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Relating VFCs on thin compactifications

Abstract

Many moduli spaces that occur in geometric analysis admit “Fredholm-stratified thin compactifications”, and hence admit a relative fundamental class (RFC), as defined previously by the authors. We extend these results, emphasizing the naturality of the RFC, eliminating the need for a stratification, and proving three compatibility results: the invariants defined by the RFC agree with those defined by pseudo-cycles, the RFC is compatible with cutdown moduli spaces, and the RFC agrees with the virtual fundamental class (VFC) constructed by Pardon via implicit atlases in all cases where both are defined.

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Correspondence to Eleny-Nicoleta Ionel.

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Communicated by Jean-Yves Welschinger.

Appendices

Appendix A: Comparison of homology theories

This first appendix records needed facts about the Steenrod, Čech and Borel–Moore homology and the corresponding cohomology theories, and provides references.

A.1: Spaces

We will consider five categories: the category \({\mathcal A}\) of Hausdorff spaces and continuous maps, the subcategories \({\mathcal {A}}_{{\tiny C}}\) of compact spaces and \({\mathcal {A}}_{{\tiny CM}}\) of compact metric spaces, the category \({\mathcal {A}}_{{\tiny LC}}\) of locally compact spaces and proper continuous maps, and the subcategory \({\mathcal {A}}_{{\tiny EC}}\subset {\mathcal {A}}_{{\tiny LC}}\) of locally compact, separable metric spaces with finite (covering) dimension. Every n-dimensional separable metric space is homeomorphic to a subset of \({\mathbb {R}}^{2n+1}\) [10, Theorem V 3], and using local compactness one can lift this to a homeomorphism into \({\mathbb {R}}^{2n+2}\) with a closed image [5, IV.8.2 and 8.3]. Thus objects in \({\mathcal {A}}_{{\tiny EC}}\) can be regarded as closed subsets X of euclidean space (“euclidean closed”).

For finite-dimensional Hausdorff manifolds, the properties of being \(\sigma \)-compact and second countable are equivalent, and any manifold with these properties is separable, paracompact, metrizable, and in the category \({\mathcal {A}}_{{\tiny EC}}\).

A.2: Homology

Steenrod homology \({}^s{\mathrm {H}}_*\) and Čech homology \({\check{{\mathrm {H}}}}_*\) are introduced in Sect. 1. For the intersection theory done in Sects. 4 and 5, it is useful to also use Borel–Moore homology \(H^{{\tiny BM}}_*\). Throughout, we restrict attention to constant coefficients in \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\).

  • \(\bullet \) Steenrod (also called Steenrod-Sitnikov) homology is defined on \({\mathcal {A}}_{{\tiny LC}}\) using chains that are dual to compactly supported, finite-value Alexander-Spanier cochains [15, § 4]. It can also be defined on locally compact metrizable spaces using “canonical coverings” [22, 23]; see also Milnor’s construction on compact pairs [17]. It has an extension to paracompact spaces in \({\mathcal {A}}\) called strong homology [14, Chapter 19].

  • \(\bullet \) Čech homology is defined for pairs in \({\mathcal {A}}\) using nerves of covers. It is only a partially exact homology theory, but it has the Continuity Property (1.9) [7, pages 260-261] and is exact for finitely triangulable spaces [7, § IX.9].

  • \(\bullet \) Borel–Moore homology \(H^{{\tiny BM}}_*\) is defined on \({\mathcal {A}}_{{\tiny LC}}\) using sheaves [2, 3, 12] or on \({\mathcal {A}}_{{\tiny EC}}\) using singular cohomology [8, Chapter 19], [9, Appendix B]. It has all of the properties listed in Sect. 1 for Steenrod homology. In particular, for each open set \(U\subseteq X\) there is a natural restriction map

    $$\begin{aligned} \rho _U: H^{{\tiny BM}}_*(X)\rightarrow H^{{\tiny BM}}_*(U) \end{aligned}$$

    corresponding to (1.1) and an exact sequence corresponding to (1.2) (cf. [12, IX.2.1]).

Steenrod and Borel–Moore satisfy the Eilenberg–Steenrod axioms, so are naturally isomorphic to singular homology on the category of triangulable spaces. Both are single space theories in the sense of [7, § X.7] (see page vii in [15] and Corollary V.5.10 and the cautionary note V.5.12 [2]). In particular, the theorems in Section X.7 of [7] show that

$$\begin{aligned} \begin{aligned} {}^s{\mathrm {H}}_*(X, A)= {}^s{\mathrm {H}}_*(X\setminus A), H^{{\tiny BM}}_*(X, A)= H^{{\tiny BM}}_*(X{\setminus }A) \end{aligned}\end{aligned}$$
(A.1)

for any closed pair (XA). Steenrod and Borel–Moore homologies satisfy Minor’s modified continuity property: if \(\{X_\alpha \}\) is an inverse system in \({\mathcal {A}}_{{\tiny CM}}\) with \(X = \varprojlim X_\alpha \), then there is a natural exact sequence

