Examensarbete i matematik, 30 hp
Handledare och examinator: Tobias Ekholm
Juni 2011
An Introduction to Lefschetz Coincidence
Theory with an Application to Differential
Equations
1. Introduction 4
2. Preliminaries 5
2.1. Notation and Necessary Definitions . . . 5
2.2. The Lefschetz Number, the Intersection Number and the Euler Class . . . 8
2.2.1. The Lefschetz Number . . . 8
2.2.2. The Intersection Number and the Euler Class . . . 9
2.2.3. Connections . . . 10
2.3. Definitions necessary for Chapter 4 . . . 11
3. Lefschetz Coincidence Theory 12 3.1. Introduction . . . 12
3.2. Two Important Theorems . . . 19
4. Application to Differential Equations 24 4.1. Theoretical Background . . . 24
4.2. Application . . . 26
Fixed-point theorems and theories are one of the most successful applications of Math-ematics, being used almost everywhere from Economics to Engineering, most notably probably for qualitative solutions of differential equations.
This thesis concerns itself mainly with Lefschetz Coincidence Theory and its applica-tion to Ordinary Differential Equaapplica-tions (ODEs). First, we will introduce notaapplica-tion and assumed theory, before introducing the Lefschetz Number and those parts of Intersection Theory in more depth, insofar they are relevant. We will then introduce the Lefschetz Coincidence Number L(f, g) before stating and proving Theorems VI.14.4 and VI.14.5 in [5], which can be found in chapter 3.2. After this, we will present an application of this theory to differential equations and their solutions.
In particular, the first part of this thesis is based heavily on Part VI of [5] and especially on chapter VI.14.
I would like to thank my supervisor, who showed great patience and understanding throughout the writing process. I would also like to thank my girlfriend and my family, who helped me throughout the writing process and who were very understanding while suffering greatly.
In this chapter, we are going to present necessary preliminaries and definitions as well as introducing the Lefschetz Number.
2.1. Notation and Necessary Definitions
we will assume familiarity with basic topological and algebraical definitions such as those of a manifold or of homology and cohomology as well as homotopy and homotopy equi-valence.
The most important definition for this paper is the following:
Definition A fixed point of a function f ∶ X → Y is a point x ∈ X such that f(x) = x. If f, g∶ X → Y are two functions, then a coincidence is a point x ∈ X such that f(x) = g(x). Definition Let f ∶ X → Y be a map. For any p-simplex σ ∶ ∆p → X in X, the
composition f ○ σ ∶ ∆p → Y is a singular p-simplex in Y and extends uniquely to a
homomorphism f∆∶ ∆p(X) → ∆p(Y ) by f∆(∑σnσσ) = ∑σnσ(f ○ σ).
Definition Let Mn be an oriented, closed manifold. Then we have for all integers k, Hk(M) ≅ Hn−k(M)
where the isomorphism is the Poincaré duality.
Definition Let d ∶ X → X × X be the diagonal map d(x) = (x, x). Then the cup product is the homomorphism
∪ ∶ Hp(X) ⊗ Hq(X) → Hp+q(X)
denoted by α ∪ β = d∗(α × β). On the chain-cochain level, this is equivalent to the following:
∪ ∶ ∆p(X) ⊗ ∆q(X) → ∆p+q
with (f ∪ g)(c) = f ⊗ g(θ(d∆(c))) where θ ∶ ∆∗(X × Y ) → ∆∗(X) ⊗ ∆∗(Y ) is the natural
canonical chain map (x, y) ↦ x ⊗ y in degree 0. For more details about this map, see e.g. Theorem VI.1.2 in [5].
Definition We define the cap product on the chain-cochain level as follows:
∩ ∶ ∆p(X) ⊗ ∆
n(X) → ∆n−p(X)
by f ∩ c = (1 ⊗ f)∆c. This induces a cap product in homology:
∩ ∶ Hp(X) ⊗ H
n(X) → Hn−p(X).
