THE NON-CRITICALITY OF CERTAIN FAMILIES OF
AFFINE VARIETIES
by
Cornel Pintea
Abstract. In this paper we show that certain closed families of affine varieties of a Euclidean space and some families of spheres of a higher dimensional sphere as well, are not critical sets for certain special real valued functions.
Mathematics Subject Classification: Primary 55N10; Secondary 55R05;
Key words and phrases: affine varieties, homology and homotopy groups, critical points.
1. Introduction
In this paper we first get some topological information on the complement of certain families of affine varieties of a Euclidean space, expressed in terms of homology groups. In the last section we use these information to prove that the considered families of affine varieties, are not properly critical for certain special real valued functions. The family of spheres, as a closed subset of an m+1 dimensional sphere, produced as the closure of the inverse image by certain stereographic projections of a considered family of affine varieties, is also shown to be non-critical for the same class of special real valued functions. Let us finaly mention that the (non)-criticality problem of closed sets in the plane and in the three dimensional space has been studied before by M. Grayson, C. Pugh and A. Norton in [2,3] while the non-criticality of a family of fibers over a closed countable subset of the base space of a fibration has been studied before in [6]. In the previous papers [4,5] the non-criticality problem of finite and countable subsets of certain manifolds with respect to mappings having manifolds as target spaces is considered and studied.
2. Preliminary Results
We start this section by considering certain families of affine varieties of the
1.1. Proposition If {Ai}i≥1 is a family of affine varieties of the Euclidean
space
R
m+1 of various dimensions such thati
≠
j
⇒
δ
(
A
i,
A
j)
>
0
for some c>0,where
δ
(
A
i,
A
j)
is the distance betweenA
i andA
j, then(
i)
m q j i i m
q
R
A
H
R
A
H
\
1\
1 1 1 + ≥ ≥ +
⊕
≅
U
for all q≥1.Proof. Using Poincaré duality and the cohomology sequence, with compact supports, of the pair
(
,
)
1 1
U
≥ + i i mA
R
, we have:
0
≅
H
q+1(
R
m+1)
Hq(Rm+1)≅0|∫ |∫
which provides us the isomorphism
≅
≥ + + − ≥−
U
U
1 1 1 1
,
i i m q m c i i q mc
A
H
R
A
H
.Therefore, using the duality theorem [8, Theorem 6.9.10] and the above isomorphism, we have successively
) ( ) , ( ) ( ) \ ( . 1 1 1 1 . 1 . 1 1 i q m c j i i m q m c i i q m c i i m
q R A H A H R A H A
H − ≥ ≥ + + − ≥ − ≥
+
U
≅U
≅U
≅⊕
.Further on we will show, following the same way, that
).
,
(
)
(
1 1i m q m c i q m
c
A
H
R
A
H
−≅
− + + Indeed using Poincaré duality and the cohomology sequence, with compact supports, of the pair(
R
m+1,
A
i)
, we have:
0
(
1)
1 + +
≅
mq
R
H
(
m+1)
≅
0
q
R
H
|∫ |∫
L L→ →
U
→U
→ − + + →≥ + + − ≥ − +
− ( ) ( ) ( , ) 1( 1)
. 1 1 1 . 1 . 1 . m q m c i i m q m c i i q m c m q m
c R H A H R A H R
H
which provides us the isomorphism
H
cm−q(
A
i)
≅
H
cm−q+1(
R
m+1,
A
i).
Therefore, using the same duality theorem [8, Theorem 6.9.10] and the aboveisomorphism, we have:
), \ ( . ) , ( )
( 1 1 1
. .
i m q i
m q m c i q m
c A H R A H R A
H − ≅ − + + ≅ +
and the proof is now complete.