(A.2)

and a corresponding sequence with \({}^s{\mathrm {H}}_*\) replaced by \(H^{{\tiny BM}}_*\) [17, Thm. 4], [16, p.81], [2, V.5.15]. There are many axiomatic characterizations of Steenrod homology (cf. [4, 11]. In particular, Theorem 1 of [11] asserts:

  • Up to natural isomorphism, \({}^s{\mathrm {H}}_*(X;G)\) is the unique homology theory on \({\mathcal {A}}_{{\tiny C}}\) that satisfies the Eilenberg–Steenrod axioms, is functorial in both X and G, and has the continuity property (1.9) for infinitely divisible abelian groups G.

A.3: Natural transformations

For any closed subset X of a paracompact n-dimensional oriented manifold N and \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\), there is a natural isomorphism (a version of Steenrod duality)

$$\begin{aligned} H^k(N, N{\setminus }X;R) \overset{\cong }{\longrightarrow } {}^s{\mathrm {H}}_{n-k}(X;R) \end{aligned}$$
(A.3)

(see [15, Theorem 11.15], using the isomorphism from the Alexander-Spanier to the singular cohomology of \((N, N{\setminus }X)\) in the proof of [24, Cor. 6.9.6]). There is a similar isomorphism (Poincaré-Alexander duality)

$$\begin{aligned} \begin{aligned} H^k(N, N{\setminus }X;R) \overset{\cong }{\longrightarrow } H^{{\tiny BM}}_{n-k}(X;R) \end{aligned}\end{aligned}$$
(A.4)

for Borel–Moore homology [25, Thm. 10.4] or [12, IX.4.7]. Taking \(N={\mathbb {R}}^n\), (A.3) and (A.4) together show that there is a natural isomorphism \(\beta \) from Steenrod to Borel–Moore homology on the category \({\mathcal {A}}_{{\tiny EC}}\). In fact, on \({\mathcal {A}}_{{\tiny EC}}\) with \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\), there is a commutative diagram of natural transformations

(A.5)

where \(\gamma \) is as in [22, Theorem 4], \(\psi =\beta \circ \varphi \) and \(\alpha = \gamma \circ \beta ^{-1}\) (\(\varphi \) factors through compactly supported homology: see page 291 and the first arrow on page 308 in [15].) Furthermore,

  1. (i)

    The isomorphism \(\beta \) carries the restriction map (1.1) to the corresponding restriction map in Borel–Moore homology, and carries the exact sequence (1.2) to the corresponding Borel–Moore sequence by matching both to the exact sequence

    $$\begin{aligned} \begin{aligned} \cdots \longrightarrow H^{k-1}(N{\setminus }X) \overset{\delta }{\longrightarrow } H^k(N, N{\setminus }X) \overset{j^*}{\longrightarrow } H^k(N) \overset{\iota ^*}{\longrightarrow } H^k(N{\setminus }X) \longrightarrow \cdots \end{aligned}\nonumber \\ \end{aligned}$$
    (A.6)

    in Alexander-Spanier (or equivalently singular) cohomology; cf. Theorems 11.15 and 8.1 in [15], and diagram IX.3.5 in [12]. (Steenrod and Borel–Moore homologies are also isomorphic on the category \({\mathcal {A}}_{{\tiny CM}}\) [3, § 5].)

  2. (ii)

    \(\gamma \) is surjective, and is an isomorphism if \(R={\mathbb {Q}}\), or if \(R={\mathbb {Z}}\) and either (i) \(k=\mathrm {dim}\, X\) or (ii) X is a compact and locally contractible [22, Thm. 4].

  3. (iii)

    \(\varphi \) and \(\psi \) are isomorphisms if X is compact and locally contractible [15, § 9.6], [2, § V.12].

  4. (iv)

    \(\alpha \) is an isomorphism for locally contractible spaces X in \({\mathcal {A}}_{{\tiny CM}}\) [2, V.5.19].

In particular, the natural transformations in (A.5) give isomorphisms

$$\begin{aligned} \begin{aligned} H^{{\tiny BM}}_*(X; {\mathbb {Q}})\ \cong \ {}^s{\mathrm {H}}_*(X; {\mathbb {Q}})\ \cong \ {\check{{\mathrm {H}}}}_*(X; {\mathbb {Q}}) \end{aligned}\end{aligned}$$
(A.7)

for compact metric spaces X, and

(A.8)

for compact manifolds N and for finite polyhedra.

