cL under the projection π : L [0, ) cL. Let
× ∞ →
ρ : cL [0, ) be defined by ρ ([z,t]) := t. We call ρ the defining
cL cL cL
→ ∞
function of the cone. For any ε > 0 we denote by cL(ε) the open subset
[z,t] cL t < ε .
{ ∈ | }
Definition 2.1. (cf. [7,Definition1.1]) Asecond-countable locally compact
Hausdorff topological space Mm is called a pseudomanifold with isolated
conical singularity of dimension m, if there is a finite set S (or SMm) of
isolated singular points s Mm such that:
i
∈
1. Mm s is an open smooth manifold Mreg of dimension m.
i i
\∪ { }
2. For each singular point s there is an open neighborhood of s
s
N
together with a homeomorphism φ : cL (ε ), where L is a closed
s s s s s
N →
smooth manifold, ε > 0, and the restriction of φ to s is a smooth
s s s
N \{ }
diffeomorphism on its image.
3. If s ,s S, then either = , or s = s .
0 1 ∈ Ns0 ∩Ns1 ∅ 0 1
4 HOˆNGVAˆNLEˆ,PETRSOMBERGANDJIRˇ´IVANZˇURA
The smooth manifold Mreg := Mm S is called the regular stratum of
\
Mm, and L (or simply L) is called the singularity link of a singular point
s
s. The map φ : cL(ε ) is called a singular chart (around a singular
s s s
N →
point s). We also denote by (ε) the preimage φ−1(cL(ε)) for 0 < ε ε .
Ns s ≤ s
Let us notice that that we assume the singularity link L to be compact.
s
For the simplicity of exposition we suppose in this note that Mreg and L
s
are orientable and Mreg is connected.
Example 2.2. 1. Let M be a smooth manifold with boundary ∂M = L
which is a disjoint union of k compact connected components L . An easy
i
way to construct a pseudomanifold w.i.c.s. is to glue to M the closed cone
c¯L := L [0,1]/L 0 , or to glue to M the union of the closed cones c¯L
i
× ×{ }
along the boundary ∂M = L 1 = (L 1 ).
i i
×{ } ∪ ×{ }
2. ThequadricQ = z Cm+1 m+1z2 = 0 withisolated singularity
m { ∈ | i=1 i }
at 0 is a pseudomanifold w.i.c.s.. PThe quadric Q is a cone over L =
m
Q S2m+1(√2), where S2m+1(√2) is the sphere of radius √2 in Cm+1.
m
∩
It is easy to see that L consists of all pairs (x,√ 1y) Sm(1) Sm(1)
Rm+1 √ 1Rm+1 = Cm+1 such that x,y = 0.−Hence∈L is diff×eomorph⊂ic
⊕ − h i
to the real Stiefel manifold V = SO(m+1)/SO(m 1).
2,m+1
−
3. Anysmoothmanifoldwithkmarkedpointsisapseudomanifoldw.i.c.s.
with singular points being the marked points.
Nowletusintroducethenotionofasmoothstructureonapseudomanifold
w.i.c.s. by refining the Mostow’s concept [24, 1]. We denote by C∞(Xreg)
§
(resp. C∞(Xreg)) the space of smooth functions on Xreg (resp. the space of
0
smooth functions with compact support in Xreg). Note that any function
f C∞(Xreg) has a unique extension to a continuous function j f on X
∈ 0 ∗
by setting j f(x) := 0 if x X Xreg. The image j (C∞(Xreg)) is a
∗ ∈ \ ∗ 0
sub-algebra of C0(X).
Definition 2.3. A smooth structure on a pseudomanifold w.i.c.s. M is
a choice of a subalgebra C∞(M) of the algebra C0(M) of all real-valued
continuous functions on M satisfying the following three properties.
1. C∞(M)is a germ-definedC∞-ring, i.e. it is theC∞-ringof all sections
of a sheaf SC∞(M) of continuous real-valued functions (for each open set
U M there is a collection C∞(U) of continuous real-valued functions on
⊂
U such that the rule U C∞(U) defines the sheaf SC∞(M), moreover, for
any n if f , ,f C∞7→(U) and g C∞(Rn), then g(f , ,f ) C∞(U)
1 n 1 n
··· ∈ ∈ ··· ∈
[24, 1]).
