L2 -invariants and Morse - Smale flows on manifolds
We study the homotopy invariants of free cochain and Hilbert complexes. These L2 -invariants are applied to the calculations of exact values of minimal numbers of closed orbits of some indexes of nonsingular Morse - Smale flows on manifolds of large dimensions.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509395971997696 |
|---|---|
| author | Sharko, V. V. Шарко, В. В. |
| author_facet | Sharko, V. V. Шарко, В. В. |
| author_sort | Sharko, V. V. |
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| datestamp_date | 2020-03-18T19:51:20Z |
| description | We study the homotopy invariants of free cochain and Hilbert complexes.
These L2 -invariants are applied to the calculations of exact values of minimal numbers of closed orbits of some indexes of nonsingular Morse - Smale flows on manifolds of large dimensions.
|
| first_indexed | 2026-03-24T02:40:26Z |
| format | Article |
| fulltext |
UDC 517.938.5
V. V. Sharko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
L2-INVARIANTS AND MORSE – SMALE FLOWS
ON MANIFOLDS
L2
-INVARIANTY TA POTOKY MORSA – SMEJLA
NA MNOHOVYDAX
We study the homotopy invariants of free cochain and Hilbert complexes. These L2-invariants are applied to the
calculations of exact values of minimal numbers of closed orbits of some indexes of nonsingular Morse – Smale
flows on manifolds of large dimensions.
Vyvçagt\sq homotopiçni invarianty vil\nyx kolancghovyx ta hil\bertovyx kompleksiv. Ci L2
-invari-
anty zastosovugt\sq pry obçyslenni toçnyx znaçen\ minimal\nyx çysel zamknenyx orbit fiksovanyx
indeksiv nesynhulqrnyx potokiv Morsa – Smejla na mnohovydax velykyx rozmirnostej.
1. Introduction. LetMn be a closed smooth manifold. By a nonsingular Morse – Smale
flow onMn we shall mean a flow ϕt satisfying the following conditions:
1) chain-recurrent set R of ϕt consists of finitely many hyperbolic closed orbit;
2) for each pair of closed orbits of ϕt the intersection of their stable and unstable
manifolds is transversal;
3) all closed orbits of ϕt are untwisted.
Notice that usually by a nonsingular Morse – Smale flow one means a flow satisfying
the conditions 1) and 2) only.
Let ϕt be a nonsingular Morse – Smale flow on Mn. Denote by Ai, i = 0, . . . , n,
the number of closed orbits of ϕt of index i. Let also Ri = dimHi(Mn; Q). Then the
following inequalities hold true:
Ai ≥ Ri −Ri−1 + . . .+R0 (1)
for all i = 0, . . . , n, see [1 – 3]. Notice that they are not strict in general.
In this paper we study the following problem:
Problem. For a manifold Mn and i = 0, . . . , n find a nonsingular Morse – Smale
flow ϕt onMn with minimal possible value Ai of (untwisted!) closed orbits of index i.
Using numerical invariants of free cochain and Hilbert complexes of manifold Mn,
see [3, 4], we give an answer to this problem for i = 0, 1, n− 2, n− 1 and 3 ≤ i ≤ n− 4
when dimMn ≥ 6. Thus a unique unsettled case is i = 2 (and n− 3 by duality).
By definition the i-th Morse S1-numberMS1
i (Mn) of a manifoldMn is the minimal
number of closed orbits of index i taken over all nonsingular Morse – Smale flows on
manifoldMn.
It is convenient to define the following function ρ: Z→ N by ρ(x) = x for x ≥ 0 and
ρ(x) = 0 for x < 0.
Let Mn, n ≥ 6, be a closed manifold with zero Euler characteristic and with
π1(Mn) = π. Then the Morse S1-numbers of the manifold Mn are given by the fol-
lowing formulas:
MS1
0 (Mn) =MS1
n−1(M
n) = 1,
MS1
1 (Mn) =MS1
n−2(M
n) = µ(π)− 1,
MS1
i (Mn) = Ŝi+1
(2) (Mn) + ρ
[
(−1)i
i∑
j=0
(−1)j dimN [G]
(
Hj
(2)(M
n)
)]
for 3 ≤ i ≤ n− 4, where µ(π) is the minimal number of generators of π.
c© V. V. SHARKO, 2007
522 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 523
2. Stable invariants of finitely generated modules and L2-modules. We give sev-
eral definitions and results about finitely generated modules over group rings. Most of the
facts are also valid for modules over a wider class of rings.