A.4: Cohomology

Singular, Čech and Alexander-Spanier cohomology are all defined on the category of pairs in \({\mathcal A}\); we will restrict attention to paracompact pairs and constant coefficients \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\). The last two are naturally isomorphic [24, 6.8.8 and Exercise 6.D.3], so we use the notation \({\check{{\mathrm {H}}}}^*\) for both. The corresponding theories based on compactly supported cochains are also isomorphic, and will be denoted by \({\check{{\mathrm {H}}}}^*_c\). In Part II of [15], Massey also uses locally finite-valued cochains for closed pairs, but these give the same theory as Alexander-Spanier cohomology by [15, Thm. 8.1]. Thus for paracompact pairs (XA) and \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\), it suffices to consider three theories: \({\check{{\mathrm {H}}}}^*\), \({\check{{\mathrm {H}}}}^*_c\) and singular cohomology \(H^*\). These are related by natural transformations

(A.9)

which preserve cup products [2, Thm. III.2.1 and Cor. III.4.12] and [24, § 6.5, § 6.9]. Furthermore,

  1. (i)

    \(\mu \) is an isomorphism if X is compact [24, 6.6.9].

  2. (ii)

    \(\nu \) is an isomorphism if X and A are locally contractible (e.g. manifolds) and A is either closed [2, Theorem III.2.1] or open (by the proof of 6.9.6 in [24]).

For paracompact, locally compact spaces X, Steenrod homology pairs with \({\check{{\mathrm {H}}}}^*_c\) in the sense that there is a Kronecker pairing

$$\begin{aligned} \begin{aligned} {}^s{\mathrm {H}}_*(X)\otimes {\check{{\mathrm {H}}}}_c^*(X) \rightarrow R \end{aligned}\end{aligned}$$
(A.10)

(cf. [15, § 4.8]), and an exact sequence

$$\begin{aligned} \begin{aligned} 0 \rightarrow \text{ Ext }\left( {\check{{\mathrm {H}}}}^{k+1}_c(X), R\right) \longrightarrow {}^s{\mathrm {H}}_k(X;R)\overset{\tau }{\longrightarrow } \text{ Hom }\left( {\check{{\mathrm {H}}}}^{k}_c(X), R\right) \rightarrow 0 \end{aligned}\end{aligned}$$
(A.11)

that is natural in both X and R [15, Cor. 4.18 and p.371]. The same sequence holds with \({}^s{\mathrm {H}}_*\) replaced by \(H^{{\tiny BM}}_*\) [3, Theorem 3.3]. In particular, there is a natural transformation

$$\begin{aligned} \begin{aligned} \tau : {}^s{\mathrm {H}}_k(X;R) \longrightarrow {\check{{\mathrm {H}}}}^{k}_c(X;R)^\vee \end{aligned}\end{aligned}$$
(A.12)

where \({ }^\vee \) denotes dual, i.e. \(\text{ Hom }(\cdot , R)\). By (A.11), \(\tau \) is an isomorphism for \(R={\mathbb {Q}}\). For compact metric spaces, it factors through Čech homology, as follows.

Lemma A.1

On the category \({\mathcal {A}}_{{\tiny CM}}\) with \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\), \( {\check{{\mathrm {H}}}}_c^*(X)= {\check{{\mathrm {H}}}}^*(X)\) and there is a commutative diagram of natural transformations

(A.13)

with \(\gamma \) as in (A.5) and \(\sigma \) as defined below. For \(R={\mathbb {Q}}\), all three maps are isomorphisms.

Proof

First note that the compact metric space X can be written as the inverse limit of a system \(\{X_{\ell }\}\) in the category of finite polyhedra (cf. [16, p.82] and [24, 6.6.7]). The homology maps induced by the inclusions \(X\rightarrow X_\ell \) define maps \(\iota _*, j_*\) and \({\check{\iota }}_*\) into inverse systems as in the diagram below. But \(\gamma \) is an isomorphism for finite polyhedra, thus we get a diagram (A.13) when X is replaced by \(X_\ell \), and \(\sigma \) is defined to be \(\tau \circ \gamma ^{-1}\). By the naturality of \(\gamma \) and \(\tau \), there are induced maps \(\gamma _{\bullet }\), \(\tau _{\bullet }\) and \(\sigma _{\bullet }\) between the inverse systems, and the front left and back squares in the diagram commute. Furthermore, \(\gamma _\bullet \) is an isomorphism, and the bottom triangle commutes.

The map \({\check{\iota }}_*\) is an isomorphism by Čech continuity, and \(j_*\) is an isomorphism by the continuity of Čech cohomology and the fact that \(\varprojlim \mathrm {Hom}(\cdot , R)=\mathrm {Hom}(\varinjlim \cdot , R)\). The first statement of the lemma follows by defining \(\sigma \) to be the composition \({\check{\iota }}_*^{-1} \sigma _{\bullet }j_*\).