§
2. C∞(M) C∞(Mreg).
|Mreg
⊂
3. j (C∞(Mreg)) C∞(M).
∗ 0 ⊂
We refer the reader to [23] for the theory of C∞-algebras.
Lemma2.4. Anysmooth structureonapseudomanifold w.i.c.s. M satisfies
the following partially invertibility. If f C∞(M) is nowhere vanishing,
∈
then 1/f C∞(M).
∈
SMOOTH STRUCTURES ON CONICAL PSEUDOMANIFOLDS 5
Proof. Assume that f C∞(M) is nowhere vanishing. It suffices to show
∈
thatlocally1/f isasmoothfunction. Sincef = 0,shrinkinganeighborhood
6
U of x if necessary, we can assume that there is an open interval ( ε,ε)
−
which has no intersection with f(U). Now there exists a smooth function
ψ : R R such that
→
a) ψ = Id,
|(U)
b) ( ε/2,ε/2) does not intersect with ψ(R).
Clea−rly G : R R defined by G(x) = ψ(x)−1 is a smooth function.
→
Note that 1/f(y) = G(f(y)) for all y U. This completes the proof of our
claim. ∈ (cid:3)
Example 2.5. 1. LetM˜ beanorientablesmoothmanifoldwithaconnected
orientable boundary ∂M = L and M obtained by M˜ by collapsing L to a
point, see Example 2.2.1. Let C∞(M˜) be the canonical smooth structure
on M˜. Denote by π : M˜ M the surjective continuous map which is 1-1
on M˜ L to its image Mr→eg. We set
\
C∞(M) := f C0(M) π∗(f) C∞(M˜) .
w { ∈ | ∈ }
It is easy to see that C∞(M) satisfies the conditions in Definition 2.3. We
w
call M˜ the canonical resolution of M.
2. Let L = Sn and X be the blowup of the point of origin 0 Rn+1 , i.e.
X = (x,l) Rn+1 RPn x l . Letπ :X Rn+1 = cLbeth∈eprojection
{ ∈ × | ∈ } →
on the first factor. We set C∞(cL) := f C0(cL)π∗(f) C∞(X) . It is
rp { ∈ | ∈ }
easy to see that C∞(cL) is a smooth structure according to Definition 2.3.
rp
3. Let L = S2n+1 and X be a blowup of the point of origin 0 Cn+1
and π : X cL = Cn+1 be the canonical projection. Using this res∈olution
(X π cL) w→e define another smooth structure C∞(cL) on cL which clearly
→ cp
also satisifies the condition in Definition 2.3.
4. Let M be a pseudomanifold w.i.c.s. and M˜ be a smooth manifold. We
callM˜ a resolution of M ifthereexistsacontinuoussurjectivemapπ :M˜
M suchthattherestriction ofπ toM˜ π−1(S )isasmoothdiffeomorphis→m
M
\
on its image. Using the same construction as in examples above we define a
resolvable smooth structure C∞M on M. We observe that there are many
M˜
non-diffeomorphic resolutions of a given conical pseudomanifold, which lead
to different smooth structures on M, e.g. Examples 2.5.2, 2.5.3.
5. Let C∞(M) and C∞(M) be smooth structures on M. Then C∞(M)
1 2 1 ∩
C∞(M) is a smooth structure on M.
2
Remark 2.6. If L is a standard sphere, then cL has the standard smooth
structure, which appears in a family of natural smooth structures on cL,
see Remark 2.16.1. Note that the larger the space C∞(M) is, the smaller is
the space of smooth mappings from a smooth manifold N to M. Thus our
choice of a smooth structure on M should be guided by our desire to have
a large space or a small space of smooth mappings from a smooth manifold
N to M.
6 HOˆNGVAˆNLEˆ,PETRSOMBERGANDJIRˇ´IVANZˇURA
Definition 2.7. Let M and N be conical pseudomanifolds provided with
smooth structures C∞(M) and C∞(N) respectively. A continuous map
σ : M N is called a smooth map, if σ∗(f) C∞(M) for all f C∞(N).