Definition Let c be a 0 chain. Hence, c = ∑xnxx over points x ∈ X where nx = 0
except for a finite number of x. Then define (c) = ∑xnx∈ Z. Thus, ∶ ∆0(X) → Z is a
homomorphism. Furthermore, induces a homomorphism ∗ ∶ H0(X) → Z. Both and
Definition Let M be a compact, orientable connected m−manifold with boundary ∂M and let R be a commutative ring. A fundamental class of M, [M], is an element of Hn(M; R) whose image in Hn(M, M/x; R) is a generator. Note that this corresponds to
a choice of orientation of M.
Definition Let M be a compact, orientable connected m−manifold with boundary ∂M. Then the ’intersection product’ is defined as follows:
● ∶Hi(M) ⊗ Hj(M) → Hi+j−n(M), or
● ∶Hi(M, ∂M) ⊗ Hj(M) → Hi+j−n(M), or
● ∶Hi(M, ∂M) ⊗ Hj(M, ∂M) → Hi+j−n(M, ∂M), by
a● b = D−1(D(b) ∪ D(a)) = (D(b) ∪ D(a)) ∩ [M] = D(b) ∩ (D(a) ∩ [M]) = D(b) ∩ a which is equivalent to
D(a ● b) = D(b) ∪ D(a).
Definition If f ∶ Nn → Mm is a map from a compact, oriented n−manifold N to a compact, oriented m−manifold M, taking ∂N to ∂M, then
f!∶ Hn−p(N) → Hm−p(M) and f!∶ Hn−p(N, ∂N) → Hm−p(M, ∂M) are both defined by
f!= DMf∗D−1N
Furthermore,
f!∶ Hn.p(M) → Hm−p(N) and f!∶ Hn−p(M, ∂M) → Hn−p(N, ∂N)
are both defined by
f!= DN−1f∗DM.
f! and f! are called transfer or shriek maps.
Definition Let Nn, Mm be orientable manifolds. Then f ∶ N → M has the graph Γ= {(x, f(x)) ∈ N × M}.
2.2. The Lefschetz Number, the Intersection Number and the
Euler Class
In this section, we are defining the Lefschetz Number and the Intersection Number. We will also state a few important theorems for both of them and presenting their connection. The connection between the Lefschetz Number and the Euler Class is also going to be mentioned. For proofs and more see [5].
2.2.1. The Lefschetz Number
Definition Let K be a finite simplicial complex and consider homology with coefficients in a field Λ. Let f ∶ ∣K∣ → ∣K∣ be a map and let f∗ ∶ Hi(∣K∣ ; Λ) → Hi(∣K∣ ; Λ) be the
induced map in homology and tri(f∗) its trace in degree i. The we define the Lefschetz
number of f to be
LΛ(f) = ∑ i
(−1)itr
i(f∗) ∈ Λ. (2.1)
One of the most famous theorems regarding the Lefschetz number is the Lefschetz-Hopf Fixed Point Theorem, first stated in 1926 in [12].
Theorem 2.2.1 Let K be a finite simplicial complex and let f ∶ ∣K∣ → ∣K∣ be a map. If L(f) ≠ 0, then f has a fixed point.
2.2.2. The Intersection Number and the Euler Class
Definition Let Nn be a smoothly embedded submanifold of Ww, both oriented and with N meeting ∂W transversely in ∂N, if there are boundaries. Then the Euler class of a normal bundle to N in W is
χWN = (iWN) ∗ (τNW) ∈ Hw−n(N) Since τW
N ∈ Hw−n(W ), clearly if Hw−n(W ) = 0, then χWN = 0.
Definition Let W = N × N and let d ∶ N → ∆ ⊂ W be the diagonal map. Then put τ = τ∆N xN ∈ Hn(N × N).1 Also set χ = d∗(τ) = d∗(τ∣∆) = d∗(χN×N∆ ) ∈ Hn(N), the Euler
class of the tangent bundle of N.
In order to compute this ’class’ we use the following theorem, which will lead us eventually to the definition of the intersection number.
