Morse Functions on Cobordisms
We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of exact values of Morse numbers of smooth cobordisms.
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| author | Sharko, V. V. Шарко, В. В. |
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UDC 517.938.5
V. V. Sharko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
MORSE FUNCTIONS ON COBORDISMS
ФУНКЦIЇ МОРСА НА КОБОРДИЗМАХ
We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the
calculation of exact values of Morse numbers of smooth cobordisms.
Вивчаються гомотопiчнi iнварiанти схрещених i гiльбертових комплексiв. Цi iнварiанти використову-
ютьcя для пiдрахунку точних значень чисел Морса гладких кобордизмiв.
1. Introduction. Let Wn bea smooth manifold. Вy definition the i-th Morse number
Mi(W
n) of a manifold Wn is the minimal number of critical points of index i taken
over all Morse functions on Wn.
It is known [8, 17] that for closed smooth manifolds of dimension greater than 6
the i-th Morse numbers are invariant of the homotopy type. There is a very complicated
unsolved problem: find exact values of Morse numbers for every i ([17] for more
details).
In [18] using new homotopy invariants of free cochain complexes and Hilbert
complexes of non simply-connected closed manifolds Wn, n ≥ 8, we gave exact values
of i-th Morse numbers for 4 ≤ i ≤ n− 4.
The Morse number M(Wn) of a manifold Wn is the minimal number of critical
points of all indexes taken over all Morse functions on Wn.
In this paper we calculate exact values of Morse numbers for some cobordisms
(Wn, V n−10 , V n−11 ).
2. Crossed modules [2]. A G-crossed module is a triple (C, ∂,G), where C and G
are groups, ∂ : C −→ G is a homomorphism and G acts on C from the left (the action
will be denoted by gc). Furthermore, the homomorphism ∂ should satisfy the following
conditions:
a) ∂(gc) = g(∂)g−1 for all g ∈ G, c ∈ C,
b) cdc−1 = (∂) · d for all c, d ∈ C.
Thus, if G acts on itself by conjugation action, then a) says that ∂ is a G-homomorphism.
The following statements are immediate consequences of the definition:
1) K = Ker ∂ is contained in the center of G,
2) N = Im ∂ is a normal subgroup of G.
Let Q = G/N. The action of G on C induces a natural structure of Z[Q]-module on the
center of C, and K = Ker ∂ is a submodule of this module. Moreover the action of G
on C induces the structure of Z[Q]-module on Cab = C/[C,C]. Obvious and important
special cases are: 1) the case when C is a Z[G]-module (so ∂ = 0) and 2) the case when
C is a normal subgroup of G (so ∂ is the inclusion).
A morphism (α, β) from the crossed module (C, ∂,G) to (C ′, ∂′, G′) is a pair of
group homomorphisms α : C −→ C ′ and β : G −→ G′ such that β · ∂ = ∂′ · α and
α(g · c) = β(g) · α(c) (g ∈ G, c ∈ C).
Let CM denote the category of crossed modules. If β = Id on G = G′, we say that
α is a G-morphism and denote this category by CMG.
An important case of crossed module is so-called free crossed module defined by
J. H. C. Whitehead [2].
c© V. V. SHARKO, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1 119
120 V. V. SHARKO
A G-crossed module (C, ∂,G) is called a free crossed module with indexed basis
(ci∈I) ⊂ C if it satisfies the following universal property: given a G′-crossed module
(C ′, ∂′, G′), an indexed subset (c′i∈I) ⊂ C ′ and homomorphism f : G −→ G′ such that
f(∂(ci)) = ∂′(c′i) for each i ∈ I, then there is a unique homomorphism g : C −→ C ′
such that g(ci) = c′i for each i ∈ I and the pair (f, g) is a morphism of crossed modules.
The following fundamental theorem is also due to J. H. C. Whitehead [2].
Theorem. Let X be a path-connected CW-complex, and Y a CW-complex
obtained from X attaching two-dimensional cell. Then π2(Y,X, x) is a free crossed
π1(X,x)-module with basis corresponding to the cells so attached.
Fix a group G. A G-crossed module C is said to be projective if it is projective in
the category CMG, that is to say, for any surjective morphism of G-crossed modules
f : A → B and any g : C → B in CMG, there is an h : C → A in CMG such that
f · h = g.
Let crossed module (C, ∂,G), N = Im ∂, Q = G/N and Cab = C/[C,C].
J. G. Ratcliffe proved that (C, ∂,G) is a projective crossed module if and only if
Cab = C/[C,C] is a projective module Z[Q]-module and mapping the two-dimensional
homology groups
∂∗ : H2(C)→ H2(N)
induced by the homomorphism ∂ : C → G is trivial.