→ ∈ ∈
Remark 2.8. Denote by i the inclusion Mreg M. Condition (1) in
→
Definition 2.3 implies that i is a smooth map. Since the kernel of the ho-
momorphism i∗ : C∞(M) C0(Mreg) is zero, we can regard C∞(M) as a
→
subalgebra of C∞(Mreg). In the same way we can regard C∞(Mreg) as a
0
subalgebra of C∞(M).
The existence of a smooth partition of unity on a smooth space is an
important condition for the validity of many theorems in analysis and ge-
ometry, for example it is used in the proof of Lemma 4.7 below. In [24]
Mostow analyzed several consequences of the existence of a smooth parti-
tion of unity. We will show that our smooth structures satisfy the existence
of partition of unity.
Lemma 2.9. Let s S and let U be a neighborhood of s. Then there exists
∈
a function f C∞(M) such that
∈
(1) 0 f 1 on M;
≤ ≤
(2) f(s)= 1;
(3) f = 0 outside U.
Proof. Obviously, thereexists ε > 0such that (ε) U. We willconstruct
s
N ⊂
the required function f in several steps using a singular chart φ : (ε)
s s
N →
cL(ε).
In the first step we define an auxiliary smooth function χ C∞((0,ε)).
∈ 0
It is defined in the following way.
1 1 2
χ(a) = 0 for a (0, ε], 0 < χ(a) < 1 for a ( ε, ε),
∈ 5 ∈ 5 5
2 3 3 4
χ(a) = 1 for a [ ε, ε], 0< χ(a) < 1 for a ( ε, ε),
∈ 5 5 ∈ 5 5
4
χ(a) = 0 for a [ ε,ε).
∈ 5
Inthesecondstepwedefineacontinuousfunctionχ C0(M)bysetting
M
∈
χ (x) := χ ρ (φ (x)) for x (ε),
M cL s s
◦ ∈ N
χ(x) := 0 for x (Mreg φ−1(L (0,ε)).
∈ \ s ×
Note that χ is a smooth function on Mreg with compact support, and
M
consequently an element of C∞(M).
In the third step we define a new function ψ C0(M). We set
∈
2
ψ(x) := 1 for x (M φ−1(cL(ε)) or x φ−1((L ( ε,ε))
∈ \ s ∈ s × 5
2
ψ(x) := χ (x) for x φ−1(L (0, ε], and ψ(s) := 0.
M ∈ s × 5
SMOOTH STRUCTURES ON CONICAL PSEUDOMANIFOLDS 7
Let us show that on a neighborhood of any point x M the function ψ
∈
coincides with a function from C∞(M). If x (M φ−1(cL(ε)) or x
∈ \ s ∈
φ−1((L (2ε,ε))), then on a neighborhood of x the function ψ coincides
s × 5
with the constant function 1 C∞(M). If x φ−1(L (0, 2ε]), then on a
∈ ∈ s × 5
neighborhood of x the function ψ coincides with the function χ C∞(M).
∈
Finally on a neighborhood of the point s the function ψ coincides with a
constant function 0 C∞(M). Consequently ψ C∞(M), and then also
f = 1 ψ C∞(M)∈. This function has all the re∈quired properties. (cid:3)
− ∈
Lemma 2.10. For every compact subset K M and every neighborhood U
⊂
of K there exists a function f C∞(M) such that
∈
(1) f 0 on M;
≥
(2) f > 0 on K;
(3) f = 0 outside U.
Proof. For each point x K we take its open neighborhood V in such a
x
∈
way that V V, and we take a function f C∞(M) described in Lemma
x x
⊂ ∈
2.9(note that Lemma 2.9 trivially holds for any regular point x Mreg).