Theorem 2.2.2 Take coefficients of (co)homology in some field. Let B= {α} be a basis of H∗(N). Let {α○} be the dual basis of H∗(N), i.e. ⟨α○∪ β, [N]⟩ = δα,β, noting that
deg(α○) = n − deg(α). Then τ = ∑
α∈B(−1)
deg(α)α○× α ∈ Hn(N × N)
and therefore also
χ= d∗τ = ∑
α∈B(−1)
deg(α)α○∪ α
1
This theorem has the following, straightforward corollary.
Corollary 2.2.3 The evaluation of the Euler class on the orientation class is the Euler characteristics, i.e. ⟨χ, [N]⟩ = χ(N).
Definition 2.2.4 Suppose now that f∶ N → N is a map of a closed, oriented, connected manifold into itself. Let (1 " f) ∶ N → N × N denote the composition (1 × f) ○ d, where d∶ N → ∆ ⊂ W = N ×N is the usual diagonal map. Then (1"f) = {(x, f(x)) ∈ N × N} = Γ is the graph of f . Orient Γ by [Γ] = (1 " f)∗[N] and W = N × N by [W ] = [N] × [N].
Define the ’intersection’ of [Γ] and [∆] as
[∆] ⋅ [∆] = ∗([Γ] ● [∆])
2.2.3. Connections
We are now in a position to state very interesting connections between the Lefschetz number, the Euler Class and the Intersection Number.
Theorem 2.2.5 The Lefschetz Number L(f) = [Γ] ⋅ [∆], the intersection number. Now, clearly, if f has no fixed points, Γ ∩ ∆ = ∅ and thus L(f) = 0, as expected from Theorem 2.2.1.
We also get the following, alternative definition of the Lefschetz Number as a Corollary from the above theorem.
Corollary 2.2.6 If N is a smooth, closed, orientable manifold and f ∶ N → N is smooth and such that the differential f∗x does not have 1 as an eigenvalue at any fixed point
x∈ N of f, then
L(f) = ∑
x
sign(det(I − f∗x))
In fact, both Theorem 2.2.5 and Corollary 2.2.6 are due to Lefschetz. Now we will state two simple propositions which are used in Chapter 4
Proposition 2.2.7 Let idX ∶ X → X be the identity function on a space X. Then L(idX) = χ(X)
Proposition 2.2.8 The Lefschetz number is homotopy equivalent.
2.3. Definitions necessary for Chapter 4
The following definitions are taken from [6] and translated by myself.
Definition Let X be a space. Assign a vector to each point x ∈ X, vx, say. Then the
space of all vectors v = {vx∣x ∈ X} is a vector field over X.
Definition Let X be a space. A flow is a mapping ϕt∶ X × R → X where t is the time parameter in R. In addition, for all x ∈ X, the following must hold:
• ϕ0(x) = x
• ϕt(ϕs(x)) = ϕs+t(x) for all t, s ∈ R.
Now fix x ∈ X and define γ(x) = {ϕt(x)}
t∈R. Then the codomain of γ(x) is the orbit of
x. The orbit is T-periodic if there exists a T ∈ R+ such that ϕt+T(x) = ϕt(x) for all t ∈ R
In this chapter we are going to introduce the Lefschetz Coincidence Theory and prove two important theorems which are going to be used in the next chapter in order to present an application to Differential Equations. This chapter is based on chapter IV.14 of [5].
3.1. Introduction
In general, a fixed point of a function f ∶ M → M is just a coincidence between f and idM ∶ M → M ∶ x ↦ x. Thus, a coincidence theory is a generalisation, both in terms of
the functions involved and the spaces they map to.
Definition Let Nn and Mm be closed orientable manifolds and let f, g ∶ N → M. Then fN → M has graph Λf = {(x, f(x)) ∈ N × M} and g ∶ N → M has graph Λg with
equivalent definition. Both graphs are submanifolds of N ×M. The Lefschetz Coincidence Number of f and g is defined as
L(f, g) = [Γf] ⋅ [Γg] = ∗([Γf] ● [Γg]) ∈ Z (3.1)
where [Γf] = (1 " f)∗[N], [Γg] = (1 " f)∗[N] and e∗∶ H0(N × M) → Z is the
Remark Note that the definition does not depend on the orientations of N and M, as long as we orient N × M by the product orientations. We can also define the Lefschetz Coincidence Number in case one of N or M is nonorientable. For a more detailed treatment of the nonorientable case, see Appendix A.