Let Z be the ring of integers and C the field of complex numbers. Let G be a dis-
crete countable group. Denote its integer and complex group rings by Z[G] and C[G]
respectively. Each group ring admits an augmentation epimorphism ε: Z[G] → Z (ε:
C[G] → C) defined by ε
(∑
i
αigi
)
= Σiαi. Denote by I[G] the kernel of the epimor-
phism ε. The ring C[G] has also an involution ∗: C[G]→ C[G] given by
(∑
i
αigi
)∗
=
=
∑
i
αig
−1
i , where α is the conjugate to α ∈ C. Define the trace tr: C[G] → C by
tr
(∑k
i
αigi
)
= α1, where α1 is the coefficient at g1 = e, the unit of the group G.
The ring C[G] has also an inner product
〈∑
i
αigi,
∑
i
βigi
〉
=
∑
i
αiβi. Then for
each r ∈ C[G] its norm |r| can be defined by |r| = tr(rr∗)1/2. Let L2(G) be the com-
pletion of C[G] with respect to this norm. Then L2(G) has a structure of a Hilbert space
(with inner product given by the same formula as for the group ring C[G]) and elements
of G constitute its orthonormal basis. Notice that C[G] acts faithfully and continuously
by left multiplication on L2(G)
C[G]× L2(G)→ L2(G),
therefore we can regard C[G] as a subset of the set B(L2(G)) of bounded linear operators
on L2(G). A a week closure of C[G] in B(L2(G)) is called the von Neumann algebra of
G and denoted by N [G]. The map N [G] → L2(G) given by w → w(e) turns out to be
injective and this allows us to identify N [G] with a subspace of L2(G).
Thus algebraically we have C[G] ⊂ N [G] ⊂ L2(G). The involution and the trace
map on C[G] extends toN [G] by the same formulas. Moreover, the trace map can also be
extended to the space Mn(N [G]) of (n × n)-matrices over von Neumann algebra N [G]
by tr(W ) =
∑n
i=1
wii, whereW = (wij) is a matrix with entries in N [G].
Following Cohen [5] we will now define a notion of Hilbert N [G]-module. Let E =
= N
⋃
∞, where∞ is the first infinite cardinal. For each n ∈ E letL2(G)n be the Hilbert
direct sum of n copies of L2(G). Thus L2(G)n is a Hilbert space. The von Neumann
algebraN [G] acts on L2(G)n from the left, whence L2(G)n is a leftN [G]-module called
a free L2(G)-module of range n.
The left HilbertN [G]-moduleM is a closed left C[G]-submodule of L2(G)n for some
n ∈ E. If n ∈ N, then Hilbert N [G]-moduleM is called finitely generated.
Following [5, 6] we will say that a Hilbert N [G]-submodule of M is a closed left
C[G]-submodule of M , a Hilbert N [G]-ideal is a Hilbert N [G]-submodule of L2(G),
and a Hilbert N [G]-homomorphism f : M → N between Hilbert N [G]-modules is a
continuous left C[G]-map.
Let M be a Hilbert N [G]-module and let p : L2(G)n → L2(G)n be a orthogonal
projection ontoM ⊂ L2(G)n. Then the number
dimN [G](M) = tr(p) =
n∑
i=1
〈p(ei), ei〉L2(G)n
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
524 V. V. SHARKO
is called a von Neumann dimension of M , where ei = (0, . . . , 1(g), . . . , 0) is a standard
basis in L2(G)n. It is known that dimN [G](V ) is a nonnegative real number [6].
In what follows we will assume, unless otherwise stated, that Λ is an associative ring
with unit e and M is a left finitely generated Λ-module. Rings for which the rank of a
free module is uniquely defined are called IBN -rings. It is known that the group rings
Z[G] and C[G] are IBN -rings. In the present paper, we consider only IBN -rings. For
a module M let µ(M) be the minimal number of its generators. If M is zero, then
µ(M) = 0. Evidently, µ(M
⊕
Fn) ≤ µ(M) + n, where Fn is a free module of rank n.
There are examples (of stably-free modules) when this inequality is strict [3]. Recall that
a Λ-module M is called stably-free if the direct sum of M with some free Λ-module Fk
is free.
A ring Λ is said to be Dedekind-finite if, for any λ1, λ2 ∈ Λ, relation λ1 · λ2 = 1
implies λ2 ·λ1 = 1. A ring Λ is stably-finite if the matrix rigsMn(Λ) are Dedekind-finite
for all n ∈ N. The terminology here follows the usage of workers in operator algebras.
Definition 1. Let d be a function from the category of Λ-modulesM (not necessarily
over group rings) to the set of nonnegative integers N0. We say that this function d is weak
additive if the following conditions holds true:
a) d(M) = d(N) if modulesM and N are isomorphic;
b) d(M) = 0 if and only ifM = 0;
c) d(M
⊕
Fn) = d(M) + n for any free module Fn of rank n ∈ N.