The following important theorem is due to M. Dyer [2].
Theorem. Let X be a connected CW-subcomplex of a connected 2-complex Y,
where π1(X,x) = G and x ∈ X is base point. Then the triple (π2(Y,X, x), ∂, π1(X,x))
is a projective crossed module.
(The homomorphism ∂ : π2(Y,X, x)→ π1(X,x) is taken from the exact homotopy
sequence of the pair (Y,X).)
A projective crossed chain complex is sequence of groups and homomorphisms
e←− π ∂1←− G ∂2←− C2
∂3←− C3 ←− . . .
∂n←− Cn
such that:
a) (C2, ∂2, G) is a projective G-crossed module,
b) for each i ≥ 3 the module Ci is a projective Z[π]-module, ∂i is a homomorphism
of Z[π]-modules, ∂2 commutes with the action of the group G and ∂3(C3) is a Z[π]-
module,
c) ∂i · ∂i+1 = 0.
Obviously, G acts on each Ci, i ≥ 2. A crossed chain complex is said to be of dimension
n if Ci = 0 for i > n.
With any projective crossed chain complex (Ci, ∂i, G) associate the chain complex
projective Z[π]-modules
∂ab
2←− Cab2
∂3←− C3 ←− . . .
∂n←− Cn.
3. Stable invariants of finite generated modules and L2-modules. In what
follows, M will be a left finitely generated Λ-module over a certain associative ring
Λ with unit. Rings for which the rank of the free module is uniquely defined are called
IBN -rings.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
MORSE FUNCTIONS ON COBORDISMS 121
Denoting the minimum number of the generators of the module M by µ(M), we get
µ(M
⊕
Fn) < µ(M) + n, where Fn is a free module of rank n. There exist examples
(stably-free modules) when the strict inequality holds.
Recall that a Λ-module M is called stably-free if the direct sum of M and a free
Λ-module Fk is free. We assume that if the module M is zero, then µ(M) = 0.
Definition 3.1. For a finitely generated module M over IBN -ring Λ let us define
the following function (stable minimal generators of the module M) [17]
µs(M) = lim
n−→∞
(
µ(M ⊕ Fn)− n)
)
.
If a ring Λ is Hopfian then for any Λ-module M µs(M) = 0 if and only if M = 0.
Recall that a ring Λ is called Hopfian, if every epimorphism of a free Λ-module Fn
on itself is an isomorphism. It is clear, that for any non-zero module M we have
0 < µs(M) 6 µ(M).
Denote the ring of integers by Z and the field of complex numbers by C. Let G
be a discrete group. Denote its integer group ring by Z[G] and the group ring over the
field C by C[G]. It is known that the group rings Z[G] and C[G] are IBN -rings. From
theorems of Kaplansky and Cockroft it follows that the group rings Z[G] and C[G] are
Hopfian.
In the ring C[G] there exists an involution ∗ : C[G] → C[G],
(∑
i
αigi
)∗
=
=
∑
i
αig
−1
i , where α denotes the conjugation in C. This involution satisfies the
following conditions:
a) (r∗)∗ = r;
b) (αr1 + βr2)∗ = αr∗1 + βr∗2 , (α, β ∈ C);
c) (r1r2)∗ = r?2r
?
1 .
We can define the trace tr : C[G] −→ C by the rule tr
(∑k
i
αigi
)
= α1, where α1
is the coefficient of g1 = e, which is the identity of the group G. It is obvious that the
trace satisfies the following conditions:
a) tr(e) = 1;
b) tr is C-linear mapping;
c) tr(r1r2) = tr(r2r1);
d) tr(rr∗) = 0, and if tr(rr∗) = 0, then r = 0.
In the ring C[G] there is an inner product
〈∑
i
αigi,
∑
i
βigi
〉
=
∑
i
αiβi.
The norm for an element from C[G] may be defined by |r| = tr(rr∗)1/2. Consider
a completion of the ring C[G] with respect to this norm and denote it by L2(G). Then
L2(G) is a Hilbert space (the inner product assigns the same formula as for the group
ring C[G]). The Hilbert space L2(G) has an orthonormal basis consisting of all elements
of the group G. Now C[G] acts faithfully and continuously by left multiplication on
L2(G)satisfies the following condition
C[G]× L2(G) −→ L2(G),
so we may regard C[G] ⊆ B(L2(G)),where B(L2(G)) denotes the set of bounded linear
operators on L2(G). Let N [G] denote the (reduced) group von Neumann algebra of G:
thus by definition N [G] is a week closure of C[G] in B(L2(G)). Therefore the map
w → w(e) allows us to identify N [G] with a subspace of L2(G), where w ∈ N [G] and
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
122 V. V. SHARKO
e is unit element of the group G. Thus algebraically we have C[G] ⊂ N [G] ⊂ L2(G).