For \(R={\mathbb {Q}}\), the \(\lim ^{\! 1}\) term in (A.2) and the \(\text{ Ext }\) term in (A.11) vanish. Hence \(\iota _*\) and \(\tau \), and therefore \(\gamma \) and \(\sigma \), are isomorphisms (cf. [20, Remark 5.0.2]).

\(\square \)

Thus on \({\mathcal {A}}_{{\tiny CM}}\), \(\sigma \) defines a Kronecker pairing \({\check{{\mathrm {H}}}}_*(X)\otimes {\check{{\mathrm {H}}}}^*(X) \rightarrow R\) in Čech theory, and this is related to (A.10) by

$$\begin{aligned} \begin{aligned} \langle \gamma (b),\, \alpha \rangle \,=\, \langle b,\, \alpha \rangle \qquad \forall b\in {}^s{\mathrm {H}}_*(X), \ \alpha \in {\check{{\mathrm {H}}}}^*(X). \end{aligned}\end{aligned}$$
(A.14)

A.5: Borel–Moore via embeddings

Fulton and MacPherson developed a simplified version of Borel–Moore homology on \({\mathcal {A}}_{{\tiny EC}}\) for use in algebraic geometry; see Chapter 19 of [8] or Appendix B of [9]. Fixing coefficients in a field or in \({\mathbb {Z}}\), one defines the embedded Borel–Moore homology of a closed subset X of \({\mathbb {R}}^m\) by formula (A.4):

$$\begin{aligned} \begin{aligned} {\overline{H}}_k(X)\ \overset{\mathrm{def}}{=}\ \ H^{m-k}({\mathbb {R}}^m,{\mathbb {R}}^m\setminus X), \end{aligned}\end{aligned}$$
(A.15)

(cf. [8, §19.1, eq (1)]). Similarly, for any open subset U of X, one sets

$$\begin{aligned} {\overline{H}}_k(U)\ =\ {\overline{H}}_k(X, A) \ =\ H^{m-k}({\mathbb {R}}^m\setminus A,{\mathbb {R}}^m{\setminus }X). \end{aligned}$$

where \(A=X{\setminus }U\) (cf. [9, (30) in Appendix B]). Using facts about singular cohomology, one can verify that the groups \({\overline{H}}_*\) have all of the properties of Borel–Moore homology (cf. [8], Section 19.1 and especially Example 19.1.1). For a direct correspondence, note that for an object X in \({\mathcal {A}}_{{\tiny EC}}\), the choice of a closed embedding \(\iota :X\rightarrow {\mathbb {R}}^m\) with image \(X^e\) induces natural isomorphisms

(A.16)

where the second is the composition of (A.4) and (A.15).

A.6: Cap products

For any closed subset X of a locally compact Z, we have a localized (sheaf theoretic supported) cap product

$$\begin{aligned} \begin{aligned} H^{{\tiny BM}}_n(Z) \otimes H^k(Z, Z{\setminus }X) \xrightarrow {\cap }H^{{\tiny BM}}_{n-k}(X), \end{aligned}\end{aligned}$$
(A.17)

on Borel–Moore homology with coefficients in any commutative Noetherian ring, cf. [12, IX.3.1, § IX.5 and § II.9], [8, § 19.1]. This has several naturality properties (cf. [12, § IX.3]), including:

  • For a proper map \(f:(Z', X')\rightarrow (Z, X)\) of closed, locally compact pairs,

    $$\begin{aligned} \begin{aligned} f_*(a' \cap f^*\xi )= f_*a' \cap \xi . \end{aligned}\end{aligned}$$
    (A.18)
  • For closed subsets \(X\mathop {\hookrightarrow } \limits ^i Z\) and \(Y\subseteq Z\),

    $$\begin{aligned} \begin{aligned} (a \cap \xi )\cap i^*\eta = a\cap (\xi \cup \eta ) \quad \text{ in } H^{{\tiny BM}}_*(X\cap Y). \end{aligned}\end{aligned}$$
    (A.19)
  • If \(X\subseteq Z\) is closed and \(U\mathop {\hookrightarrow } \limits ^j Z\) is open, then the restriction to U satisfies

    $$\begin{aligned} \begin{aligned} \rho _{U\cap X}(a \cap \xi )= \rho _U(a)\cap j^*\xi \quad \text{ in } H^{{\tiny BM}}_*(U\cap X). \end{aligned}\end{aligned}$$
    (A.20)

These formulas hold for all \(a \in H^{{\tiny BM}}_*(Z)\), \(a' \in H^{{\tiny BM}}_*(Z')\), \(\xi \in H^*(Z, Z{\setminus }X)\) and \(\eta \in H^*(Z, Z{\setminus }Y)\).