∈
Finally, we take an open neighborhood W V of x such that f > 1.
x ⊂ x x|Wx 2
Because K is compact, we can find a finite number of x ,...,x in K such
1 r
that
W W K.
x1 ∪···∪ xr ⊃
Now it is sufficient to set f = f + +f . (cid:3)
x1 ··· xr
Lemma 2.11. Let U be a locally finite open covering of M. Then
i i∈I
{ }
there exists a locally finite open covering V (with the same index set)
i i∈I
such that V¯ U . { }
i i
⊂
Proof. The proof is standard. (cid:3)
Proposition 2.12. Let U be a locally finite open covering of M such
i i∈I
that each U has a compa{ct c}losure U¯ . Then there exists a partition of unity
i i
f subordinate to U .
i i∈I i i∈I
{ } { }
Proof. Let V be the same covering as in Lemma 2.11. Let W be
i i∈I i i∈I
an open cov{eri}ng such that V¯ W W¯ U . According to L{emm}a 2.10
i i i i
⊂ ⊂ ⊂
for every i I there exists a function g C∞(M) such that
i
∈ ∈
(1) g 0 on M;
i
(2) g >≥ 0 on V¯;
i i
(3) g = 0 outside W .
i i
Because V suppg U for every i I, the sum g = g is well
i ⊂ i ⊂ i ∈ i∈I i
definedand everywhere positive. Since our algebra C∞(M) isPgerm-defined,
g belongs to C∞(M), and according to the partial invertibility property in
Lemma 2.4 1/g C∞(M). Consequently, defining f = g /g, we obtain the
i i
desired partition∈of unity. (cid:3)
Corollary 2.13. Smooth functions on M separate points on M.
8 HOˆNGVAˆNLEˆ,PETRSOMBERGANDJIRˇ´IVANZˇURA
Proof. Let x ,x M, x = x . We take an ε-neighborhood (ε) of x
1 2 ∈ 1 6 2 Nx2 2
such that x (ε). Then it suffices to take a function f from Lemma
2.9 and we h1a6∈veNfx(2x ) = 0 and f(x ) = 1. (cid:3)
1 2
Next we would like to define a notion of a locally smoothly contractible
differentiable structure on M. For this purpose we shall have to take a
product U(x) [0,1], where U(x) is an open neighborhood of x M, and
× ∈
endow it with a differentiable structure. Though the product U(x) [0,1]
×
need not be a pseudomanifold w.i.c.s., we can use the same concept of a
smooth structure as Mostow used [24, 3]. We say that C∞(M) is locally
§
smoothly contractible, if for any x M there exists an open neighborhood
∈
U(x) x together with a smooth homotopy σ :U(x) [0,1] U(x) joining
∋ × →
the identity map with the constant map U(x) x [24, 5]. Note that there
7→ §
is anaturalsmoothstructureC∞(U(x) [0,1]) generated by C∞(U(x)) and
×
C∞([0,1]) [24, 3], more precisely, the sheaf SC∞(U(x) [0,1]) is generated
§ ×
by π∗(SC∞([0,1])) and π∗(SC∞(U(x)))), where π and π is the projection
1 2 1 2
from U(x) [0,1] to [0,1] and U(x) respectively. In particular, π and π
1 2
×
are smooth maps.
DenotebyC∞ (L [0, ))thesubalgebrainC∞((L [0, ))consisting
L×{0} × ∞ × ∞
offunctionstakingconstantvaluesalongL 0 . ClearlyC∞ (L [0, ))
×{ } L×{0} × ∞
is isomorphic (as R-algebra) to C∞(cL).
w
Lemma2.14. 1. Afunctionf(x,t) C∞(L [0, ))belongstoC∞ (L
∈ × ∞ L×{0} ×
[0, )) if and only if f can be written as f(x,t) = t g(x,t) + c, where
g ∞C∞(L [0, )) and c R. ·
∈ × ∞ ∈
2. Let C∞(cL) C∞(cL) be the subalgebra consisting of all functions
e ⊂ w
f on which can be written as f([x,t]) = g(tf (x), ,tf (x)) for some g
1 k
C∞(Rk) and f C∞(L). Then C∞(cL) is a loca··ll·y smoothly contractibl∈e
i ∈ e
smooth structure on cL.
Proof. 1) The “if” assertion in the first statement is obvious. Let us prove
the “only if” assertion. For any f C∞(L [0, )) we have
∈ × ∞
1 df(x,tr) 1 df(x,tr)
f(x,t) = f(x,0)+ dr = f(x,0)+t dr.