The involution and the trace map on N [G] may be defined exactly as for the ring C[G].
For the set Mn(N [G]) of n × n matrices over von Neumann algebra N [G], the trace
map can be extended by setting tr(W ) =
∑n
i=1
wii, where W = (wij) is a matrix with
entries in N [G].
Let L2(G)n denote the Hilbert direct sum n copies of L2(G), so L2(G)n is a Hilbert
space. The von Neumann algebra N [G] acts on L2(G)n from the left, so L2(G)n is a
left N [G]-module called a free Hilbert N [G]-module of rank n. The left Hilbert N [G]-
module M is a closed left C[G]-submodule of L2(G)n for some n. By definition an
Hilbert N [G]-submodule of M is a closed left C[G]-submodule of M, an L2(G)-ideal
is an Hilbert N [G]-submodule of L2(G), and homomorphism f : M −→ N between
Hilbert N [G]-modules is a continuous left C[G]-map [3].
Let M be a Hilbert N [G]-module and let p : L2(G)n → L2(G)n be an orthogonal
projection onto M ⊂ L2(G)n. Von Neumann dimension of Hilbert N [G]-module M
is called the number dimN [G](M) = tr(p) =
∑n
i=1
〈p(ei), ei〉L2(G)n . Here ei =
= (0, . . . , g, . . . , 0) is standard basis in L2(G)n. It is known that dimN [G](V ) is non-
negative real number [10].
Definition 3.2. Let M be a finite generated Z[G]-module, consider Hilbert N [G]-
module L2(G)
⊗
Z[G]M and define following number
S(M) = µs(M)− dimN [G]
L2(G)
⊗
Z[G]
M
.
Lemma 3.1. For any finite generated Z[G]-module M, the number S(M) is non-
negative.
The proof in [18].
4. Stable invariants of homomorphisms. The next results can be found in [18].
Consider a Λ-homomorphism f : Fk → Ft, where Fk, Ft are free modules of ranks k
and t respectively over ring Λ. The homomorphism f is a splitting along a submodule
F p ⊆ Fk, if there is a presentation of f of the form f = fp
⊕
ft : F p
⊕
F k−p →
→ F̃p
⊕
F̃t−p, such that f |Fp
⊕
0 = fp : F p → F̃p, f |0⊕
Fk−p
= ft : F k−p → F̃t−p,
where fp is an isomorphism.
From now in this situation we will suppose that submodules F p, F k−p, F̃p, F̃t−p
are free.
Definition 4.1. The number p above is called the rank of a splitting f = fp
⊕
ft.
The rank R(f) of a homomorphism f is the maximal value of possible ranks of splittings
of f.
Definition 4.2. Stabilization of a homomorphism f : Fk → Ft by a free module
Fp is a homomorphism fst(p) : Fk
⊕
Fp → Ft
⊕
Fp, such that fst(p)|Fk
⊕
0 = f,
fst(p)|0⊕
Fp
= Id.
A thickening of a homomorphism f : Fk → Ft by free modules Fm and Fn is the
homomorphism fth(m,n) : Fk
⊕
Fm → Ft
⊕
Fn, such that fth(m,n)|Fk
⊕
0 = f,
fth(m,n)|0⊕
Fm
= 0.
Definition 4.3. The stable rank Sr(f) of a homomorphism f : Fk → Ft is the limit
of values of
Sr(f) = lim
m,n,p→∞
(
R(fth(m,n)st(p))− p
)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
MORSE FUNCTIONS ON COBORDISMS 123
This limit always exists. There are examples of stably free modules with Sr(f) >
> R(f). For any homomorphism f : Fk → Ft the following equality holds: Sr(fst(v)) =
= Sr(f) + v.
Consider a composition of homomorphisms of free modules Fm
f−−−−→ Fn
g−−−−→
g−−−−→ Ft, such that g · f = 0 (condition (∂)). We say, that the homomorphisms f and
g are splitting along submodules F p ⊆ Fm and F q ⊆ Fn if there are presentations of f
and g of the form
0 −−−−→ F p
f1−−−−→ F̃p −−−−→ 0⊕ ⊕
Fm−p
f2−−−−→ Fn−p−q
g2−−−−→ Ft−q⊕ ⊕
0 −−−−→ Fq
g1−−−−→ F̃q −−−−→ 0
such that f |Fp
⊕
0 = f1, g|0⊕
0
⊕
F q
= g1. We admit that the module F p or F q to be
zero module. In the sequel we will suppose that submodules F p, F q, Fm−p, Ft−q, Fn−p−q
are free.
