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Zeroth canonical homomorphism from singular to Milnor–Thurston homology is injective
Topology and its Applications 248 (2018) 143–148
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Zeroth canonical homomorphism from singular to Milnor–Thurston homology is injective Janusz Przewocki Adam Mickiewicz University, Faculty of Mathematics and Computer Science, ul. Umultowska, 87, 61-614 Poznań, Poland
a r t i c l e
i n f o
Article history: Received 2 March 2018 Received in revised form 27 August 2018 Accepted 29 August 2018 Available online 31 August 2018
a b s t r a c t Milnor–Thurston homology also known as “measure homology” is a generalisation of singular homology where chains are defined to be measures (possibly supported on infinitely many singular simplices). In this paper we prove that the canonical homomorphism from singular to Milnor–Thurston homology is injective in dimension zero for metrizable spaces. © 2018 Elsevier B.V. All rights reserved.
MSC: 55N35 Keywords: Algebraic topology Homology theory Milnor–Thurston homology Measure homology
1. Introduction The basic ideas of Milnor–Thurston homology theory were introduced in [1, Chapter 6] in the context of hyperbolic geometry. Its precise definition was later formulated by Zastrow [2] and by Hansen [3] under the name “measure homology”. The main feature of this homology theory is that, by extending definition of a chain, it provides more freedom when constructing chains representing given homology classes. The chains are defined to be measures concentrated on the space of singular simplices instead of linear combinations of singular simplices as in singular homology theory. This homology theory satisfies the usual axioms of Eilenberg and Steenrod [2, Proof of the coincidence theorem 5.0], so it is isomorphic to singular homology for spaces homotopy equivalent to CW-complexes. For more general class of spaces calculation of Milnor–Thurston homology is by no means straightforward. Spaces that are not homotopy equivalent to CW-complexes tend to have complicated structure of its algebraic-topology invariants (see for example [4–6]). The reason behind this phenomenon is due to finiteE-mail address: [email protected]. https://doi.org/10.1016/j.topol.2018.08.016 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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J. Przewocki / Topology and its Applications 248 (2018) 143–148
ness of algebraic constructions (e.g. discrete groups are described by finite words of generators, chains in singular homology groups are finite linear combinations). On the other hand, Milnor–Thurston homology is a theory where chains supported on infinite number of simplices are admitted. Therefore, investigation of its behaviour for general topological spaces is valuable. This research problem was undertaken in [7] and [8]. In particular, it has been proved that for Peano continua (i.e. locally connected metric compacta) zeroth Milnor–Thurston homology group coincides with singular homology via the natural transformation, so called canonical homomorphism (see Section 2 for definition), between these functors [8, Theorem 3.2]. Moreover, even for more general spaces the canonical homomorphism is proved to be a monomorphism, provided our space has Borel path-components [8, Theorem 4.1]. This cumbersome assumption of Borel path-components occurred to be necessary, since there is a space where the canonical homomorphism is not injective [8, Theorem 5.8]. In this paper we are answering the question: what are more reasonable assumptions on our space implying that the canonical homomorphism is injective? More precisely, we prove that for all metrizable spaces the canonical homomorphism is a monomorphism. 