Remark Let f ∶ N → N and let g = idN = 1. Then L(f, 1) = [Γf] ⋅ [Γid] = [Γf] ⋅ [∆] =
L(f), the last step following from Theorem 2.2.5 and Γid= (1 " 1)∗[N] = ((1 × 1) ○ d)∗[N] =
(d)∗[N] = [∆].
Now if L(f, g) ≠ 0, we see that, similar to the Lefschetz Number, this implies a coin-cidence between f and g.
Formulas for shriek maps and the intersection product
First of all, we are going to state and prove a few formulas regarding shriek maps, induced maps in (co)homology and the intersection product1.
Proposition 3.1.1 Let f∶ Nn→ Mm be a map of oriented manifolds. Then we have 1. f∗(b ∩ [N]) = f!(b) ∩ [M] and f!(a ∩ [M]) = f∗(a) ∩ [N];
2. f![M] = [N];
3. f!(a ∩ b) = f∗(a) ∩ f!(b); 4. n= m ⇒ f∗f!= deg (f) = f!f∗;
5. n= m ⇒ f!f∗= deg (f) on im (f!) and f∗f!= deg (f) on im (f∗);
6. (fg)!= g!f! and (fg)!= f!g!.
1
Proof 1. For the first part, we compute
f!(b) ∩ [M] = DMf∗D−1N(b) ∩ [M]
= f∗D−1N(b) = f∗(b ∩ [N]) .
Similarly, for the second part we have
f!(a ∩ [M]) = DN−1f ∗
DM(a ∩ [M])
= DN−1f∗(a)
= f∗(a) ∩ [N] .
4. f∗f!(a) = f∗D−1Nf∗DM(a) = f∗(f∗DM(a) ∩ [N]) = DM(a) ∩ f∗[N] = DM(a) ∩ deg(f) [M] = deg(f)a. and f!f∗(a) = DMf∗d−1N(a) = DMf∗(f∗(a) ∩ [N]) = DM(a ∩ f∗[N] O) = DM(a ∩ [M]) deg(f) = a deg(f).
5. For the first part we have
f!f∗f!(a) = f!(deg(f)(a)) = deg(f)f!(a)
where the first step follows from 4. Similarly, for the second part we have
6. (fg)!= D −1 N(fg)∗DM = DN−1g∗f∗DM = DN−1g ∗ DMD−1Nf ∗ DM = g!f!. and (fg)!= D M(fg)∗D−1N = DMf∗g∗DN−1 = DMf∗D−1NDMg∗D−1N = f!g!.
Proposition 3.1.2 Let f∶ Nn→ Mm be a map of oriented manifolds. Then f!(a ● b) = f!(a) ● f!(b).
If n= m, then
Proof We have the following f!(a) ● f!(b) = D−1Nf∗DM(a) ● DN−1f∗DM(b) = D−1N (f∗DM(b) ∪ f∗DM(a)) = D−1Nf ∗(D M(b) ∪ DM(a)) = D−1Nf∗DM(a ● b) = f!(a ● b).
Now let n = m. Then we have
f∗(a) ● f∗(b) = DM(f∗(a) ∪ f∗(b)) = DMf∗(a ∪ b) = DMf∗D−1N(a ● b) = f!(a ● b) = f!f∗f ∗(a ● b) = deg(f)f∗(a ● b).
Proposition 3.1.3 Let f ∶ Kk→ Nn and g∶ Ll → Mm be maps of oriented manifolds. Then we have
3.2. Two Important Theorems
This constitutes the main part of this thesis; Here we will be stating and proving two theorems, useful for computing the coincidence number L(f, g) of two functions f, g ∶ Nn→ Mm.