Definition 4.4. The number p + q will be called the common rank of a splitting
of homomorphisms f and g along submodules F p ⊆ Fm and F q ⊆ Fn. The common
rank Cr(f, g) of the homomorphisms f and g is a maximal value of common ranks of a
splitting of f and g.
Definition 4.5. The stabilization of a composition of homomorphisms of free
modules Fm
f−−−−→ Fn
g−−−−→ Ft, satisfying the condition (∂) by free modules Fp
and Fq is the following composition of homomorphisms
0 −−−−→ Fp
id−−−−→ Fp −−−−→ 0⊕ ⊕
Fm
f−−−−→ Fn
g−−−−→ Ft⊕ ⊕
0 −−−−→ Fq
id−−−−→ Fq −−−−→ 0.
We will denote it by (fst(p), gst(q)).
Definition 4.6. Consider a composition of homomorphisms f and g Fm
f−−−−→
f−−−−→ Fn
g−−−−→Ft, satisfying the condition (∂). The thickening of this composition by
free modules Fp and Fq is the following composition of homomorphisms Fm
⊕
Fp
fth(p)−−−−→
fth(p)−−−−→ Fn
gth(q)−−−−→ Ft
⊕
Fq, such that fth(p)|Fm
⊕
0 = f, fth(p)|0⊕
Fp
= 0, gth(q) =
= g. It will be denoted by (fth(p), gth(q)).
Definition 4.7. The stable common rank Scr(f, g) of the composition of homomor-
phisms of free modules Fm
f−−−−→ Fn
g−−−−→ Ft, satisfying the condition (∂) is the
limit of values of common ranks
Scr(f, g) = lim
p,q,v,w→∞
(
Cr(fth(p)st(v), gth(q)st(w))− v − w
)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
124 V. V. SHARKO
This limit always exists. There are examples of stably free modules showing that
Scr(f, g) ≥ Cr(f, g).
Lemma 4.1. For arbitrary composition of homomorphisms f and g Fm
f−−−−→
f−−−−→ Fn
g−−−−→ Ft, satisfying the condition (∂) the following equality holds true:
Scr(fst(x), gst(y)) = Scr(f, g) + x+ y.
Definition 4.8. The stable common rank from the left ( from the right) Scrl(f, g)
(Scrr(f, g)) of the composition of homomorphisms of free modules Fm
f−−−−→ Fn
g−−−−→
g−−−−→ Ft, satisfying condition (∂) is the following limit of values of common ranks:
Scrl(f, g) = lim
p,v,w→∞
(
Cr(fth,l(p)st(v), gst(w))− v − w
)
(
Scrr(f, g) = lim
q,v,w→∞
(
Cr(fst(v), gth,r(q)st(w))− v − w
))
.
Remark 4.1. For stable common rank from the left (from the right) Scrl(f, g)(
Scrr(f, g)
)
of the composition of the homomorphisms satisfying the condition (∂) the
analogues of Lemma 4.1 hold true.
Definition 4.9. The defect D(f, g) of the composition of homomorphisms of free
modules Fm
f−−−−→ Fn
g−−−−→ Ft, satisfying condition (∂) is the following number:
D(f, g) = Sr(f) + Sr(g)− Scr(f, g).
Remark 4.2. If in composition of the homomorphisms f and g the module Fn/f(Fm)
is stable free, but non free, then all way D(f, g) > 0.
Lemma 4.2. Consider two compositions of homomorphisms of free modules
Fm
f−−−−→ Fn
g−−−−→ Ft, and
0 −−−−→ Fv
id−−−−→ Fv −−−−→ 0⊕ ⊕ ⊕
Fm
⊕
Fp
fth,l(p)−−−−−→ Fn
gth,r(q)−−−−−→ Ft
⊕
Fq⊕ ⊕ ⊕
0 −−−−→ Fw
id−−−−→ Fw −−−−→ 0
satisfying the condition (∂) (the numbers p, q, v, w are nonnegative). Then the following
equality holds true:
D(f, g) = D
(
fth,l(p)st(v), gth,r(q)st(w)
)
.