2. Preliminaries To construct Milnor–Thurston homology theory let us first recall some notions from measure theory. Let Ω be a topological space. The smallest σ-algebra generated by open sets is called Borel σ-algebra, and its elements are Borel sets. A set function on a given σ-algebra is called a signed measure if it satisfies the usual conditions (σ-additivity and vanishing on the empty set), but without the requirement of being non-negative. A signed measure that takes only non-negative values is called unsigned measure. Moreover, we use adjectives Borel and/or finite if a measure is defined on the Borel σ-algebra and/or it has finite values only. We say that D ⊂ Ω is a carrier of a signed Borel measure μ if for any Borel set B ⊂ Ω \ D we have μ(B) = 0. Finally, let us define M (Ω) to be the set of finite signed Borel measures on Ω. Throughout this paper δω will denote Dirac measure concentrated on ω ∈ Ω, i.e. the measure taking value one on all Borel subsets that contain ω, and value zero otherwise. It is immediate to see that any linear combination of Dirac measures is an element of M (Ω). Now, we can define Milnor–Thurston homology groups. Let us start with a space C(Δk , X) of singular simplices, i.e. a space of continuous functions from the standard simplex Δk to a given topological space X. We endow C(Δk , X) with the compact-open topology. A group of k-dimensional chains shall be defined as Ck (X) := {μ ∈ M (C(Δk , X)) | there exists a compact carrier for μ}. Measure ∂μ assigned to μ by the boundary operator ∂ : Ck (X) → Ck−1 (X) is defined to be:
(∂μ)(B) :=
k (−1)i μ(∂i−1 (B)),
for B ⊂ C(Δk−1 , X),
(1)
i=0
where ∂i : C(Δk , X) → C(Δk−1 , X) is a map induced by the inclusion Δk−1 → Δk as ith face. Remark. In this paper we are concerned with zeroth homology groups, so we use only ∂i : C(Δ1 , X) → C(Δ0 , X), for i = 0, 1. We identify the standard 1-simplex Δ1 with [0, 1], and C(Δ0 , X) with X itself. It is a convention that ith face of Δk is opposite to ith vertex. From that follows that for σ ∈ C(Δ1 , X) we get ∂0 σ = σ(1) and ∂1 σ = σ(0).
J. Przewocki / Topology and its Applications 248 (2018) 143–148
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The sequence of real vector spaces Ck (X) with boundary operator (1) is a chain-complex [2, Corollary 2.9], and kth Milnor–Thurston homology group Hk (X) is defined to be kth homology group of this complex. For a more thorough discussion of this homology theory and its properties see [2]. Finally, we see that there is an obvious identification of singular chains and Milnor–Thurston chains Ck (X) → Ck (X), n i=1
αi σi →
n
αi δσi ,
i=1
which commutes with boundary operators. This identification yields the canonical homomorphism Hk (X) → Hk (X). 3. Proof of the main theorem Let X denote a Polish space and let M(X) be a σ-algebra of sets measurable with respect to every σ-finite Borel measure on X. Zero measure on a σ-algebra F is a set function that has value zero on every set in F. Lemma 1. Let μ be an unsigned finite Borel measure on X. There is a unique unsigned measure that is an extension of μ to any σ-algebra F ⊂ M(X). Proof. Every set A ∈ M(X) is measurable with respect to μ. Thus μ∗ (A) = μ∗ (A). However, the outer and inner measures are, respectively, the upper and the lower bounds of the possible values that extension of μ can have. Consequently, the extension of μ to F ⊂ M(X) is unique. 2 Lemma 2. The only extension to a σ-algebra F ⊂ M(X) of the zero measure on the Borel σ-algebra is the zero measure on F. Proof. Notice that is the obvious corollary of the previous lemma if we consider only unsigned measures. On the other hand, let ν be a signed measure on F that is zero on Borel sets. There is the Hahn decomposition [9, p. 122]: ν = ν+ − ν− , where ν± are unsigned measures. From the fact, that ν is zero on Borel sets, we see that ν+ and ν− are both extensions of the same unsigned Borel measure. Thus, by the previous lemma, we see that ν+ = ν− , which yields ν = 0. 2 Lemma 3. Let μ be a signed Borel measure on X. Then there is a unique extension of μ to any σ-algebra F ⊂ M(X). Proof. The natural candidate for the extension is μ∗+ − μ∗− , where μ+ − μ− is the Hahn decomposition of μ (cf. proof of Lemma 2). It is unique, for if ν is any other extension, then ν − (μ∗+ − μ∗− ) is an extension of the zero measure, so it is zero by Lemma 2. 2 . Let f : Ω1 → Ω2 be If μ is a signed Borel measure on X we shall denote its extension to M(X) by μ a function between sets Ω1 and Ω2 and let F be a σ-algebra of sets in Ω1 . The family of subsets A ⊂ Ω2 satisfying f −1 (A) ∈ F is a σ-algebra, which follows from the fact that operations on sets behave well under preimages. Now, if f : X → Y is a continuous map between Polish spaces, the family Mf (Y ) := {A ∈ M(Y ) | f −1 (A) ∈ M(X)} is also a σ-algebra as an intersection of two σ-algebras. Moreover, the continuity of f implies that Mf (Y ) is nonempty.