Z dr Z d(tr)
0 0
Clearly 1 df(x,tr) dr C∞(L [0, )). This proves the first statement.
0 d(tr) ∈ × ∞
2) It iRs easy to see that C∞(cL) satisfies the first condition in Definition
e
2.3. WeobservethatC∞(cL)alsosatisfiesthesecondconditionofDefinition
e
2.3, i.e. j (f) C∞(cL) for any f C∞(cL), since this assertion is a
∗ ∈ e ∈ 0
consequence of Remark 2.16.4 below. To prove the second statement of
Lemma 2.14 it suffices to show that the map
F : cL(1) [0,1] cL, ([x,t],λ) [x,λt]
× → 7→
is a smooth map. Equivalently we have to show that any function F∗(f),
f C∞(cL(1)), belongs to the germ-defined C∞-ring C∞(cL(1) [0,1])
∈ e ×
generated by C∞(cL(1)) and C∞([0,1]). Repeating the previous argument,
e
SMOOTH STRUCTURES ON CONICAL PSEUDOMANIFOLDS 9
we can write f([x,t]) = g(tf (x), ,tf (x)), where f C∞(L) and g
1 k i
C∞(Rk). Clearly (F∗(f))([x,t],λ)·=··g(λtf (x), ,λtf (∈x)) can be writte∈n
1 k
···
asafunctionG(λ,tf (x), ,tf (x)), henceitbelongstoC∞(cL(1) [0,1]).
1 k
··· × (cid:3)
Now we show a geometric way to construct a nice locally smoothly con-
tractible smooth structure on a conical pseudomanifold M.
Definition2.15. AEuclideansmooth structureonapseudomanifoldw.i.c.s.
M is defined by a smooth embedding I : L Sl(1) Rl+1 and a trivial-
s s
→ ⊂
ization φ : cL for each s S as follows. Let Iˆ denote the induced
s s s M s
embedding Nof c→L Rl+1. A co∈ntinuous function on M is called smooth,
s
→
if it is smooth on Mreg and its restriction to is a pull back of a smooth
s
N
function on Rl+1 via Iˆ φ for all s.
s s
◦
By composing an embedding I with an isometric embedding g :
s l,l+k
Sl(1) Sl+k(1) Rl+k+1 we get another embedding I : L Sl+k(1).
s,+k s
→ ⊂ →
Denote by Iˆ the inducedembeddingcL Rl+k+1. Itis easy to see that
s,+k s
→
thesmoothstructuresdefinedbyI andI areequivalent. Thismotivates
s s,+k
us to give the following concept.
Two smooth embeddings I1 : L Sk1(1) Rk1+1 and I2 : L
Sk2(1) Rk2+1 are called Eucslideanse→quivalent, i⊂f there exists a sdiffeoms o→r-
⊂
phism Θ : Rk1+k2+2 Rk1+k2+2 such that Θ Iˆ1 = Iˆ . Two
→ ◦ s,+k2+1 s,+k1+1
Euclidean smooth structures are called Euclidean equivalent, if the corre-
sponding embeddings I are Euclidean equivalent. The embedding Iˆ φ is
s s s
◦
called a smooth chart around singular point s S .
M
∈
Remark 2.16. 1. A pseudomanifold w.i.c.s. may have more than one
Euclidean smooth structure. For example, conical pseudomanifolds cL(z =
1)andcL(z = 0) arenotisomorphic, whereL(z = θ)is thecircle in S2(1)
2 ⊂
R3(x,y,z) defined by the equation z = θ ( 1,1). This is proved by
∈ −
observing that the function f(x,y,z) = (x2 + y2 + z2)1/2 is smooth on
cL(z = 1) butit is not smooth on cL(z = 0). Using thediffeomorphism T :
2 α
R3 R3, z αz,α = 0, we conclude that all cL(z = α) are diffeomorphic,
→ 7→ 6
if 0 < α < 1. Note that the “smallest” smooth structure on cS1 is the
| |
isolated smooth structure cL(z = 0).