Theorem 3.2.1 For f, g ∶ Nn → Mn maps of closed oriented manifolds, the Lefschetz Coincidence number is given by each of the following formulas. (Traces are computed with coefficients in the rationals, or in Zp where L(f, g) must be reduced mod p. The
subscript i on tr indicates the trace on the ith degree (co)homology.)
L(f, g) = ∑ i (−1)itr i(f∗g!) (3.2) L(f, g) = ∑ i (−1)itr i(g!f∗) (3.3) L(f, g) = ∑ i (−1)itr i(f∗g!) (3.4) L(f, g) = ∑ i (−1)itr i(g!f∗) (3.5) L(f, g) = ∗(g " f)![∆M] (3.6) L(f, g) = (−1)nL(g, f) (3.7) For the proof of this theorem we need the following lemma:
Lemma 3.2.2 With the notation of Theorem 3.2.1,
Proof Hn−p(N) f ! ÐÐÐ→ Hn−p(M) ÐÐÐ→ Hg∗ n−p(N) ×× ×Ö∩[N] ×× ×Ö∩[M] ×× ×Ö∩[N] Hp(N) f∗ ÐÐÐ→ Hp(M) g! ÐÐÐ→ Hp(N)
is commutative. Thus result.
Proof Let γf = DN×M[Γf] and γg = DN×M[Γg]. Furthermore, let B = {α} be a basis
2. First of all, we know that tr(AB) = tr (BA) . (3.8) Thus, L(f, g) = ∑ i (−1)itr(f∗g!) by (3.2) = ∑ i (−1)i tr(g!f∗) by (3.8).
3. Using Lemma 3.2.2 and (3.3) and (3.7) as well as (3.8), we get the desired result. 4. Using once again (3.8), (3.5) follows directly from (3.4).
5. ∗(g " f)![∆M] = ∗(DN−1(g " f) ∗D M×M[∆M]) = ∗((g " f)∗(τ) ∩ [N]) = ⟨(g " f)∗(τ), [N]⟩ = ∑ β (−1)deg(β)⟨g∗(β○) ∪ f∗(β), [N]⟩ = ∑ β (−1)deg(β)⟨g∗(β○), f∗(β) ∩ [N]⟩ = ∑ β (−1)deg(β)⟨β○, g∗(f∗(β) ∩ [N])⟩ = ∑ β (−1)deg(β)⟨β○, g!f∗(β) ∩ [M]⟩ = ∑ β (−1)deg(β)⟨β○∪ g!f∗(β), [M]⟩ = ∑ i (−1)itr i(g!f∗) = L(f, g)
6. Using f∗g∗= (−1)ng∗f∗, we get L(f, g) = [Γf] ⋅ [Γg] = (1 " f)∗[N] ⋅ (1 " g)∗[N] = (−1)n((1 " g) ∗[N] ⋅ (1 " f)∗[N]) = (−1)n([Γ g] ⋅ [Γf]) = (−1)nL(g, f).
Theorem 3.2.3 If Nnand Mnare smooth closed oriented n-manifolds and f, g∶ N → M are smooth and such that the difference of differentials g∗− f∗ is nonsingular at each coincidence point of f and g, then
L(f, g) = ∑
x
sign det(g∗− f∗)x, (3.9)
where the sum is over all coincidences x∈ Nn of f and g.
In short, this ’counts’ all coincidences, depending on their orientation.
Proof Let g∗− f∗ be nonsingular and as in the theorem. Then Γf and Γg are transverse
at least one x in their intersection. Furthermore, we can attach the orientation ±1 to x. This orientation is equivalent to ±sign det(g∗− f∗), depending on n.2
Corollary 3.2.4 Let f, g∶ Nn→ Mn and h∶ Kn→ Nn. Then we have 1. L(fh, gh) = deg(h)L(f, g).
2. L(f, f) = deg(f)χ(M).
3. If in addition g is homotopic to a homeomorphism with degree ±1, then L(f, g) = ±L(f, g−1).
2
Proof 1. From 3.1.1(6) we get tr((fh)∗(gh!)) = tr(f∗h∗h!g!) = deg(h) tr(f∗g!), from
which the result follows. 2.