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J. Przewocki / Topology and its Applications 248 (2018) 143–148
Lemma 4. Consider continuous map f : (X, M(X)) → (Y, Mf (Y )) between Polish spaces. Let μ be a signed Borel measure, then f μ = f μ, where · denotes the extension to the respective σ-algebra. Proof. The measure f μ is well-defined on Mf (Y ) since every preimage with respect to f is in M(X) by definition. Let A be a Borel set in Y . Then by continuity f −1 (A) is Borel in X. Therefore f μ (A) = f μ(A). So, f μ is an extension of f μ. By the uniqueness of extension (cf. Lemma 3) we get the assertion of our lemma. 2 Lemma 5. Let μ1 and μ2 be Borel measures on a Polish space X. Then μ 1 ± μ 2 = μ 1 ± μ2 . Proof. It is an obvious consequence of Lemma 3.
2
Theorem 6. Let X be a Polish space and each element of the family {Aα}nα=1 of Borel subsets lie in different n path components of X. Then there is no measure ν ∈ C1 (X) such that ∂ν = α=1 μα , where each μα is a nonzero measure with its carrier in Aα , provided that μβ (Aβ ) = 0, for some index β. Proof. Suppose there exists ν ∈ C1 (X), such that ∂ν = α μα . The space C(Δ1 , X) is a Polish space [10, Theorem 4.19], so there is an extension ν of ν to M(C(Δ1 , X)). Let us take a path-component A containing Aβ and take any x ∈ A. Then ∂0−1 ({x}) is a Borel subset of C(Δ1 , X). Therefore, A = ∂1 (∂0−1 ({x})) is an analytic set as a continuous image of a Borel set in a Polish space. From that we see that A is also universally measurable [10, Theorem 21.10]. Continuous inverse image of an analytic set is analytic [10, Proposition 14.4], hence A ∈ F := M∂0 (X) ∩ M∂1 (X). Observe that we have ∂0−1 (A) = ∂1−1 (A). And from that follows ∂0 ν(A) − ∂1 ν(A) = ν(∂0−1 (A)) − ν(∂1−1 (A)) = 0.
(2)
However, by Lemma 4 and Lemma 5 we have ∂0 ν(A) − ∂1 ν(A) = ∂ 0 ν(A) − ∂1 ν(A) = ∂0 ν − ∂1 ν(A) =
μα (A) = μβ (Aβ ) = 0. α
That contradicts equation (2). 2 Theorem 7. Let X be a metric space, then the canonical homomorphism between zeroth singular homology and zeroth Milnor–Thurston homology is injective. n Proof. Let us take a 0-chain i=1 αxi xi , for xi ∈ X and αxi ∈ R, whose homology class is in the kernel of n the canonical homomorphism. That means that there exists ν ∈ C1 (X) such that i=1 αxi δxi = ∂ν. There is a compact carrier D of ν. Consider the evaluation function e : Δ1 × D → X. Let X := e(Δ1 × D). It is a compact subspace of X and every compact metric space is Polish. In addition, we can see that ν is in the image of the inclusion C1 (X ) → C1 (X), which follows from the observation that D ⊂ C(Δ1 , X ) ⊂ C(Δ1 , X). Let us take a partition {xi }ni=1 = α Xα such that each Xα is an intersection of {xi }ni=1 with some path component of X. Now, we have n i=1
αxi δxi =
α x∈Xα
αx δx .