A.7: Fundamental classes

Let N be an oriented topological n-manifold (a Hausdorff space locally homeomorphic to \({\mathbb {R}}^n\)). Its orientation determines fundamental classes

$$\begin{aligned} \begin{aligned}{}[N]\in {}^s{\mathrm {H}}_n(N;{\mathbb {Z}}) [N]\in H^{{\tiny BM}}_n(N;{\mathbb {Z}}) \end{aligned}\end{aligned}$$
(A.21)

in Steenrod [15, § 4.9] and Borel–Moore homology [12, IX.4.6]. The restriction of [N] to an open set \(U\subseteq N\) is the fundamental class of U:

$$\begin{aligned} \rho _U[N]=[U]. \end{aligned}$$

On each component \(N_\alpha \) of N, the orientation determines isomorphisms \({}^s{\mathrm {H}}_n(N_\alpha ;{\mathbb {Z}}) = {\mathbb {Z}}= H^{{\tiny BM}}_n(N_\alpha ;{\mathbb {Z}})\) under which \([N_\alpha ]\) corresponds to 1. The naturality of the transformation \(\beta \) in (A.5) with respect to restriction maps then implies that the two fundamental classes (A.21) correspond under \(\beta \).

For any closed subset Y of N, the cap product (A.17) with the fundamental class is an isomorphism

$$\begin{aligned} D: H^{k} (N, N {\setminus }Y) \underset{\cong }{\xrightarrow { [N]\cap \;\;}}H^{{\tiny BM}}_{n-k}(Y) \end{aligned}$$
(A.22)

which is precisely (A.4) cf. [12, IX.4.7]. There is corresponding isomorphism with values in Steenrod homology. In particular, if N is compact, taking \(Y=N\) gives the Poincaré duality isomorphisms

$$\begin{aligned} \begin{aligned} D: H^k(N) \overset{\cong }{\longrightarrow } H^{{\tiny BM}}_{n-k}(N) D: H^k(N) \overset{\cong }{\longrightarrow } {}^s{\mathrm {H}}_{n-k}(N). \end{aligned}\end{aligned}$$
(A.23)

A.8: Submanifolds

Suppose that N is a paracompact oriented \(C^1\)n-manifold and

$$\begin{aligned} V\hookrightarrow N \end{aligned}$$

is a properly embedded oriented submanifold of codimension k. By identifying a neighborhood U of V in N with the total space of the normal bundle to V and using excision, the Thom class of the normal bundle defines a singular cohomology class

$$\begin{aligned} \begin{aligned} u=u_{V, N}\in H^{k}(U, U{\setminus }V) = H^{k}(N, N{\setminus }V) \end{aligned}\end{aligned}$$
(A.24)

Following Fulton [8, § 19.2], we call u the orientation class of V in N. Then [V] and [N] are related

$$\begin{aligned} \begin{aligned}{}[V]\ =\ [N] \cap u_{V, N} \quad \text{ in } H^{{\tiny BM}}_{n-k}(V) \end{aligned}\end{aligned}$$
(A.25)

as in [12, IX.4.9]. In particular, [V] corresponds to \(u_{V,N}\) under the duality (A.22). The naturality of Thom class gives a naturality property of u: if a \(C^1\) map \(f:M\rightarrow N\) of oriented manifolds is transverse to V, then \(W=f^{-1}(V)\) is an oriented submanifold of M with orientation class

$$\begin{aligned} \begin{aligned} u_{W,M} = f^*u_{V,N}. \end{aligned}\end{aligned}$$
(A.26)

A.9: Intersection pairing

For closed subsets X and Y of a manifold N as in §A.8, there is a cup product in singular cohomology

$$\begin{aligned} H^{n-k}(N, N{\setminus }X) \otimes H^{n-\ell }(N, N{\setminus }Y) \overset{\cup }{\longrightarrow } H^{2n-k-\ell }(N, N\setminus (X\cap Y)) \end{aligned}$$

(cf. [24, § 5.6], noting that \(\{N-X, N-Y\}\) is an excisive pair by [24, 4.6.4]). The duality (A.22) translates the cup product into the cap product

$$\begin{aligned} \begin{aligned} H^{{\tiny BM}}_k(X) \otimes H^{n-\ell }(N, N{\setminus }Y) \overset{\cap }{\longrightarrow } H^{{\tiny BM}}_{k+\ell -n} (X\cap Y) \end{aligned}\end{aligned}$$
(A.27)

by the formula:

$$\begin{aligned} b \cap \alpha = D\big ( D^{-1}b \cup \alpha \big ). \end{aligned}$$

Note that the righthand side is \(b\cap i^*\alpha \) for the cap product (A.17), where i is the inclusion of X into N (cf. (A.19) with \(a=[N]\), \(b=D\xi =[N]\cap \xi \), and \(\eta =\alpha \)). Thus this cap product depends only on the restriction of \(\alpha \) to X.