L(f, f) = L(idMf, idMf) = deg(f)L(idM, idM) = deg(f)χ(M)
3. Let h be the homeomorphism homotopic to g. Then
4.1. Theoretical Background
The Application to Differential Equation is following from the proof of the following theorem. First, however, we need to define the differential equation we are working with. Definition We define γ as the solution of the Ordinary Differential Equation
˙γ(τ) = V (γ(τ)), (4.1)
where V is a vector field.
Theorem 4.1.1 Let v be a time-dependent 1-periodic vector field on CPn, i.e.
v∶ CPn× R → T CPn, v(x, t) ∈ TxCPn, v(x, t) = v(x, t + 1). (4.2) This vector field has at least n+ 1 periodic orbits of period 1.
To prove this, we need the following two Lemmata.
Lemma 4.1.2 Let S1 = R/Z and define the vector field V = ∂t∂ + v(x, t) on S1× CPn. Furthermore, define the return map of the flow of V as φ ∶ CPn → CPn with φ(x) = π1(γ(1)) where π1 ∶ S1 × CPn → CPn is the projection to the second factor. Then
Lemma 4.1.3 φ, as defined in Lemma 4.1.2, is homotopic to the identity . Remark The idea of the proof is simple:
First, we will show Lemma 4.1.3 by deforming the vector field to 0. Then, by applying Lefschetz, we can prove Lemma 4.1.2, from which Theorem 4.1.1 will be immediately obvious.
In addition, Lemma 4.1.2 gives us the desired application to Ordinary Differential Equations for free.
Proof of Lemma 4.1.3First, as is required, we deform the vector field V to 0 in order to get ˜V = 0×CPn. Thus, the identity on ˜V is idV˜ = (0, idCPn). Thus, we can ignore the first
coordinate and simply show that φ ≃ idCPn (which is equivalent to (0, φ) ≃ (0, idCPn)).
It is important to note that the flow at point t = 0 is equal to the identity, i.e. v(x, 0) = x by definition. Thus, γ(τ) = τ if t = 0. However, φ is defined as the second factor of γ(1). Nevertheless, since we are dealing with the flow, there is a function, Γl, say, such that
Γ0 = γ(0) = id and Γ1 = γ(1). Thus, by deforming the vector field to 0 in the first
coordinate, we see immediately that φ is homotopic to the identity.
Proof of Lemma 4.1.2 This Lemma can be easily shown by calculating L(φ, idCPn) = L(φ) which was defined in (2.1).
In order to calculate L(φ), we use first Proposition 2.2.7 to determine L(id). Then, by Proposition 2.2.8, L(f) = L(id). Hence L(f) = L(id) = χ(CPn) = n + 1.
Proof of Theorem 4.1.1 Since the return map of the flow, φ, has at least n + 1 fixed points, this means that there are at least n + 1 periodic orbits of period 1 in v, the time dependent 1-periodic vector field on CPn. The length of the period is due to the vector
4.2. Application
non-orientable case
In this appendix, we will give a quick overview of the Lefschetz Coincidence Theory in the non-orientable case, assuming dim M = dim N, following closely [3]. For a more detailed treatment, see for example [3] or [8].
For this appendix, we have the general case where M, N are not assumed to be either compact or non-orientable. Furthermore, f is compact, while g is proper and orientable. Furthermore, let R be a field.
In the case of (possibly) non-orientable manifolds, the Lefschetz Coincidence Number can be defined using the so-called Leray trace, which is defined for a map h of a vector space E into itself with finite-dimensional quotient space E/N(h), where N(h) is the subspace of elements mapped to zero under the action of some powers of h. For the purposes of this appendix, we denote the Leray trace by Trh.
In addition, it is important to note that there are three cases we have to look at: 1. g(∂M) ⊂ ∂N.
2. f(∂M) ⊂ ∂N.