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147
From Theorem 6 it follows that for each α we have x∈Xα αx δx (Xα ) = 0. Therefore the sum of coefficients αx is zero, and α x is homologous to zero, since Xα is contained in a single path component. n x∈Xα x Consequently, i=1 αxi xi is trivial in singular homology. 2 Let X be a Polish space and P ⊂ X is its path-component. Since R = {(σ, t, x) ∈ C(Δn , X) × Δn × X | σ(t) = x} is a Borel relation, we can see C(Δn , P ) is an analytic subset of the Polish space C(Δn , X). Indeed, let π1 and π3 denote the projections of C(Δn , X) × Δn × X onto C(Δn , X) and X, respectively. Then C(Δn , P ) = π3 (π1−1 (P )) which is clearly an analytic subset. The above observation is sufficiently powerful to prove the following Theorem 8. Suppose that Polish space X has finitely many path-components Pi , for i = 1, ..., n. Then H∗ (X) ∼ =
n
H∗ (Pi ).
i=1
However, the fact that C(Δn , P ) is an analytic subset does not imply that the canonical homomorphism for higher dimensional homology groups is injective. For a space X with only one path-component fact that C(Δn , P ) = C(Δn , X) is analytic is trivial, but possible proof of injectivity of the canonical homomorphism is certainly not (see for example calculations done for the Warsaw Circle in [7]). Finally, it is important to mention that Theorem 8 does not imply Theorem 7 in its full generality. Theorem 7 does not require space X to have finitely many path-components, nor to be Polish. Proof of Theorem 8. It is sufficient to prove that C∗ (X) ∼ =
n
C∗ (Pi )
i=1
as chain-complexes. n Let μ ∈ Ck (X) then we define s(μ) ∈ i=1 Ck (Pi ) with the formula s(μ) =
n
si (μ),
i=1
where si (μ)(Bi ) = μ (Bi ), for Bi a Borel subset of C(Δk , Pi ). The above map is well-defined, since Bi being a Borel subset of analytic subset C(Δk , Pi ) is analytic in C(Δk , X). n n n Similarly, for i=1 μi ∈ i=1 Ck (Pi ) we define t( i=1 μi ) ∈ Ck (X) with the following formula n n
t( μi )(B) = μi (B ∩ C(Δk , Pi )), for B a Borel subset of C(Δk , X). i=1
i=1
Let us notice that both s and t are linear maps. Moreover, we see that t(s(μ)) = μ. Indeed, by additivity of μ we have: t(s(μ))(B) =
n i=1
s(μ)(B ∩ C(Δk , Pi )) =
n i=1
μ (B ∩ C(Δk , Pi )) = μ(B),
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n where B is a Borel subset of C(Δk , X). Next, we see that ith summand of s(t( i=1 μi )) is n n
t( μi )(Bi ) = μ i (Bi ) = μi (Bi ), for Bi a Borel subset of C(Δk , Pi ). i=1
i=1
n n The last equality is due to the fact that μj (Bi ) = 0 for i = j. From that we see that s(t( i=1 μi )) = i=1 μi . Finally, notice that both s and t commute with the boundary operator. 2 Acknowledgements The research was supported by the National Science Centre grant 2015/19/B/ST1/01458. References [1] W. Thurston, Geometry and topology of three-manifolds, http://www.msri.org/publications/books/gt3m, 1978. [2] A. Zastrow, On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory, Pac. J. Math. 186 (2) (1998) 369–396, https://doi.org/10.2140/pjm.1998.186.369. [3] S.K. Hansen, Measure homology, Math. Scand. 83 (2) (1998) 205–219, https://doi.org/10.7146/math.scand.a-13851. [4] M.G. Barratt, J. Milnor, An example of anomalous singular homology, Proc. Am. Math. Soc. 13 (1962) 293–297, https:// doi.org/10.2307/2034486. [5] K. Eda, Free σ-products and non-commutatively slender groups, J. Algebra 148 (1) (1992) 243–263, https://doi.org/10. 1016/0021-8693(92)90246-I. [6] K. Eda, K. Kawamura, The singular homology of the Hawaiian earring, J. Lond. Math. Soc. (2) 62 (1) (2000) 305–310, https://doi.org/10.1112/S0024610700001071. [7] J. Przewocki, Milnor–Thurston homology groups of the Warsaw Circle, Topol. Appl. 160 (13) (2013) 1732–1741, https:// doi.org/10.1016/j.topol.2013.07.001. [8] J. Przewocki, A. Zastrow, On the coincidence of zeroth Milnor–Thurston homology with singular homology, Fund. Math. 243 (2) (2018) 109–122, https://doi.org/10.4064/fm893-6-2018, arXiv:1403.3341. [9] P.R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, NY, 1950. [10] A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Zeroth canonical homomorphism from singular to Milnor–Thurston homology is injective Janusz Przewocki Adam Mickiewicz University, Faculty of Mathematics and Computer Science, ul. Umultowska, 87, 61-614 Poznań, Poland
a r t i c l e
i n f o
Article history: Received 2 March 2018 Received in revised form 27 August 2018 Accepted 29 August 2018 Available online 31 August 2018
a b s t r a c t Milnor–Thurston homology also known as “measure homology” is a generalisation of singular homology where chains are defined to be measures (possibly supported on infinitely many singular simplices). In this paper we prove that the canonical homomorphism from singular to Milnor–Thurston homology is injective in dimension zero for metrizable spaces. © 2018 Elsevier B.V. All rights reserved.