Applying (A.22) again yields a natural intersection pairing in Borel–Moore homology

$$\begin{aligned} \begin{aligned} H^{{\tiny BM}}_k(X) \otimes H^{{\tiny BM}}_\ell (Y) \overset{\bullet }{\longrightarrow } H^{{\tiny BM}}_{k+\ell -n} (X\cap Y) \end{aligned}\end{aligned}$$
(A.28)

given by

$$\begin{aligned} \begin{aligned} a \bullet b \, =\, D\big (D^{-1}b\cup D^{-1}a\big ) = b\cap D^{-1}a. \end{aligned}\end{aligned}$$
(A.29)

(The order reversal is needed to obtain the correct signs, cf. [2, § V.11]).

The intersection pairing (A.28) is natural with respect to the restriction map \(\rho _U\) to any open subset U of N: the naturality of the cup product and (A.1) translates into the identity

$$\begin{aligned} \begin{aligned} \rho _U (a\bullet b)= \rho _U( a )\bullet \rho _U (b). \end{aligned}\end{aligned}$$
(A.30)

In particular, if U is any neighborhood of \(X\cap Y\), then the restriction to U induces the identity on \(H^{BM}_*(X\cap Y)\) and hence

$$\begin{aligned} \begin{aligned} a\bullet b= \rho _U( a )\bullet \rho _U (b). \end{aligned}\end{aligned}$$
(A.31)

Thus the intersection localizes on any open neighborhood of \(X\cap Y\).

Example A.2

Let X and Y be properly embedded oriented submanifolds of an oriented \(C^1\) manifold N. If X and Y intersect transversally, then \(X\cap Y\) is an oriented manifold, and

$$\begin{aligned}{}[X]\bullet [Y]\ =\ [X\cap Y] \qquad \text{ in } H^{{\tiny BM}}_*(X \cap Y). \end{aligned}$$

The proof exactly as in the proof of Theorem VI-11.9 of [1], using formulas (A.19), (A.25), (A.29) and the naturality of Thom classes, and interpreting all terms as elements of Borel–Moore homology.

More generally, consider proper maps \(f:M\rightarrow N\) and \(g:P \rightarrow N\) between oriented manifolds. Set

$$\begin{aligned} \begin{aligned} Z\ =\ \big \{ (x,y)\in N\times N\,\big |\, f(x)=g(y)\big \} \end{aligned}\end{aligned}$$
(A.32)

and define \(h:Z\rightarrow N\) by \(h(x, y)=f(x)=g(y)\).

Lemma A.3

Suppose that maps f and g as above are transverse and have complementary dimensions in N. Then Z is a 0-dimensional manifold, and has an induced orientation such that

$$\begin{aligned} \begin{aligned} f_*[M]\bullet g_*[P] \ =\ h_*[Z] \end{aligned}\end{aligned}$$
(A.33)

in \(H^{{\tiny BM}}_0(X\cap Y;{\mathbb {Z}})\), where X and Y are the images of f and g.

Proof

The assumptions that f and g are transverse with complementary dimension imply that Z is a discrete set of points and that f, g, and \(f\times g\) are immersions at each point \(z=(x,y)\in Z\). Hence we can find disjoint open neighborhoods \(U_p\) of the points \(p\in X\cap Y\) so that, for \(U=\mathop {\bigsqcup } \limits U_p\), we have

$$\begin{aligned} f^{-1}(U)= \bigsqcup _{x\in f^{-1}(X\cap Y)} V_x, \quad \text{ and } \quad g^{-1}(U)= \bigsqcup _{y\in g^{-1}(X\cap Y)} W_y \end{aligned}$$

where \(\{V_x\}\) (resp. \(\{W_y\}\)) are disjoint neighborhoods of x in M (resp. y in P), and where f and g restrict to a proper embeddings \(f:V_x \rightarrow U\) and \(g:W_y \rightarrow U\).

As in (A.31), the lefthand side of (A.33) localizes, and hence is a locally finite sum of local intersections

$$\begin{aligned} f_*[M]\bullet g_*[P] \ =\ \sum _{(x, y)\in Z} f_*[V_x] \bullet g_*[W_y]\ =\ \sum _{(x, y)\in Z} \left[ f(V_x) \cap g(W_y)\right] , \end{aligned}$$

where the second equality is obtained by applying Example A.2. But \(f(V_x) \cap g(W_y)\) is exactly h(z), and is oriented as the local intersection of f and g. This identification induces an orientation \([z]\in H^{{\tiny BM}}_0(z;{\mathbb {Z}})\) for each \(z\in Z\), and hence an orientation on Z. The lemma follows. \(\square \)