Definition 4.10. The defect from the left ( from the right) Dl(f, g)
(
Dr(f, g)
)
of
a composition of homomorphisms of free modules Fm
f−−−−→ Fn
g−−−−→ Ft, satisfying
condition (∂) is the following number:
Dl(f, g) = Srl(f) + Sr(g)− Scrl(f, g)(
Dr(f, g) = Sr(f) + Srr(g)− Scrr(f, g)
)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
MORSE FUNCTIONS ON COBORDISMS 125
Remark 4.3. For the defect from the left (from the right) Dl(f, g)
(
Dr(f, g)
)
of
a composition of homomorphisms f and g satisfying condition (∂) the analogues of
Lemma 4.2 and Remark 4.2 hold true.
5. Homotopy invariants of cochain complexes. If (C, d) : C0 d0−→ C1 d1−→ . . .
. . .
dn−1
−→ Cn is a free cochain complex over a ring Λ, then the numbers Dr(d0),Dl(dn−1),
Dr(d0, d1), Dl(dn−2, dn−1), D(di, di+1) are defined for 1 ≤ i ≤ n − 3. In [18] proof
that they are invariants of the homotopy type of a cochain complex (C, d).
Definition 5.1. Let (C, d) : C0 d0−→ C1 → . . .
dn−1
−→ Cn be a free cochain complex.
Then cochain complex (C(i), d(i)) : C0 d0−→ C1 → . . .
di−1
−→ Ci is called i-th skeleton of
cochain complex (C, d).
Let (C∗, d∗)) : C0
d1−→ C1 → . . .
dn−→ Cn, be a sequence of free Hilbert N [G]-
modules and bounded C[G]-map such that di+1 ◦ di = 0. It is called a Hilbert complex.
The reduced cohomology of Hilbert complex (C∗, d∗)), it is a collection of Hilbert
N [G]-modules Hi
(2)(C
∗, d∗) = Ker di/ Im di−1.
Definition 5.2. Consider a free cochain complex over Z[G]
(C∗, d∗) : C0 d0−→ C1 → . . .
dn−1
−→ Cn.
Hilbert complex L2(G)
⊗
Z[G]
C∗, Id
⊗
Z[G]
d∗
:
L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d
0
−→ L2(G)
⊗
Z[G]
C1 → . . .
Id
⊗
Z[G] d
n−1
−→ L2(G)
⊗
Z[G]
Cn
of free Hilbert N [G]-modules is the Hilbert complex generated by Z[G]-complexes.
Consider the i-th skeletons of these complexes(
C∗(i), d∗(i)
)
: C0 d0−→ C1 → . . .
di−1
−→ Ci,
L2(G)
⊗
Z[G]
C0
Id
⊗
Z[G] d
0
−→ L2(G)
⊗
Z[G]
C1 −→ . . .
Id
⊗
Z[G] d
i−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 reduced cohomology of the i-th skeleton of the
Hilbert complex L2(G)
⊗
Z[G]
C∗(i), Id
⊗
Z[G]
d∗(i)
.
Definition 5.3 [16, 17]. For the cochain complex (C∗, d∗) over Z[G] set
Ŝi(2)(C
∗, d∗) = µs(Γ
i)− dimN [G] Γ̂i.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
126 V. V. SHARKO
Lemma 5.1. The numbers Ŝi(2)(C
∗, d∗) are non-negative for every i.
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
∗, ∂∗).
Definition 5.4. The Morse number of a cochain complex
(C, d) : C0 d0−→ C1 → . . .
dn−1
−→ Cn
over a ring Λ is the numberM(C, d) =
∑n
i=0
µ(Ci).
Definition 5.5. The homotopy Morse number Mh(C, d) of a cochain complex
(C, d) over a ring Λ is the minimum of Morse numbers taken over all cochain complexes
homotopy equivalent to (C, d).
For a cochain complex (C, d) denote by D0(C, d) = Dr(d0, d1), Dn−2(C, d) =
= Dl(dn−2, dn−1), Di(C, d) = D(di, di+1).
Theorem 5.1. Let (C, d) : C0 d0−→ C1 → . . .
dn−1
−→ Cn, n ≥ 4, be a free cochain
complex over a group ring Z[G] such that if Di(C, d) 6= 0, then Di+1(C, d) = 0 for
0 ≤ i ≤ n− 2. Then the homotopy Morse number of (C, d) equal:
Mh(C, d) = 2
∑n−2
i=0
(Di(C, d)) + 2
∑n−1
i=1
(
Ŝi(2)(C, d)
)
+
+
∑n−1
i=0
dimN [G]
Hi(L2(G)
⊗
Z[G]
C, id
⊗
Z[G]
d)
+ 2µ(Hn(C, d))−
−dimN [G]
Hn(L2(G)
⊗
Z[G]
C, id
⊗
Z[G]
d)
.