MSC: 55N35 Keywords: Algebraic topology Homology theory Milnor–Thurston homology Measure homology
1. Introduction The basic ideas of Milnor–Thurston homology theory were introduced in [1, Chapter 6] in the context of hyperbolic geometry. Its precise definition was later formulated by Zastrow [2] and by Hansen [3] under the name “measure homology”. The main feature of this homology theory is that, by extending definition of a chain, it provides more freedom when constructing chains representing given homology classes. The chains are defined to be measures concentrated on the space of singular simplices instead of linear combinations of singular simplices as in singular homology theory. This homology theory satisfies the usual axioms of Eilenberg and Steenrod [2, Proof of the coincidence theorem 5.0], so it is isomorphic to singular homology for spaces homotopy equivalent to CW-complexes. For more general class of spaces calculation of Milnor–Thurston homology is by no means straightforward. Spaces that are not homotopy equivalent to CW-complexes tend to have complicated structure of its algebraic-topology invariants (see for example [4–6]). The reason behind this phenomenon is due to finiteE-mail address: [email protected]. https://doi.org/10.1016/j.topol.2018.08.016 0166-8641/© 2018 Elsevier B.V. All rights reserved.
144
J. Przewocki / Topology and its Applications 248 (2018) 143–148
ness of algebraic constructions (e.g. discrete groups are described by finite words of generators, chains in singular homology groups are finite linear combinations). On the other hand, Milnor–Thurston homology is a theory where chains supported on infinite number of simplices are admitted. Therefore, investigation of its behaviour for general topological spaces is valuable. This research problem was undertaken in [7] and [8]. In particular, it has been proved that for Peano continua (i.e. locally connected metric compacta) zeroth Milnor–Thurston homology group coincides with singular homology via the natural transformation, so called canonical homomorphism (see Section 2 for definition), between these functors [8, Theorem 3.2]. Moreover, even for more general spaces the canonical homomorphism is proved to be a monomorphism, provided our space has Borel path-components [8, Theorem 4.1]. This cumbersome assumption of Borel path-components occurred to be necessary, since there is a space where the canonical homomorphism is not injective [8, Theorem 5.8]. In this paper we are answering the question: what are more reasonable assumptions on our space implying that the canonical homomorphism is injective? More precisely, we prove that for all metrizable spaces the canonical homomorphism is a monomorphism. 2. Preliminaries To construct Milnor–Thurston homology theory let us first recall some notions from measure theory. Let Ω be a topological space. The smallest σ-algebra generated by open sets is called Borel σ-algebra, and its elements are Borel sets. A set function on a given σ-algebra is called a signed measure if it satisfies the usual conditions (σ-additivity and vanishing on the empty set), but without the requirement of being non-negative. A signed measure that takes only non-negative values is called unsigned measure. Moreover, we use adjectives Borel and/or finite if a measure is defined on the Borel σ-algebra and/or it has finite values only. We say that D ⊂ Ω is a carrier of a signed Borel measure μ if for any Borel set B ⊂ Ω \ D we have μ(B) = 0. Finally, let us define M (Ω) to be the set of finite signed Borel measures on Ω. Throughout this paper δω will denote Dirac measure concentrated on ω ∈ Ω, i.e. the measure taking value one on all Borel subsets that contain ω, and value zero otherwise. It is immediate to see that any linear combination of Dirac measures is an element of M (Ω). Now, we can define Milnor–Thurston homology groups. Let us start with a space C(Δk , X) of singular simplices, i.e. a space of continuous functions from the standard simplex Δk to a given topological space X. We endow C(Δk , X) with the compact-open topology. A group of k-dimensional chains shall be defined as Ck (X) := {μ ∈ M (C(Δk , X)) | there exists a compact carrier for μ}. Measure ∂μ assigned to μ by the boundary operator ∂ : Ck (X) → Ck−1 (X) is defined to be:
(∂μ)(B) :=
k (−1)i μ(∂i−1 (B)),
for B ⊂ C(Δk−1 , X),
(1)
i=0
where ∂i : C(Δk , X) → C(Δk−1 , X) is a map induced by the inclusion Δk−1 → Δk as ith face. Remark. In this paper we are concerned with zeroth homology groups, so we use only ∂i : C(Δ1 , X) → C(Δ0 , X), for i = 0, 1. We identify the standard 1-simplex Δ1 with [0, 1], and C(Δ0 , X) with X itself. It is a convention that ith face of Δk is opposite to ith vertex. From that follows that for σ ∈ C(Δ1 , X) we get ∂0 σ = σ(1) and ∂1 σ = σ(0).
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The sequence of real vector spaces Ck (X) with boundary operator (1) is a chain-complex [2, Corollary 2.9], and kth Milnor–Thurston homology group Hk (X) is defined to be kth homology group of this complex. For a more thorough discussion of this homology theory and its properties see [2]. Finally, we see that there is an obvious identification of singular chains and Milnor–Thurston chains Ck (X) → Ck (X), n i=1
αi σi →
n
αi δσi ,
i=1
which commutes with boundary operators. This identification yields the canonical homomorphism Hk (X) → Hk (X). 3. Proof of the main theorem Let X denote a Polish space and let M(X) be a σ-algebra of sets measurable with respect to every σ-finite Borel measure on X. Zero measure on a σ-algebra F is a set function that has value zero on every set in F. Lemma 1. Let μ be an unsigned finite Borel measure on X. There is a unique unsigned measure that is an extension of μ to any σ-algebra F ⊂ M(X). Proof. Every set A ∈ M(X) is measurable with respect to μ. Thus μ∗ (A) = μ∗ (A). However, the outer and inner measures are, respectively, the upper and the lower bounds of the possible values that extension of μ can have. Consequently, the extension of μ to F ⊂ M(X) is unique. 2 Lemma 2. The only extension to a σ-algebra F ⊂ M(X) of the zero measure on the Borel σ-algebra is the zero measure on F. Proof. Notice that is the obvious corollary of the previous lemma if we consider only unsigned measures. On the other hand, let ν be a signed measure on F that is zero on Borel sets. There is the Hahn decomposition [9, p. 122]: ν = ν+ − ν− , where ν± are unsigned measures. From the fact, that ν is zero on Borel sets, we see that ν+ and ν− are both extensions of the same unsigned Borel measure. Thus, by the previous lemma, we see that ν+ = ν− , which yields ν = 0. 2 Lemma 3. Let μ be a signed Borel measure on X. Then there is a unique extension of μ to any σ-algebra F ⊂ M(X). Proof. The natural candidate for the extension is μ∗+ − μ∗− , where μ+ − μ− is the Hahn decomposition of μ (cf. proof of Lemma 2). It is unique, for if ν is any other extension, then ν − (μ∗+ − μ∗− ) is an extension of the zero measure, so it is zero by Lemma 2. 2 . Let f : Ω1 → Ω2 be If μ is a signed Borel measure on X we shall denote its extension to M(X) by μ a function between sets Ω1 and Ω2 and let F be a σ-algebra of sets in Ω1 . The family of subsets A ⊂ Ω2 satisfying f −1 (A) ∈ F is a σ-algebra, which follows from the fact that operations on sets behave well under preimages. Now, if f : X → Y is a continuous map between Polish spaces, the family Mf (Y ) := {A ∈ M(Y ) | f −1 (A) ∈ M(X)} is also a σ-algebra as an intersection of two σ-algebras. Moreover, the continuity of f implies that Mf (Y ) is nonempty.