2.2. Remark If we replace in proposition 1.1 the Euclidean space
R
m+1 with ann-connected manifold Mm+1 and the family
{ }
A
i i≥1 of affine varieties with a closed submanifold N having the connected components{ }
N
i i≥1 of various dimensions, then the similar isomorphism(
i)
qj i
i
q
M
N
H
M
N
H
\
\
1 1
⊕
≥ ≥
≅
U
still holds for 1≤q≤n-1.2.3. Corollary Let {Ai}i≥1 be a family of affine varieties of the Euclidean space Rm+1
of various dimensions not exceeding m-1 such that
i
≠
j
⇒
δ
(
A
i,
A
j)
>
0
for somec>0. In these conditions
U
1 1
\
≥ +
i i
m
A
R
is connected and for any1
≤
q
≤
m
−
1
wehave
q
J
i i m
q
m
R
A
Z
H
≅
≥ +
−
(
\
)
1 1
U
where by
J
q we denote the set{
j
≥
1
:
dim
A
j=
q
}
.Proof. Indeed, the connectedness follows easily by using a weak version of Thom theorem. Concerning the stated isomorphism we first apply proposition 1.1 to deduce that
(
\
)
,
\
11 1
1
i m q m j i
i m
q
m
R
A
H
R
A
H
− +≥ ≥
+
and then observe that
R
m+1\
A
i has the homotopy type of the sphere m q.
S
− □If
S
m+1 is the (m+1)-dimensional sphere andp
∈
S
m+1 a point, consider the stereographic projectionϕ
p :Sm+1\{ }
p →p┴,
p p x p p x x x p , 1 , ) ( − − =
ϕ
, wherep┴:={q∈Sm+1| q,p =0} is the hyperplane of
R
m+2 orthogonal on p. It is obviously a diffeomorphism, its inverse being: 1 −
p
ϕ
p┴
→
S
m+1\
{
p
}
,(
)
.
1
||
||
1
||
||
2
)
(
2 2 1+
−
+
=
−y
p
y
y
y
pϕ
Let α:
R
n
→
R
m+2 be an embedding. It is easy to see that N=α(R
n)is a closed submanifold ofR
m+2 iflim
||x||→∞||
α
(
x
)
||
=
∞
.Consequently, for a closed submanifold N of Rm+2 iffeomorphic with some Euclidean space which is contained in p┴ for some
p
∈
S
m+1, we have that)
(
1N
p −ϕ
={ }
p ∪ϕ
p−1(N).2.4. Corollary Let {Ai}i≥1 be a family of affine varieties of the Euclidean space 1 +
m
R
of various dimensions such that
2
≤
dim
A
i≤
m
−
1
andi
≠
j
⇒
δ
(
A
i,
A
j)
>
0
for some c>0. Assume also that for any i≥1 there existsr
(
i
)
∈
{
1
,...,
l
}
with the property that Ai ⊂ pr⊥(i) for some 1,...,
∈
m+1l
S
p
p
and such thatϕ
− ( )∩ϕ
−1 ( )=φ
) ( 1
)
(i i r j j
r N N
for
i
≠
j
. In these conditions the equalitiesU
1 1 ) ( ( ) ≥ − i i i r Aϕ
=U
1 1 ) ( ( ) ≥ − i i i r A
ϕ
={
}
U
1 1
) (
1
,....,
(
)
≥ −
∪
i i i r lA
p
p
ϕ
hold as well as the isomorphism
≅
≥ − + −U
1 1 ) ( 1dim
\
(
)
j j j r m A
m
S
A
H
i
ϕ
Z(JdimAi)
.