Proof. The number Di(C, d) arises in this theorem because in definition of the
number Si(2)(C, d) we take the number µs(Γi) but not the number µ(Γi). For example,
in view of Remark 4.2 if the module Ci/di−1(Ci−1) is stable free but non free, then
Di(C, d) > 0. Therefore, if Di(C, d) > 0 then in dimension i and i − 1 we have
additional free modules of the rank Di(C, d) such that they not give contribution in Ŝi(2).
From conditions of theorem it follow that in the homotopy type of (C, d) any
cochain complex (D, d) : D0 d0−→ D1 → . . .
dn−1
−→ Dn such that Di0(C, d) = 0 satisfies
the following condition µ(Di0/d
i0−1Di0−1) = µs(Di0/d
i0−1Di0−1) for i0. From [17]
it follow that in the homotopy type of (C, d) there exist minimal cochain complex in
dimension i0 such that µ(Ci0) = Ŝi0(2)(C, d)+Ŝi0+1
(2) (C, d)+dimN [G]
(
Hi0(L2(G)
⊗
Z[G]⊗
Z[G] C, id
⊗
Z[G] d)
)
.
The value of Morse number of cochain complex may be find direct calculation.
Theorem 5.1 is proved.
6. Applications. It is well known that all chain complexes constructed from cellular
decompositions of the non-simply connected CW-complex K have the same homotopy
type. Therefore it follows directly from the previous or from [9,17] that the numbers
Ŝi(2)(K) and D̂0
r(K), D̂n−1l (K), D0
r(K), Dn−2l (K), Di(K) for 1 ≤ i ≤ n − 2. are
invariants of the homotopy type of the CW-complex K.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
MORSE FUNCTIONS ON COBORDISMS 127
For a smooth manifold Wn there is an approach to the construction of cochain
complex via Morse functions. The details can be found in [15].
Let (Wn, V n−10 , V n−11 ) be a compact smooth manifold with boundary ∂Wn =
= V n−10 ∪ V n−11 . Let π = π1(Wn) be the fundamental group of the manifold Wn.
Denote by p : (W̃n, Ṽ n−10 , Ṽ n−11 ) → (Wn, V n−10 , V n−11 ) the universal covering space.
Here Ṽ n−1i = p−1(V n−1i ). Let us choose on Wn an ordered Morse function f : Wn →
→ [0, 1], f−1(0) = V n−10 , f−1(1) = V n−11 and a gradient-like vector field ξ [17].
Using the mapping p, lift f and ξ to W̃n, and denote a lifted function and a vector field
by f̃ and ξ̃ respectively. Using f, ξ (f̃ , ξ̃) construct chain complexes of abelian group
C∗(W
n, f, ξ):
C∗(W
n, f, ξ) : C0
d1←− C1 ← . . .
dn←− Cn,
C∗(W̃
n, f̃ , ξ̃) : C̃0
d̃1←− C̃1 ← . . .
d̃n←− C̃n,
where Ci = Hi(Wi,Wi−1, Z), C̃i = Hi(W̃i, W̃i−1, Z) and W̃i = f̃−1[0, ai], Wi =
= f−1[0, ai] are submanifolds containing all critical points of indices less than or equal
to i. For the generators of the chain groups Ci (Ĉi) one can take middle disks of
critical points of index i constructed by the vector field ξ (ξ̂). The fundamental group
π = π1(Wn) acts on manifolds W̃n. Making use of this actions, we can turn the chain
group C̃i into finitely generated modules over ring Z[π]. Making use of the involution,
we turn the right Z[π]-module C(i) = HomZ[π](Ci, Z[π]) into a left one and construct
the following free cochain complex
C∗(W̃n, f̃ , ξ̃) : C̃(0) d̃(0)−→ C̃(1) → . . .
d̃(n−1)
−→ C̃(n).
Taking the tensor product of C∗
(
W̃n, f̃ , ξ̃
)
and L2(π) as Z[π]-module, we obtain the
cochain complex of abelian groups which can be used for the definition the numbers
Ŝi(2)(W
n, V n−10 ) and Di(Wn, V n−10 ).
On cobordism (Wn, V n−10 , V n−11 ) using (f, ξ) construct crossed projective chain
complexes Ccr∗ (Wn, f, ξ):
e← π ← π1(V n−10 )
d2←− π2(Wn, V n−10 )
d̃(3)←− C̃3 d̃4←− . . . d̃
(n−1)
←− C̃(n).
and using (−f,−ξ) construct crossed projective chain complexes Ccr∗ (Wn,−f,−ξ):
e← π ← π1(V n−11 )
d2←− π2(Wn, V n−11 )
d̃(3)←− D̃3 d̃4←− . . . d̃
(n−1)
←− D̃(n).