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Lemma 4. Consider continuous map f : (X, M(X)) → (Y, Mf (Y )) between Polish spaces. Let μ be a signed Borel measure, then f μ = f μ, where · denotes the extension to the respective σ-algebra. Proof. The measure f μ is well-defined on Mf (Y ) since every preimage with respect to f is in M(X) by definition. Let A be a Borel set in Y . Then by continuity f −1 (A) is Borel in X. Therefore f μ (A) = f μ(A). So, f μ is an extension of f μ. By the uniqueness of extension (cf. Lemma 3) we get the assertion of our lemma. 2 Lemma 5. Let μ1 and μ2 be Borel measures on a Polish space X. Then μ 1 ± μ 2 = μ 1 ± μ2 . Proof. It is an obvious consequence of Lemma 3.
2
Theorem 6. Let X be a Polish space and each element of the family {Aα}nα=1 of Borel subsets lie in different n path components of X. Then there is no measure ν ∈ C1 (X) such that ∂ν = α=1 μα , where each μα is a nonzero measure with its carrier in Aα , provided that μβ (Aβ ) = 0, for some index β. Proof. Suppose there exists ν ∈ C1 (X), such that ∂ν = α μα . The space C(Δ1 , X) is a Polish space [10, Theorem 4.19], so there is an extension ν of ν to M(C(Δ1 , X)). Let us take a path-component A containing Aβ and take any x ∈ A. Then ∂0−1 ({x}) is a Borel subset of C(Δ1 , X). Therefore, A = ∂1 (∂0−1 ({x})) is an analytic set as a continuous image of a Borel set in a Polish space. From that we see that A is also universally measurable [10, Theorem 21.10]. Continuous inverse image of an analytic set is analytic [10, Proposition 14.4], hence A ∈ F := M∂0 (X) ∩ M∂1 (X). Observe that we have ∂0−1 (A) = ∂1−1 (A). And from that follows ∂0 ν(A) − ∂1 ν(A) = ν(∂0−1 (A)) − ν(∂1−1 (A)) = 0.
(2)
However, by Lemma 4 and Lemma 5 we have ∂0 ν(A) − ∂1 ν(A) = ∂ 0 ν(A) − ∂1 ν(A) = ∂0 ν − ∂1 ν(A) =
μα (A) = μβ (Aβ ) = 0. α
That contradicts equation (2). 2 Theorem 7. Let X be a metric space, then the canonical homomorphism between zeroth singular homology and zeroth Milnor–Thurston homology is injective. n Proof. Let us take a 0-chain i=1 αxi xi , for xi ∈ X and αxi ∈ R, whose homology class is in the kernel of n the canonical homomorphism. That means that there exists ν ∈ C1 (X) such that i=1 αxi δxi = ∂ν. There is a compact carrier D of ν. Consider the evaluation function e : Δ1 × D → X. Let X := e(Δ1 × D). It is a compact subspace of X and every compact metric space is Polish. In addition, we can see that ν is in the image of the inclusion C1 (X ) → C1 (X), which follows from the observation that D ⊂ C(Δ1 , X ) ⊂ C(Δ1 , X). Let us take a partition {xi }ni=1 = α Xα such that each Xα is an intersection of {xi }ni=1 with some path component of X. Now, we have n i=1
αxi δxi =
α x∈Xα
αx δx .
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From Theorem 6 it follows that for each α we have x∈Xα αx δx (Xα ) = 0. Therefore the sum of coefficients αx is zero, and α x is homologous to zero, since Xα is contained in a single path component. n x∈Xα x Consequently, i=1 αxi xi is trivial in singular homology. 2 Let X be a Polish space and P ⊂ X is its path-component. Since R = {(σ, t, x) ∈ C(Δn , X) × Δn × X | σ(t) = x} is a Borel relation, we can see C(Δn , P ) is an analytic subset of the Polish space C(Δn , X). Indeed, let π1 and π3 denote the projections of C(Δn , X) × Δn × X onto C(Δn , X) and X, respectively. Then C(Δn , P ) = π3 (π1−1 (P )) which is clearly an analytic subset. The above observation is sufficiently powerful to prove the following Theorem 8. Suppose that Polish space X has finitely many path-components Pi , for i = 1, ..., n. Then H∗ (X) ∼ =
n
H∗ (Pi ).