Definition 2. For a finite generated module M over IBN -ring Λ let us define the
following function:
µs(M) = lim
n→∞
(µ(M ⊕ Fn)− n).
Lemma 1. The function µs(M) is well defined and is weak additive for modules
over stably-finite rings.
Proof. Condition a) is obvious. Let us prove b). Suppose that µs(M) = 0 for some
non-zero module M . Then there exists n ∈ N such that for the module N = M
⊕
Fn
we have µ(N) = n. Therefore, there is an epimorphism f: Fn → N of a free module Fn
of rank n onto the module N . In addition, there exists a canonical epimorphism p: N =
= M
⊕
Fn → Fn with the kernel equal to M . Let K be the kernel of the epimorphism
p ◦ f: Fn → Fn. It follows from the construction of f and p thatK �= 0. Moreover, p ◦ f
is an epimorphism onto a free module, therefore it splits, whenceK ⊕Fn = Fn. Since Λ
is stably-finite, we obtain thatK = 0. The condition c) is proved in [7].
Corollary 1. The function µs(M) is week additive for modules over the rings Z[G]
and C[G].
Proof. It follows from theorems of Kaplansky and Cockroft [3] that the group rings
Z[G] and C[G] are hopfian.
Remark 1. It is clear, that for any non-zero moduleM we have that
0 < µs(M) � µ(M).
The difference
µ(M)− µs(M)
estimates how much times the addition a free module of rank one to the modulesM
⊕
kΛ,
k = 0, 1, . . . , does not increase by one the number µ(M
⊕
kΛ). There are also inequali-
ties
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 525
µ(M
⊕
N) � µ(M) + µ(N),
µs(M
⊕
N) � µs(M) + µs(N).
It is not hard to construct examples of projective modules in which strict inequalities hold
true.
Lemma 2. For every finitely generated module M over IBN -ring Λ there exists
n ∈ N such that for the module N = M
⊕
nΛ and for allm ≥ 0 we have that
µ(N
⊕
mΛ) = µ(N) +m.
Moreover, µ(N) is additive for the module N .
Proof. An existence of such a number n is proved in [3]. It is clear, that if for a
module N we have
µ(N
⊕
mΛ) = µ(N) +m,
then µ(N) = d(N) by the virtue of the definition of the function d(N).
Let N be a submodule of the free module Fk. Following H. Bass we define f-rank of
the pair (N,Fk) to be the largest nonnegative integer r such thatN contains a direct sum-
mand of Fk isomorphic to free module Fr. We shall denote this number by f-rank(N,Fk).
By definition f-rank of (N,Fk) is called additive if
f-rank(N ⊕ Fm, Fk ⊕ Fm) = f-rank(N,Fk) +m.
We note that for any submoduleN of the free module Fk there exist a positive integerm0
such that f-rank of (N ⊕Fm, Fk⊕Fm) is additive for allm > m0, see [3] (Lemma III.7).
3. Homotopy invariants of cochain complexes. It is known that the homology
(cohomology) of a free chain (cochain) complex over the ring of integers determines its
homotopy type. But for a free chain (cochain) complex over arbitrary rings this is not the
case, one should require the existence of a chain (cochain) map that induces homology
(cohomology) isomorphisms.
Definition 3. Let (C∗, d∗) : C0
d1←− C1
d2←− . . . dn←− Cn be a free chain complex.
Then the following chain complex:(
C∗(i), d∗(i)
)
: C0
d1←− C1
d2←− . . . di−→ Ci
is called the i-th skeleton of the chain complex (C∗, d∗).
It is well known that the Euler characteristic χ(C∗, d∗) =
∑
(−1)iµ(Ci) is an in-
variant of the homotopy type of the chain complex (C∗, d∗). But in general the i-th Euler
characteristics of homotopy equivalent chain complexes (C∗, d∗) and (D∗, ∂∗) may differ
each from other.
Definition 4. Let (C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Cn be a free chain complex
and
χi(C∗, d∗) = (−1)iχ
(
C∗(i), d∗(i)
)
.
The following number:
χa
i (C∗, d∗) = min
{
χi(D∗, ∂∗) | (D∗, ∂∗) is homotopy equivalent to (C∗, d∗)
}
will be called the i-th Euler characteristics of the chain complex (C∗, d∗).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
526 V. V. SHARKO
For cochain complexes the definitions are similar.
Theorem 1. Let (C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Cn be a free chain complex.