A.10: Intersection numbers

With constant coefficients \(R={\mathbb {Z}}\) or \({\mathbb {Q}}\), the intersection of classes \(a\in H^{{\tiny BM}}_k(X;R)\) and \(b\in H^{{\tiny BM}}_\ell (Y;R)\) of complementary dimension (i.e. \(k+\ell =n=\mathrm {dim}\, N\)) is a 0-dimensional class

$$\begin{aligned} a\bullet b\in H^{{\tiny BM}}_0 (X\cap Y;R). \end{aligned}$$

If \(X\cap Y\) is compact, there is an augmentation map \( \varepsilon : H^{{\tiny BM}}_*(X\cap Y;R)\rightarrow R \) (induced by the map from \(X\cap Y\) to a point), and the intersection number is defined by

$$\begin{aligned} \begin{aligned} a\cdot b = \varepsilon (a\bullet b) \in R. \end{aligned}\end{aligned}$$
(A.34)

This can be written in many other ways:

$$\begin{aligned} \begin{aligned} a\cdot b \ =\ \varepsilon (b \cap \alpha ) \ =\ \langle b\cap \alpha ,1 \rangle \ =\ \langle b, \alpha \rangle \ =\ \int _{b}\alpha , \end{aligned}\end{aligned}$$
(A.35)

for elements \(\alpha =D^{-1}a\) of \(H^*(N, N{\setminus }X)\). In particular, when \(X=N\) is a compact oriented manifold, then \(\alpha \in {\check{{\mathrm {H}}}}_c^*(N)\) is Poincaré dual to a under (A.23) (cf. [2, V.11]).

Example A.4

Let f and g be proper maps as in Lemma A.3, and assume that the intersection of their images is compact. Then (A.32) is a compact 0-dimensional oriented manifold, consisting of finitely many points x with sign \(\varepsilon (x)=\pm 1\). In this case, using (A.33), the intersection number is

$$\begin{aligned} f_*[M]\cdot g_*[P] \ =\ \varepsilon (f_*[M]\bullet g_*[P])\ =\ \varepsilon ( h_*[Z]) \ =\ \sum _{x\in Z} \varepsilon (x) \in {\mathbb {Z}}. \end{aligned}$$

Appendix B: Dimension theory

One can define the dimension of a topological space X in several ways:

  1. (1)

    The Lebesgue covering dimension \(\mathrm {dim}\, X\) is the smallest number d so that every open cover has a refinement such that every \(x\in X\) lies in at most \(d+1\) sets of the refinement.

  2. (2)

    The (large) cohomological dimension \({\mathrm {dim}}_2 X\) is the largest k such that \({\check{{\mathrm {H}}}}^k(X, A;{\mathbb {Z}})\) is non-zero for some closed set A in X.

In addition, if X is a metric space, one has:

  1. (3)

    The Hausdorff dimension \({\mathrm {dim}}_H X\) of X is the infimum of \(\delta \ge 0\) with the following property: For any \(\varepsilon >0\), X can be covered by countably many sets \(\{A_n\}\) with \({\mathrm{diam}}(A_n)< \varepsilon \) and with \(\sum _i {\mathrm{diam}}(A_n)^\delta <\varepsilon \).

Standard results of dimension theory show that

$$\begin{aligned} {\mathrm {dim}}_2 X\ =\ \mathrm {dim}\, X \qquad \text{ and }\qquad \mathrm {dim}\, X\ \le \ {\mathrm {dim}}_H X, \end{aligned}$$
(B.1)

where the first equality holds for all non-empty paracompact Hausdorff spaces [19, 36-15 and 37-7], and the second holds for all separable metric spaces [10, p. 107]. These are related to condition (1.4) as follows.

Lemma B.1

If X is a compact Hausdorff space with \(\mathrm {dim}\, X=d\), then

$$\begin{aligned} \begin{aligned} {}^s{\mathrm {H}}_k(X;{\mathbb {Z}})\ =\ H^{{\tiny BM}}_k(X;{\mathbb {Z}})\ =\ 0 \ \quad \text{ for } \text{ all } k> d. \end{aligned}\end{aligned}$$
(B.2)

Proof

For compact X we have \({\check{{\mathrm {H}}}}^*(X)={\check{{\mathrm {H}}}}_c^*(X)\). Using (B.1) and taking A to be the empty set in the definition of \({\mathrm {dim}}_2 X\), one sees that the condition \(\mathrm {dim}\, X=d\) implies that \({\check{{\mathrm {H}}}}^k(X)=0\) for all \(k>d\). Then (B.2) follows from (A.11) and the corresponding sequence for Borel–Moore homology. \(\square \)

Lemma B.2

Suppose that X is a metric space and M is a separable metrizable topological d-manifold.

  1. (a)

    For any subspace \(S\subseteq X\), \( \mathrm {dim}\, S \le \mathrm {dim}\, X\) and \( {\mathrm {dim}}_H S \le {\mathrm {dim}}_H X\).

  2. (b)

    If X is a countable union of closed subsets \(X_i\), then \(\mathrm {dim}\, X \le \sup \, \mathrm {dim}\, X_i\) and \({\mathrm {dim}}_H X \le \sup \, {\mathrm {dim}}_H X_i\).