Proof. Indeed
U
U
1 1 ) ( 1 1 1 1 ) ( 1
)
(
\
})
,....,
{
\
(
)
(
\
≥ − + ≥ − +=
j i j r l m j j r mA
p
p
S
Aj
of 1 ( ) ) (j Aj r
−
ϕ
is a closed submanifold of m 1\
{
1,....,
l}
p
p
S
+ diffeomorphic withRdimAi, the whole union
U
1 1 ) ( ( ) ≥ − j j r Ajϕ
being also a closed submanifold of the (m-1)-connected one
S
m+1\
{
p
1,....,
p
l}
. The (m-1)-connectedness ofS
m+1\
{
p
1,....,
p
l}
follows from [4, Proposition 2.3] and it ensures us, according to remark 2.2, that
≅ ≥ − + −
U
1 1 ) ( 1 1dim ( \{ ,...., })\ ( ) j j j r l m A
m S p p A
H
i ϕ
(
( 1\{ 1,...., })\ (1)( ))
dim 1 j j r l m A m j A p p S H i − + − ≥
⊕
≅ ϕOn the other hand the inclusion of
} ,..., ˆ ,..., { \ ) ) ( \ ( ) ( \ }) ,...., { \
(Sm+1 p1 pl ϕr−(1j) Aj = Sm+1 ϕr−(1j) Aj p1 pr(j) pl
in
)
(
\
})
{
\
(
)
(
\
}
{
\
))
(
\
(
1 (1) ( ) 1 (1) m 1 r(j) r(1j) jj j r m j r j j r
m
A
p
S
A
S
p
A
S
+ϕ
−=
+ϕ
−=
+ϕ
−induces isomorphism at the level of homotopy groups in dimension less then or equal to m-1 for each
1
≤
j
≤
l
, such that, according to Whitehead theorem, it also induces isomorphism at the level of homology groups in dimension less then or equal to m-1for each 1≤i≤l. Finaly the restriction of ) (j r
p
ϕ
to (Sm+1\{pr(j)})\ϕ
r−(1j)(Aj) realizes a diffeomorphism between ( \{ })\ 1 ( )) ( ) ( 1 j j r j r m A p
S +
ϕ
− and pr⊥(j) \Aj which in its turn has the homotopy type ofS
m−dimAi. Consequently we have succesively≅
=
≥ − + − ≥ − +−
U
U
1 1 ) ( 1 1 dim 1 1 ) ( 1
dim
\
(
)
(
\
{
,....,
})
\
(
)
j j j r l m A m j j j r m A
m
S
A
H
S
p
p
A
H
i
i
ϕ
ϕ
(
)
≅
(
)
≅
(
−)
≅
− ∈ ⊥ − ∈ ⊥ − ≥⊕
⊕
⊕
j i i A i i A i A m A m J j j j r A m J j j j r A m jS
H
A
p
H
A
p
H
dim ( ) dim ( ) dim dim1 dim dim
\
\
Z(JdimAi)
.
2.5. Remark The closed subset
U
j i r i j rA
= − ) ( 1 ) ((
)
ϕ
ofS
m+1 is a union of embedded3. Application
As we have already mentioned before, in this section we use the information already obtain on the topology of the complement of a considered family {Ai}i≥1 of affine varieties in the Euclidean space
R
m+1 and those on the topology of the complement of a family of embedded spheresU
1 1
) ( ( ) ≥
−
i
i i r A
ϕ
in the higher dimensionalone
S
m+1, to show that the considered families are not critical sets for certain real valued functions.Let M,N be two differentiable manifolds and consider
}
))
(
(
)
(
|
)
,
(
{
)
,
(
=
∈
∞∩
=
φ
∞
M
N
f
C
M
N
B
f
f
C
f
CS
, where R(f) is the regularset of f, B(f)=f(C(f)) is its set of critical values while C(f) is the critical set of f. Because of the empty intersection between B(f) and f(R(f)) a mapping f from
)
,
(
M
N
C
∞ obviously separates the critical values by the regular ones.3.1. Proposition ([6])
f
∈
CS
∞(
M
,
N
)
iff(
)
1(
(
))
f
B
f
f
C
=
− . (ii) If M is aconnected differentiable manifold and
f
∈
CS
∞(
M
,
R
)