Then χa
i (C∗, d∗) = χi(C∗, d∗) if and only if f-rank of (di+1(Ci+1), Ci) is additive and is
equal to zero:
f-rank(di+1(Ci+1), Ci) = 0.
Proof. Necessity. Suppose that χa
i (C∗, d∗) = χi(C∗, d∗) but
f-rank(di+1(Ci+1), Ci) = r > 0.
The the module Ci can be represented in the form Ci = C̃i ⊕ Fr. Therefore stabilizing
the boundary homomorphisms di and di+2 via the free module Fr we can assume that the
submodule C̃i is free and there is a decomposition
Ci+1 ⊕ Fr = C̃i+1 ⊕ F̃r
such that di+1(0 ⊕ F̃r) = 0 ⊕ Fr. Canceling the fragment 0 ← Fr ← F̃r ← 0 from
(C∗, d∗) we obtain the chain complex (C̃∗, d̃∗) such that χi(C̃∗, d̃∗) < χi(C∗, d∗). It
follows that the chain complexes (C∗, d∗) and (C̃∗, d̃∗) are homotopy equivalent but
χi(C̃∗, d̃∗) < χi(C∗, d∗) which contradicts to the definition of χa
i (C∗, d∗).
If f-rank(di+1(Ci+1), Ci) is nonadditive the proof is similar.
Sufficiency. Suppose that there exists a chain complex
(C̃∗, d̃∗) : C̃0
d̃1←− C̃1
d̃2←− . . . d̃n←− C̃n
such that f-rank of
(
d̃i+1(C̃i+1), C̃i
)
additive and equal to zero but
χi(C̃∗, d̃∗) > χa
i (C∗, d∗).
Then there exists a chain complex (C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Cn which is
homotopy equivalent to (C̃∗, d̃∗) and such that χi(C∗, d∗) = χa
i (C∗, d∗). Then it follows
from necessity of our theorem that f-rank of
(
di+1(Ci+1), Ci
)
is additive and equal to
zero.
Then by Cockroft – Swan’s lemma we can stabilize the boundary homomorphisms
dj and d̃j , j = 1, 2, . . . , n, via some free modules Fkj and Fk̃j
respectively and obtain
isomorphic chain complexes
(
Cst
∗ , d
st
∗
)
and
(
C̃st
∗ , d̃
st
∗
)
. By the construction the modules
Fki ⊕ Ci ⊕ Fki+1 and Fk̃i
⊕ C̃i ⊕ Fk̃i+1
are isomorphic and therefore χi(Cst
∗ , d
st
∗ ) =
= χi(C̃st
∗ , d̃
st
∗ ). We note that
ki+1 = f-rank(dst
i+1(Ci+1 ⊕ Fki+1 ⊕ Fki+2), Ci ⊕ Fki ⊕ Fki+1) =
= f-rank(d̃st
i+1(C̃i+1 ⊕ Fk̃i+1
⊕ Fk̃i+2
), C̃i ⊕ Fk̃i
⊕ Fk̃i+1
) = k̃i+1.
Hence we get χi(C̃∗, d̃∗) = χa
i (C∗, d∗).
Theorem 1 is proved.
Remark 2. If (C∗, d∗): C0 d0
−→ C1 d1
−→ . . .
dn−1
−→ Cn is a free cochain complex
then reversing arrows in Theorem 1 we obtain that χa
i (C∗, d∗) = χi(C∗, d∗) if and only
if f-rank of (di(Ci), Ci+1) is additive and equal to zero:
f-rank(di(Ci), Ci+1) = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 527
4. The value of the i-th Euler characteristic. Let (C∗
(2), d
∗): C0
(2)
d0
−→ C1
(2)
d1
−→ . . .
. . .
dn−1
−→ Cn
(2) be a sequence of free Hilbert N [G]-modules and bounded C[G]-map such
that di+1 ◦di = 0. Such a sequence is called a Hilbert complex. The reduced cohomology
of the Hilbert complex (C∗
(2), d
∗) is the collection of L2(G)-modules Hi
(2)(C∗
(2), d
∗) =
Ker di/ Im di−1.
Definition 5. Let
(C∗, d∗): C0 d0
−→ C1 d1
−→ . . .
dn−1
−→ Cn
be a free cochain complex over Z[G]. Then the following complex:L2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
:
L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d0
−→ L2(G)
⊗
Z[G]
C1
Id
⊗
Z[G] d1
−→ . . .
Id
⊗
Z[G] dn−1
−→ L2(G)
⊗
Z[G]
Cn
of free Hilbert N [G]-modules is called a Hilbert complex generated by the Z[G]-cochain
complex (C∗, d∗).