  3. (c)

    \(\mathrm {dim}\, M = {\mathrm {dim}}_H M=d\).

  4. (d)

    If \({\overline{M}}=M\cup S\) is a thin compactification of M with \(\mathrm {dim}\, S\le d-2\), then \(\mathrm {dim}\, {\overline{M}}=d\).

Proof

For Hausdorff dimension, (a), (b) and (c) follow easily from the definition. For covering dimension, (a) and (b) are Theorems 4.1.7 and 4.1.9 of [6], respectively, and (c) is Corollary 1 of Section IV.4 of [10]. Finally, note that M can be written as a countable union of closed subsets \(B_i\) (closed balls of integer radius in some metric) each with \(\mathrm {dim}\, B_i=d\). Applying (b) to \({\overline{M}}= S\cup \bigcup B_i\) with \(\mathrm {dim}\, S\le d-2\) shows that \(\mathrm {dim}\, {\overline{M}}=d\). \(\square \)

In general, a continuous map can increase covering dimension, as occurs for Peano’s space-filling curve. One can avoid such pathologies by working with Hausdorff dimension and Lipschitz maps. Recall that a map \(f:X\rightarrow Y\) between metric spaces is called Lipschitz if there is a constant \(C>0\) such that

$$\begin{aligned} \text{ dist }(f(x), f(y))\ \le \ C\, \text{ dist }(x, y) \end{aligned}$$

for all \(x, y\in X\), and is locally Lipschitz if every point in X has a neighborhood in which f is Lipschitz. Note that if f is locally Lipschitz then its restriction to a compact set \(K\subset X\) is Lipschitz. Also note that the definition of Hausdorff dimension implies that if f is Lipschitz then

$$\begin{aligned} \begin{aligned} {\mathrm {dim}}_H f(X)\ \le \ {\mathrm {dim}}_H X. \end{aligned}\end{aligned}$$
(B.3)

Lemma B.3

Suppose that a subset S of a metric space Y is contained in the union of the images of a countable collection of maps \(\varphi _n:U_n\rightarrow Y\), where each \(U_n\) is a \(\sigma \)-compact topological manifold of dimension \(\le d\), and either

  1. (i)

    \(\varphi _n\) continuous and locally injective, or

  2. (ii)

    \(\varphi _n\) is locally Lipschitz, or

  3. (iii)

    \(\varphi _n\) is a \(C^1\) map between \(C^1\) manifolds.

Then \(\mathrm {dim}\, S \le d\) and, if S is compact, \({}^s{\mathrm {H}}_k(S) = 0\) for all \(k> d\).

Proof

Each \(U_n\) is \(\sigma \)-compact, so can be covered by a countable collection of open sets \(\{B_{mn}\}\) with compact closures \({\overline{B}}_{mn}\). Then \(\varphi _n\) restricts to maps \(\varphi _{mn}:{\overline{B}}_{mn}\rightarrow Y\).

In case (i), \(\varphi _n\) is locally injective, so we can assume, after refining the cover \(\{B_{mn}\}\), that each \(\varphi _{mn}\) is injective. Then \(\varphi _{mn}\) is a continuous injection from a compact set to a Hausdorff space, therefore a homeomorphism onto its image. Since covering dimension is a homeomorphism invariant, parts (a) and (c) of Lemma B.2 show that the image satisfies

$$\begin{aligned} \mathrm {dim}\, \, \varphi _{mn}({\overline{B}}_{mn})\ =\ \mathrm {dim}\, \, {\overline{B}}_{mn}\ \le \ d. \end{aligned}$$

In case (ii), the assumptions also imply that \(U_n\) is metrizable. After fixing a metric, inequalities (B.1), (B.3) and Lemma B.2(a,c) imply that

$$\begin{aligned} \mathrm {dim}\, \, \varphi _{mn}({\overline{B}}_{mn})\ \le \ {\mathrm {dim}}_H\, \varphi _{mn}({\overline{B}}_{mn})\ \le \ {\mathrm {dim}}_H\, {\overline{B}}_{mn}\ \le \ d. \end{aligned}$$

This inequality also holds in case (iii) because any \(C^1\) map is locally Lipschitz.

In either case, we can apply parts (a) and (b) of Lemma B.2 to conclude that

$$\begin{aligned} \mathrm {dim}\, S \ \le \ \mathrm {dim}\, \ \bigcup _{m,n} \varphi _{mn}({\overline{B}}_{mn}) \ \le \ \sup _{m,n}\mathrm {dim}\, \, \varphi _{mn}({\overline{B}}_{mn})\ \le \ d. \end{aligned}$$

The lemma then follows by applying Lemma B.1. \(\square \)

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Ionel, EN., Parker, T.H. Relating VFCs on thin compactifications. Math. Ann. 375, 845–893 (2019). https://doi.org/10.1007/s00208-019-01861-0

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