2. ClearlyanyEuclideansmoothstructureislocallysmoothlycontractible,
since the homotopy cL [0,1] cL : ([x,t],λ) [x,λ.t] is a smooth map.
× → 7→
3. In thenext section, see Proposition 3.5, we will show that for any fixed
L there are infinitely many non-equivalent Euclidean structures on cL.
4. Suppose that L is compact and C∞(cL) is a Euclidean smooth struc-
ture on cL. Let I : L Rk is defined by k smooth functions f
s i
C∞(L),i = 1,k. Then f˜(t→,x) := tf (x) are generators of the associate∈d
i i
Euclidean smooth structure C∞(cL). Thus C∞(cL) is a subalgebra of the
algebra C∞(cL).
e
10 HOˆNGVAˆNLEˆ,PETRSOMBERGANDJIRˇ´IVANZˇURA
We say that C∞(M) is finitely generated, if there is a finite number
of functions f , ,f C∞(M) such that any g C∞(M) is of form
1 k
g := gˆ(f , ,f··)·for so∈me gˆ C∞(Rk). Functions∈f , ,f are called
1 k 1 k
··· ∈ ···
generatorsofC∞(M). Remark2.16.4assertsthataEuclideansmoothstruc-
ture is finitely generated.
Proposition 2.17. Suppose that M and N are pseudomanifolds w.i.c.s.
provided with a finitely generated smooth structure. A continuous map σ :
M N is smooth, if and only if for each x M there exist a smooth chart
φ →: U(x) Rn, a smooth chart φ : U(σ∈(x)) Rm, and a smooth map
x σ(x)
σ˜ : Rn →Rm such that φ σ = σ˜ φ . Co→nsequently, two Euclidean
σ(x) x
→ ◦ ◦
smooth structures on a pseudomanifold w.i.c.s. M are Euclidean equivalent,
if and only if they are equivalent.
Proof. 1) The first assertion of Proposition 2.17 is a special case of Proposi-
tion 1.3.8 in[29], see also [23, Proposition 1.5] for an equivalent formulation.
For the convenience of the reader we give a proof of this assertion, which is
similar to the proof in the case of smooth manifolds. The “if” part is clear,
so we will prove the “only” part. Let y , ,y be coordinate functions on
1 m
Rm. By our assumption, y (φ σ) is·a··smooth function on U(x), hence
k σ(x)
◦
there exist smooth functions f on Rn such that f (φ ) = y(φ σ) for
k = 1,m. Now we define a smokoth map σ˜ : Rn Rmk byx settingσx ◦
→
σ˜(x) = (f (x), ,f (x)).
1 k
···
Clearly σ˜ satisfies the condition of our Proposition 2.17.1.
2) Let us prove the “only if” part of second assertion of Proposition 2.17.
Assume that two Euclidean smooth structures C∞(M) and C∞(M) are
1 2
Euclidean equivalent. Using the existence of a smooth partition of unity,
see Lemma 2.12, and finiteness of S , it is easy to see that C∞(M) and
M 1
C∞(M) are equivalent, i.e. there exists a homeomorphism σ :M M such
2 →
that σ∗(C∞(M)) = C∞(M). (The proof is similar to the proof for the case
1 2
of smooth manifolds M,N.)
Now we will prove the “if” part of the second assertion, i.e. we assume
that there exists a homeomorphism σ : M M such that σ∗(C∞(M)) =
→ 1
C∞(M). Let (I : L Sli Rli+1,φ : cL ) s S be
2 { i si → ⊂ si Nsi → si | i ∈ M}
embeddings defining C∞(M). Then (I ,σ φ ) s S are embed-
1 { si ◦ si | i ∈ M}
dingsdefiningC∞(M). ThisprovesthatC∞(M)andC∞(M)areEuclidean
2 1 2
equivalent. (cid:3)
Next we introducethe notion of thecotangent bundleof astratified space
X, which is similar to the notions introduced in [26], [29, B.1]. Note that
the germs of smooth functions C∞(X) is a local R-algebra with the unique
x
maximal ideal m consisting of functions vanishing at x. Set T∗(X) :=
x x
m /m2. Since the following exact sequence
x x
(2.1) 0 m C∞ j R 0
→ x → x → →