Consider the i-th skeletons of these complexes
(C∗(i), d∗(i)) : C0 d0
−→ C1 d1
−→ . . .
di−1
−→ Ci,L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
: L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d0
−→ L2(G)
⊗
Z[G]
C1
Id
⊗
Z[G] d1
−→ . . .
. . .
Id
⊗
Z[G] di−2
−→ L2(G)
⊗
Z[G]
Ci−1
Id
⊗
Z[G] di−1
−→ L2(G)
⊗
Z[G]
Ci.
Set Γi = Ci/di−1(Ci−1). It is clear that
Γ̂i = L2(G)
⊗
Z[G]
Ci/Id
⊗
Z[G]
di−1
L2(G)
⊗
Z[G]
Ci−1
is the i-th Hilbert N [G]-module of the reduced cohomology of the i-th skeleton of the
Hilbert complex L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
.
For a cochain complex (C∗, d∗) over Z[G] set
Ŝi
(2)(C
∗, d∗) = µs(Γi)− dimN [G] Γ̂i.
If (C∗, d∗) and (D∗, ∂∗) are two homotopy equivalent free cochain complexes over
the group ring Z[G] then
Ŝi
(2)(C
∗, d∗) = Ŝi
(2)(D
∗, ∂∗).
The numbers Ŝi
(2)(C
∗, d∗) are nonnegative for every i.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
528 V. V. SHARKO
Theorem 2. Let (C∗, d∗) : C0 d0
−→ C1 d1
−→ . . .
dn−1
−→ Cn be a free cochain complex
over Z[G]. Then
χa
i (C∗, d∗) =
= (−1)i
i∑
j=0
(−1)j dimN [G]
Hj
(2)
L2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
+ Ŝi+1
(2) (C∗, d∗).
Proof. Suppose that the cochain complex (C∗, d∗) : C0 d0
−→ C1 d1
−→ . . .
dn−1
−→ Cn is
such that χi(C∗, d∗) = χa
i (C∗, d∗). Consider the Hilbert complexL2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
:
L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d0
−→ L2(G)
⊗
Z[G]
C1
Id
⊗
Z[G] d1
−→ . . .
Id
⊗
Z[G] dn−1
−→ L2(G)
⊗
Z[G]
Cn
and let L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
:
L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d0
−→ L2(G)
⊗
Z[G]
C1
Id
⊗
Z[G] d1
−→ . . .
. . .
Id
⊗
Z[G] di−2
−→ L2(G)
⊗
Z[G]
Ci−1
Id
⊗
Z[G] di−1
−→ L2(G)
⊗
Z[G]
Ci
be its i-th skeleton.
It is clear from additivity of dimN [G] that
χi(C∗, d∗) =
= (−1)i
i∑
j=0
(−1)j dimN [G]
Hj
(2)
L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
=
= dimN [G]
(
Hi
(2)
(
L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
))
−
−(−1)i−1
i−1∑
j=0
(−1)j dimN [G]
(
Hj
(2)
(
L2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
))
.
Similarly to [4] one can check that
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 529
dimN [G]
Hi
(2)
L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
=
= dimN [G]
Hi
(2)
L2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
+ Ŝi+1
(2) (C∗, d∗).
Theorem 2 is proved.
5. Topological applications. Let Y be a topological space endowed with some struc-
ture K = K(Y ) of a finite CW -complex. Denote by Ki the i-th skeleton of K(Y ). Let
also n(σj) be the total number of j-cells ofK(Y ) and
χi(K(Y )) = (−1)iχ(Ki) = (−1)i
i∑
j=0
(−1)jn(σj).
Definition 6. The cellular i-th Euler characteristics of the space Y is the minimal
value of χi(K(Y )) taken over all cellular decompositionK(Y ) of Y :
χc
i (Y ) = min
{
χi(K(Y )) | K(Y ) is a cellular decomposition of Y
}
.
Remark 3. LetMn be a closed (possibly only topological) manifold having a han-
dle decomposition. Then similarly to the Definition 6 we can define the i-th handle Euler
characteristics χh
i (Mn) of the manifoldMn using handle decompositionsMn.
Evidently, that if a closed manifoldMn admits a handle decomposition, then contract-
ing each handle to its middle disk we obtain some cell decomposition ofMn. Therefore
χc
i (M
n) ≤ χh
i (Mn).
Note that for a closed simply-connected smooth manifold Mn(n > 4) the following
equality holds true:
χc
i (M
n) = χh
i (Mn) = µ(Hi(Mn,Z))− (−1)i−1
i−1∑
j=0
(−1)jµ(Hj(Mn,Q)).