Definition 6.1. The Morse number M(Wn) of a manifold Wn is the minimal
number of critical points of all indexes taken over all Morse functions on Wn.
Theorem 6.1. Let (Wn, V n−10 , V n−11 ), n ≥ 6, be a compact smooth manifold
with boundary ∂Wn = V n−10 ∪ V n−11 and π = π1(Wn) be the fundamental group of
the manifold Wn. Suppose that π(V n−1i ) −→ π1(Wn) is isomorphism, Wh(π) = 0
(Whitehead group of π) and if Di(Wn, V n−10 ) 6= 0 then Di+1(Wn) = 0 for all 1 < i <
< n− 2. The following equality holds for the Morse number of Wn :
M(Wn, V n−10 ) = 2
n−2∑
i=1
(
Di(Wn, V n−10 )
)
+ 2
n−2∑
i=3
(
Ŝi(2)(W
n, V n−10 )
)
+
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
128 V. V. SHARKO
+
n−2∑
i=2
(
dimN [π](H
i
(2)(W
n, V n−10 , Z))
)
.
Proof. Let f be an arbitrary ordered Morse function, ξ a gradient-like vector field
on Wn, and C∗
(
W̃n, f̃ , ξ̃
)
: C̃0
d̃1←− C̃1 ← . . .
d̃n←− C̃n, the chain complex associated
with them.
Denote by C∗
(
W̃n, f̃ , ξ̃
)
: C̃(0) d̃(0)−→ C̃(1) → . . .
d̃(n−1)
−→ C̃(n) the cochain complex
constructed starting from chain complex C∗
(
W̃n, f̃ , ξ̃
)
. It is clear that if chain complex
C∗
(
W̃n, f̃ , ξ̃
)
is minimal in dimension i then and cochain complex C∗
(
W̃n, f̃ , ξ̃
)
is minimal in dimension i. It is known that the operation of stabilization of the
homomorphisms d̃i, can be realized by changing Morse function and gradient-like vector
field on Wn. But the inverse operation, the elimination of contractible contractible
free chain complex of the form 0 −→ Ci −→ Ci+1 −→ 0 from the chain complex
C∗
(
W̃n, f̃ , ξ̃
)
can not always be realized by change of Morse function and gradient-
like vector field on Wn. This is possible if n ≥ 6 and Wh(π) = 0 [17]. Let
(
C, d
)
be a
chain complex homotopy equivalent to chain complex C∗
(
W̃n, f̃ , ξ̃
)
such that it Morse
number equal homotopy Morse number of C∗(W̃n, f̃ , ξ̃). By Cockroft – Swan theorem
[2] there exist contractible free chain complexes (D, ∂) and
(
D, ∂
)
such that the chain
complexes
(
C∗(W̃n, f̃ , ξ̃
⊕
D, d
⊕
∂)
)
and
(
C
⊕
D, d
⊕
∂
)
, are chain-isomorphic.
The previous notice ensures the existence of a Morse function g and gradient-like
vector field that realize the complex
(
C∗
(
W̃n, f̃ , ξ̃
⊕
D, d
⊕
∂
))
. Using elimination of
contractible contractible free chain complexes of the form 0 −→ Ci −→ Ci+1 −→ 0
and 0 −→ Ci−1 −→ Ci −→ 0 from the chain complex
(
C∗(W̃n, f̃ , ξ̃
⊕
D, d
⊕
∂)
)
we can obtain a chain complex
(
Ĉ, d̃
)
that is minimal. The conditions that n ≥ 6 and
Wh(π) = 0 ensures the existence of a Morse function g and gradient-like vector field η
that realize the complex (Ĉ, d̂). The number of critical points of Morse function g can
be computed using previous formulas from Theorem 5.1.
Theorem 6.1 is proved.
Theorem 6.2. Let (Wn, V n−10 , V n−11 ), n ≥ 6, be a compact smooth manifold with
boundary ∂Wn = V n−10 ∪ V n−11 and π = π1(Wn) be the fundamental group of Wn.