is such that R(f)=M\C(f) is also connected, then f(R(f))=(mf,Mf), wherem
inf
f
(
x
)
M x
f
=
∈ ,M
sup
f
(
x
)
M x f∈
=
andR
M
m
f
B
(
)
⊆
{
f,
f}
I
. Moreover, if M is compact, thenm
f,
M
f∈
R
and}
,
{
)
(
f
m
fM
fB
=
.3.2. Theorem Let {Ai}i≥1 be a family of affine varieties of the Euclidean space
R
m+1of various dimensions not exceeding m-1 such that
i
≠
j
⇒
δ
(
A
i,
A
j)
>
0
for somec>0.then there is no any mapping
f
∈
CS
∞(
M
,
R
)
such that C(f)=A:=U
1 ≥
i i
A
andthe restriction
A Rm
f
\ 1
|
+ is proper.Proof. Assuming that such an application exists it follows on the one hand that
}
,
{
)
(
f
m
fM
fB
⊆
and on the other hand its restriction Rm+1\C(f)
→
Im f=(
m
f,
M
f)
, pa
f(p) is a proper submersion, that is a locally trivial fibration, via Ehresmann’s theorem, whose compact fiber we are denoting by F. Its base space(
m
f,
M
f)
being contractible, it follows that the inclusion:
\
(
)
1
f
C
S
F
i
F
→
m+)) ( \ ( )
( : )
( 1
f C S F
iF q q m
q
+
→
π
π
π
are all isomorphism. Consequently, usingthe Whitehead theorem, it follows that the induced group homomorphisms ))
( \ ( )
( : )
(i H F H S 1 C f
H m
q q
F q
+
→
are also isomorphism. Because for some
i
A
J
i
≥
1
,
dim is infinite it follows thatU
11 ) ( 1
dim 1
dim
dim
(
)
(
\
(
))
(
\
(
),
≥ − +
− +
−
−
≅
≅
j
j j r m
A m m
A m A
m
F
H
S
C
f
H
S
A
H
i i
i
ϕ
the last one having a subgroup isomorphic with Z(JdimAi)
, that is a contradiction with the well known fact that any compact manifold has finitely generated homology groups.
The proof of the next theorem is completely similar.
3.3. Theorem Let {Ai}i≥1 be a family of affine varieties of the Euclidean space 1 +
m
R
of various dimensions such that
2
≤
dim
A
i≤
m
−
1
andi
≠
j
⇒
δ
(
A
i,
A
j)
>
0
for some c>0. Assume also that for any i≥1 there existsr
(
i
)
∈
{
1
,...,
l
}
with the property that Ai ⊂ pr⊥(i) for some 1,...,
∈
m+1l
S
p
p
and such thatϕ
− ( )∩ϕ
−(1)( )=φ
1)
(i i r j j
r N N
for
i
≠
j
. In these conditions there is no any mapping(
1,
)
R
S
CS
f
∈
∞ m+ such thatU
1 1
) ( ( ) )
(
≥
− ⋅
=
i
i i r A
f
C
ϕ
References
[1] Hu, S-T., Homotopy Theory, Academic Press, New York and London 1959. [2] M. Grayson $\&$ C. Pugh, Critical Sets in 3-space, Institut des Hautes Études Scientifiques, No. 77(1993), 5-61.
[3] A. Norton & C. Pugh, Critical Sets in the Plane, Michigan Math. J. 38 (1991), 441-459.
[4] Pintea, C., Differentiable Mappings with an Infinite Number of Critical Points, Proceedings of the A.M.S., vol. 128, Nr. 11, (2000), 3435-3444.
[5] Pintea, C., Some Pairs of Manifolds with Infinite Uncountable φ-Category, Topological Methods in Nonlinear Analysis, Vol. 21(2002), 101-113.
[6] Pintea, C., Closed Sets which are not
CS
∞-critical, Accepted to be published in Proc. A.M.S.[8] Whitehead, G. W., Elements of Homotopy Theory, Springer-Verlag, New-York, Heidelberg, Berlin 1978.