Now let K be a CW -complex and p: K̃ → K be the universal covering of K. Using
the map p we can lift the CW -complex structure ofK to K̃. Then the fundamental group
π = π1(K) acts free on K̃ also preserving its CW -structure. This action turns each
chain group Ci(K̃,Z) into a left module over the group ring Z[π]. It is evident that the
resulting chain module Ci(K̃,Z) is free. Moreover, lifting each i-cell of K to some cell
of K̃ we obtain a finite set of generators of Ci(K̃,Z) over Z. As a result we get a free
chain complex over the ring Z[π]:
C∗(K̃): C0(K̃,Z) d1←− C1(K̃,Z) d2←− . . . dn←− Cn(K̃,Z).
Definition 7. For a CW -complexK the following number χa
i (K) defined by
χa
i (K) = χa
i (C∗(K̃)).
is called the i-th algebraic Euler characteristics ofK.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
530 V. V. SHARKO
It is well known that any two chain complexes constructed from some cellular decom-
positions of the same topological space K have the same homotopy type. Therefore it
follows directly from the previous discussion or from [3, 8] that the numbers χa
i (K) are
invariants of the homotopy type of the cell complexK.
It is clear that for a cell complexK we have that
χa
i (K) ≤ χc
i (K).
For a smooth manifold Mn it is possible to define a cochain complex via Morse
functions (handle decomposition). The details can be found in [3]. It is proved in [8] that
the all chain complexes constructed from some Morse functions (handle decomposition)
on the manifold Mn have the same homotopy type. This means that the values of i-th
algebraic Euler characteristic χa
i (Mn) of Mn do not depend on the way of constructing
a chain complex.
If the fundamental group π = π1(K) of K is non-zero then for calculation of the
values of some χc
i (Y ) one can use L2-theory. To describe this let us recall the definition
of the integers Ŝi
(2)(K) [4].
LetCi(K̃,Z) = HomZ[G]
(
Ci(K̃,Z),Z[G]
)
. and using involution in the ring Z[G] in-
troduced the structure of left Z[G]-module on Ci(K̃,Z). Consider the following cochain
complex
C∗(K̃) = C0(K̃,Z) d0
−→ C1(K̃,Z) d1
−→ . . .
dn−1
−→ Cn(K̃,Z).
Taking the tensor product of C∗(K̃) and L2(G) as Z[G]-module we obtain the Hilbert
complex
C∗
(2)(K̃): L2(G)
⊗
Z[π]
C0(K̃,Z)
id
⊗
d0
−→ L2(G)
⊗
Z[π]
C1(K̃,Z)
id
⊗
d1
−→ . . .
. . .
id
⊗
dn−1
−→ L2(G)
⊗
Z[π]
Cn(K̃,Z).
The L2(G)-module of i-th cohomologyHi
(2)(K) of this Hilbert complex is called L2(G)-
module of i-th cohomology of the spaceK. Therefore the following Z[π]-module:
Γ̂i(K̃) = Ci(K̃,Z)/di−1
(
Ci−1(K̃,Z)
)
,
can be interpreted as the i-th cohomology module with compact support of the i-th skeleton
of K̃ and L2(G)-module
Γi(K) = L2(G)
⊗
Z[π]
Ci(K̃,Z)/ id
⊗
di−1
L2(G)
⊗
Z[π]
Ci−1(K̃,Z)
is the i-th L2(G)-module of cohomology of the i-th skeleton ofK.
Definition 8. For a cell complexK, set
Ŝi
(2)(K) = Ŝi
(2)(C
∗(K̃)) = µs(Γ̂i(K̃))− dimN [G](Γi(K)),
From our previous discussion or from [3, 4] it follows that the numbers Ŝi
(2)(K) are
invariants of the homotopy type of the cell complexK.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 531
Of course, for a smooth manifold Mn the values of the numbers Ŝi
(2)(M
n) do not
depend on the method of constructing a chain complex.
Theorem 3. Let Mn, n ≥ 6, be a closed smooth nonsimply connected manifold
with π1(Mn) = π. Then
χc
1(M
n) = χh
1 (Mn) = µ(π)− 1,
χc
2(M
n) = χh
2 (Mn),
χc
i (M
n) = χh
i (Mn) = χa
i (Mn) =
= (−1)i
i∑
j=0
(−1)j dimN [G](H
j
(2)(M
n)) + Ŝi+1
(2) (Mn)
for 3 ≤ i ≤ n− 4.
Proof. The condition n ≥ 6 allows us to construct a handle decomposition of the
manifoldMn =
⋃
Hi
j such that the free chain complex
(C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Ci,
over Z[π] corresponding this handle decomposition satisfies following conditions:
a) µ(C0) = 1;
b) µ(C1) = 1;
c) χc
2(M
n) = χh
2 (Mn);
d) f-rank(di+1(Ci+1), Ci) = 0 and is additive for 3 ≤ i ≤ n− 4.