Suppose that π2(Wn, V n−1i ) are free crossed π1(V n−1i )-modules,H2
(
W̃n, Ṽ n−1i ,Z[π]
)
are free Z[π]-modules, µ
(
H2(W̃n, Ṽ n−1i ,Z[π])
)
= µ(π2(Wn, V n−1i )), Wh(π) = 0 and
if Di(Wn) 6= 0 then Di+1(Wn) = 0 for all 3 < i < n− 3. Then the following equality
holds for the Morse number of Wn :
M(Wn, V n−10 ) = 2
n−2∑
i=2
(
Di(Wn, V n−10 )
)
+
+2
n−2∑
i=3
(
Ŝi(2)(W
n, V n−10 )
)
+
n−3∑
i=3
(
dimN [π](H
i
(2)(W
n, V n−10 , Z))
)
+
+µ
(
H2(W̃n, Ṽ n−10 ,Z[π])) + µ(H2(W̃n, Ṽ n−11 ,Z[π])
)
.
Proof. The conditions in the theorem guarantee the existence of a ordered Morse
function f : Wn → [0, 1], f−1(0) = V n−10 , f−1(1) = V n−11 without critical points of
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
MORSE FUNCTIONS ON COBORDISMS 129
indexes 0, 1, n−1, n. The Morse number of (Wn, V n−10 , V n−11 ) can be computed using
previous formulas from Theorem 5.1.
Theorem 6.2 is proved.
The estimate for Morse numbers study in works [1, 4 – 7, 10 – 19], where were use
and other approaches. In next papers we shall give the values of Morse numbers for
other class manifolds.
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cubical homotopy groupoids // Tracts Math. – 2011. – 15. – 516 p.
3. Cohen J. Von Neumann dimension and homology of covering spaces // Quart. J. Math. – 1979. – 30. –
P. 133 – 142.
4. Damian A. On stable Morse number of a closed manifold // Bull. London Math. Soc. – 2002. – 34. –
P. 420 – 430.
5. Eliashberg Y., Gromov M. Lagrangian intersections and stable Morse theory // Boll. Unione mat. ital. –
1997. – (7)11-B. – P. 289 – 326.
6. Farber M. Homological algebra of Novikov – Shubin invariants and Morse inequalities // Geom. and
Funct. Anal. – 1996. – 6. – P. 628 – 665.
7. Gromov M., Shubin M. Near-cohomology of Hilbert complexes and topology of non-siply connected
manifolds // Asterisque. – 1992. – 210. – P. 283 – 294.
8. Hajduk B. Comparing handle decomposition of homotopy equivalent manifolds // Fund. math. – 1997.
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9. Kirbu R. C., Siebenmann L. C. Foundational essays on topological manifolds, smoothings and
triangulatiions // Ann. Math. Stud. – 1977. – 88.
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Morse inequalities // Rus. J. Math. Phys. – 1996. – 4. – P. 499 – 526.
12. Novikov S. P., Shubin M. A. Morse inequalities and von Neumamm algebras // Uspekhi Mat. Nauk. –
1986. – 41. – S. 163 – 164.
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manifolds // Uspekhi Mat. Nauk. – 1986. – 41. – S. 222 – 223.
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Received 04.11.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
|
| id | umjimathkievua-article-2703 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:39Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/e8/a8afd376c2c360c9314a06605358efe8.pdf |
| spelling | umjimathkievua-article-27032020-03-18T19:34:07Z Morse Functions on Cobordisms Функцiї Морса на кобордизмах Sharko, V. V. Шарко, В. В. We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of exact values of Morse numbers of smooth cobordisms. Вивчаються гомотопiчнi iнварiанти схрещених i гiльбертових комплексiв. Цi iнварiанти використовуютьcя для пiдрахунку точних значень чисел Морса гладких кобордизмiв. Institute of Mathematics, NAS of Ukraine 2011-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2703 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 1 (2011); 119-129 Український математичний журнал; Том 63 № 1 (2011); 119-129 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2703/2162 https://umj.imath.kiev.ua/index.php/umj/article/view/2703/2163 Copyright (c) 2011 Sharko V. V. |
| spellingShingle | Sharko, V. V. Шарко, В. В. Morse Functions on Cobordisms |
| title | Morse Functions on Cobordisms |
| title_alt | Функцiї Морса на кобордизмах |
| title_full | Morse Functions on Cobordisms |
| title_fullStr | Morse Functions on Cobordisms |
| title_full_unstemmed | Morse Functions on Cobordisms |
| title_short | Morse Functions on Cobordisms |
| title_sort | morse functions on cobordisms |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2703 |
| work_keys_str_mv | AT sharkovv morsefunctionsoncobordisms AT šarkovv morsefunctionsoncobordisms AT sharkovv funkciímorsanakobordizmah AT šarkovv funkciímorsanakobordizmah |