i=1
However, the fact that C(Δn , P ) is an analytic subset does not imply that the canonical homomorphism for higher dimensional homology groups is injective. For a space X with only one path-component fact that C(Δn , P ) = C(Δn , X) is analytic is trivial, but possible proof of injectivity of the canonical homomorphism is certainly not (see for example calculations done for the Warsaw Circle in [7]). Finally, it is important to mention that Theorem 8 does not imply Theorem 7 in its full generality. Theorem 7 does not require space X to have finitely many path-components, nor to be Polish. Proof of Theorem 8. It is sufficient to prove that C∗ (X) ∼ =
n
C∗ (Pi )
i=1
as chain-complexes. n Let μ ∈ Ck (X) then we define s(μ) ∈ i=1 Ck (Pi ) with the formula s(μ) =
n
si (μ),
i=1
where si (μ)(Bi ) = μ (Bi ), for Bi a Borel subset of C(Δk , Pi ). The above map is well-defined, since Bi being a Borel subset of analytic subset C(Δk , Pi ) is analytic in C(Δk , X). n n n Similarly, for i=1 μi ∈ i=1 Ck (Pi ) we define t( i=1 μi ) ∈ Ck (X) with the following formula n n
t( μi )(B) = μi (B ∩ C(Δk , Pi )), for B a Borel subset of C(Δk , X). i=1
i=1
Let us notice that both s and t are linear maps. Moreover, we see that t(s(μ)) = μ. Indeed, by additivity of μ we have: t(s(μ))(B) =
n i=1
s(μ)(B ∩ C(Δk , Pi )) =
n i=1
μ (B ∩ C(Δk , Pi )) = μ(B),
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n where B is a Borel subset of C(Δk , X). Next, we see that ith summand of s(t( i=1 μi )) is n n
t( μi )(Bi ) = μ i (Bi ) = μi (Bi ), for Bi a Borel subset of C(Δk , Pi ). i=1
i=1
n n The last equality is due to the fact that μj (Bi ) = 0 for i = j. From that we see that s(t( i=1 μi )) = i=1 μi . Finally, notice that both s and t commute with the boundary operator. 2 Acknowledgements The research was supported by the National Science Centre grant 2015/19/B/ST1/01458. References [1] W. Thurston, Geometry and topology of three-manifolds, http://www.msri.org/publications/books/gt3m, 1978. [2] A. Zastrow, On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory, Pac. J. Math. 186 (2) (1998) 369–396, https://doi.org/10.2140/pjm.1998.186.369. [3] S.K. Hansen, Measure homology, Math. Scand. 83 (2) (1998) 205–219, https://doi.org/10.7146/math.scand.a-13851. [4] M.G. Barratt, J. Milnor, An example of anomalous singular homology, Proc. Am. Math. Soc. 13 (1962) 293–297, https:// doi.org/10.2307/2034486. [5] K. Eda, Free σ-products and non-commutatively slender groups, J. Algebra 148 (1) (1992) 243–263, https://doi.org/10. 1016/0021-8693(92)90246-I. [6] K. Eda, K. Kawamura, The singular homology of the Hawaiian earring, J. Lond. Math. Soc. (2) 62 (1) (2000) 305–310, https://doi.org/10.1112/S0024610700001071. [7] J. Przewocki, Milnor–Thurston homology groups of the Warsaw Circle, Topol. Appl. 160 (13) (2013) 1732–1741, https:// doi.org/10.1016/j.topol.2013.07.001. [8] J. Przewocki, A. Zastrow, On the coincidence of zeroth Milnor–Thurston homology with singular homology, Fund. Math. 243 (2) (2018) 109–122, https://doi.org/10.4064/fm893-6-2018, arXiv:1403.3341. [9] P.R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, NY, 1950. [10] A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.