The proof follows from Theorems 1 and 2.
Remark 4. For a closed smooth nonsimply connected manifold Mn, n ≥ 6, the
numbers χc
1(M
n), χh
1 (Mn), χc
i (M
n), and χh
i (Mn), 3 ≤ i ≤ n − 4, are invariants of
homotopy type of manifold.
6. Nonsingular Morse – Smale flows.
Definition 9. A smooth flow ϕt on smooth closed manifoldMn is called nonsingu-
lar Morse – Smale if
a) the chain-recurrent set R of ϕt consist of finite number of hyperbolic closed orbit;
b) the unstable manifold of any closed orbit has transversal intersection with the
stable manifold of any closed orbit.
A vector field X generating a nonsingular Morse – Smale flow is also called nonsin-
gular Morse – Smale. A result of K. Meyer (see [2]) says for each nonsingular Morse –
Smale vector field there exists a Lyapunov function f: Mn −→ R forX : that is a function
satisfying the following conditions:
a) X (f)y < 0 for all y that are not contained in a closed orbit;
b) dfy = 0 if and only if y is a point on a closed orbit.
We will call f self-indexing if f(y) = λ whenever y belongs to a closed orbit of
index λ.
There are two types of closed orbit: twisted and untwisted. An untwisted closed
orbit σ of index λ of a nonsingular Morse – Smale vector field X is said to be in the
standard form if there are local coordinates θ ∈ S1, x1, . . . , y1, . . . , yn−λ−1 on tubular
neighborhood of σ such that
X = x1
∂
∂x1
+ . . .+ xλ
∂
∂xλ
− y1
∂
∂y1
− . . .− yn−λ−1
∂
∂yn−λ−1
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
532 V. V. SHARKO
on this neighborhood. If X0 is a nonsingular Morse – Smale vector field onMn then there
is an arc in the space of smooth vector field on Mn, Xt, 0 ≤ t ≤ 1, such that Xt is
Morse – Smale for all 0 ≤ t ≤ 1 and closed orbits of X1 coincide with the closed orbits
of X0 and are in the standard form [2].
In what follow we will consider nonsingular Morse – Smale vector fields having only
untwisted closed orbits.
Conversely, if a manifoldMn admits a round Morse function f: Mn → R, then there
exists a nonsingular Morse – Smale vector field X onMn, such that closed orbits of index
λ ofX coincide with singular circles of index λ of the function f . By definition a function
f on Mn is said to be a round Morse function if its singular setK(f) consists of disjoint
circles and corank of the Hessian is equal to one: corankx∈K(f)f = 1 (see [3]).
It is known that under a small perturbation of a round Morse function f , each singular
circle of index λ splits into two nondegenerate critical points of indexes λ and λ + 1.
And conversely, if g : Mn → R is a Morse function having two independent (see [3])
critical points x1 and x2 of g of indexes λ and λ+1 respectively, then these points can be
replaced by one singular circle of index λ. Therefore, for the construction of nonsingular
Morse – Smale vector fields on a manifold Mn with zero Euler characteristics we may
use Morse functions.
Definition 10. The i-th Morse S1-number of a manifold Mn is the minimum num-
ber of closed orbits of index i taken over all nonsingular Morse – Smale vector fields on
Mn with untwisted closed orbits. This number will be denoted byMS1
i (Mn).
Theorem 4. Let Mn, n ≥ 6, be arbitrary closed smooth manifold with zero Euler
characteristic and with π1(Mn) = π. Then the i-th Morse S1-number of the manifold
Mn is equal:
MS1
0 (Mn) =MS1
n−1(M
n) = 1,
MS1
1 (Mn) =MS1
n−2(M
n) = µ(π)− 1,
MS1
i (Mn) = ρ(χa
i (Mn)) =
= Ŝi+1
(2) (Mn) + ρ
(−1)i
i∑
j=0
(−1)j dimN [G]
(
Hj
(2)(M
n)
)
for 3 ≤ i ≤ n− 4.
Proof. Let X be a nonsingular Morse – Smale vector field onMn such that all closed
orbits of X are untwisted. Let also f: Mn → R be a round Morse function corresponding
to X and g : Mn → R be an ordered Morse function obtaned by small pertrubation of
f . Using g we can construct a handle decomposition ofMn and from this decomposition
define the free chain complex over Z[π]:
(C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Cn.