Definition By the map θq of the space Hq(N; R) into itself we mean the following composite:
Hq(N, R) →f∗ Hq(M, R) =D Hn−q(M, ∂M; Hn(M))
→g∗ Hn−q(N, ∂N; Hn(N)) =D Hq(N; R)
Definition In case 1, we define the Lefschetz Coincidence Number as Λ′f,g= ∑q(−1)qT rθq.
Definition By the map θq of the space Hq(N, ∂N; R) into itself we mean the following composite:
Hq(N, ∂N, R) →f∗ Hq(M, ∂M, R) =DHn−q(M; Hn(M))
→g∗ Hn−q(N; Hn(N)) =DHq(N, ∂N; R)
Definition In case 2, we define the Lefschetz Coincidence Number as Λ′′f,g= ∑q(−1)qT rθq.
Theorem A.0.1 If g(∂M) ⊂ ∂N and Λ′f,g ≠ 0, then the maps f, g have a coincidence point.
Theorem A.0.2 If f(∂M) ⊂ ∂N and Λ′′f,g ≠ 0, then f, g have a coincidence point. But what about case 3? Let ∂f, ∂g denote the maps ∂M → ∂N that are restrictions to ∂M of the maps f, g. Since the manifolds ∂M, ∂N have no boundary, for them the two definitions of the Lefschetz Number coincide. The corresponding Lefschetz Number is denoted as Λ∂f,∂g and we have the following theorem:
Definition The coincidence index If,g is the image of the class µc∈ Hnc(M, (M/C) ∪ ∂M; Hn(M))
defined above under the sequence of homomorphisms
Hnc(M, (M/C) ∪ ∂M; H(M)) →d∗ Hnc(M × M, (∂M × M) ∪ ((M × M)/d(C)); Rˆ⊗Hn(M))
→(f×g)∗ H
c
n(N × intN, (∂N × intN) ∪ (N × intN/∆); Rˆ⊗Hn(N))
→⟨τ,⋅⟩R or, in short,
If,g= ⟨τ, (f × g)∗d∗(µC)⟩ .
In fact, Theorem A.0.2 is following from the next theorem.
[1] J. Andres, Chapter 1, Topological Principles for Ordinary Differential Equations in; A. Ca˜nada, P. Drábek, A. Fonda (2006) Handbook of Differential Equations: Ordinary Differential Equations Volume 3, Elsevier, North-Holland, Amsterdam. [2] M.A. Armstrong (1983) Basic Topology, Springer, New York.
[3] D.V. Artamonov, On the Lefschetz Coincidence Theorem (2009) Sbornik: Mathem-atics 200:7, 943-979.
[4] R. Bott, L.W. Tu (1982) Differential Forms in Algebraic Topology, Springer, New York.
[5] G.E. Bredon (1993) Topology and Geometry, Springer, New York.
[6] I.N. Bronstein, K.A. Semendjajew, G.Musiol, H.Mühlig (2005,2006) Handbook of Mathematics [Taschenbuch der Mathematik], Verlag Harri Deutsch, Frankfurt. [7] F.Diacu (2000) An Introduction to Differential Equations - Order and Chaos, W.H.
Freeman and Company, New York.
[8] D.L. Gonçalves and J. Jezierski (1997) Lefschetz Coincidence Formula on non-orientable Manifolds Fund. Math. 153:1, 1-23.
[9] P.A. Grillet (2007) Abstract Algebra, Springer, New York.
[10] A. Hatcher (2002) Algebraic Topology, Cambridge University Press, Cambridge. [11] J.H. Hubbard, B.H. West (1991) Differential Equations: A Dynamical Systems
Ap-proach, Springer, New York.
[12] S. Lefschetz (1926) Intersections and Transformations of Complexes and Manifolds, Trans. Am. Math. Soc. Vol. 28, 1-49.
[14] W. Walter (1998) Ordinary Differential Equations, Springer, New York.
[15] E. & M. Niedermair (2004) LATEX practice manual, [LATEX Praxisbuch], Francis
Ver-lag, Poing.
[16] R. Willms (2005) LATEX really easy, [LATEX echt einfach], Franzis Verlag, Poing.
[17] T. Zilm (2003) LATEX, vmi Buch, Frechen.