It is clear that
χc
i (M
n) = χa
i (C∗, d∗) ≤ χi(C∗, d∗)
for 3 ≤ i ≤ n − 4. The condition n ≥ 6 allows to construct a handle decomposition
Mn = ∪Hi
j ofMn such that the free complex over Z[π]
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
L2-INVARIANTS AND MORSE – SMALE FLOWS ON MANIFOLDS 533
(C∗, d∗): C0
d1←− C1
d2←− . . . dn←− Ci
corresponding to this handle decomposition satisfies the following conditions:
a) µ(C0) = 1;
b) µ(C1) = µ(π)− 1;
c) f-rank(di+1(Ci+1, Ci) = 0 and is additive for 3 ≤ i ≤ n− 4.
Using diagram technique from [3] and Theorem 3 we can construct from this handle
decomposition Mn = ∪Hi
j a round Morse function and therefore a nonsingular Morse –
Smale vector field X such that the numbers of untwisted closed orbits of X satisfy the
conditions of theorem. Homotopy invariance of the i-th Morse S1-number of Mn for
i = 0, 1, n− 1, n− 2 and 3 ≤ i ≤ n− 4 easily follows from previous discussions.
Theorem 4 is proved.
The calculation ofMS1
2 (Mn) andMS1
n−3(M
n) seems to be a difficult problem.
Acknowledges. I am grateful to the referee for constructive remark that allow to
clarify the paper.
I also thank S. Maksymenko for useful discussions.
1. Gontareva I. B., Felshtyn A. L. Analog of Morse inequality for attractor domain // Vestnik Leningrad Univ.
– 1984. – 7. – P. 16 – 20.
2. Franks J. The periodic structure of nonsingular Morse – Smale flows // Comment. math. helv. – 1978. –
53. – P. 279 – 294.
3. Sharko V. V. Functions on manifolds: algebraic and topological aspects // Transl. Math. Monogr. – Provi-
dence: Amer. Math. Soc. – 1993. – 131. – 193 p.
4. Sharko V. V. New L2-invariants of chain complexes and applications // C∗-algebras and Elliptic Theory.
Trends Math. – 2006. – P. 291 – 312.
5. Cohen J. Von Neumann dimension and homology of covering spaces // Quart. J. Math. – 1979. – 30. –
P. 133 – 142.
6. Lück W. L2-invariants: theory and applications to geometry and K-theory // Ergeb. Math. und Grenzge-
biete. – Berlin etc.: Springer, 2002. – 44. – 620 p.
7. Cockroft W., Swan R. On homotopy type of certain two-dimensional complexes // Proc. London Math.
Soc. – 1961. – 11. – P. 193 – 202.
8. Kirby R. C., Siebenmann L. C. Foundational essays on topological manifolds, smoothings and triangula-
tions // Ann. Math. Stud. – 1977. – 88.
Received 15.02.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 4
|
| id | umjimathkievua-article-3326 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:26Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/19/e3e5b3d02d04d76827d95480fd0bdd19.pdf |
| spelling | umjimathkievua-article-33262020-03-18T19:51:20Z L2 -invariants and Morse - Smale flows on manifolds L2 -інваріанти та потоки Морса-Смейла на многовидах Sharko, V. V. Шарко, В. В. We study the homotopy invariants of free cochain and Hilbert complexes. These L2 -invariants are applied to the calculations of exact values of minimal numbers of closed orbits of some indexes of nonsingular Morse - Smale flows on manifolds of large dimensions. Вивчаються гомотопічні інваріанти вільних коланцюгових та гільбертових комплексів. Ці L2 -інваріанти застосовуються при обчисленні точних значень мінімальних чисел замкнених орбіт фіксованих індексів несингулярних потоків Морса-Смейла на многовидах великих розмірностей. Institute of Mathematics, NAS of Ukraine 2007-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3326 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 4 (2007); 522-533 Український математичний журнал; Том 59 № 4 (2007); 522-533 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3326/3397 https://umj.imath.kiev.ua/index.php/umj/article/view/3326/3398 Copyright (c) 2007 Sharko V. V. |
| spellingShingle | Sharko, V. V. Шарко, В. В. L2 -invariants and Morse - Smale flows on manifolds |
| title | L2 -invariants and Morse - Smale flows on manifolds |
| title_alt | L2 -інваріанти та потоки Морса-Смейла на многовидах |
| title_full | L2 -invariants and Morse - Smale flows on manifolds |
| title_fullStr | L2 -invariants and Morse - Smale flows on manifolds |
| title_full_unstemmed | L2 -invariants and Morse - Smale flows on manifolds |
| title_short | L2 -invariants and Morse - Smale flows on manifolds |
| title_sort | l2 -invariants and morse - smale flows on manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3326 |
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