Virtual cycles of gauged Witten equation

Gang Tian; Guangbo Xu

doi:10.1515/crelle-2020-0022

Published by De Gruyter July 11, 2020

Virtual cycles of gauged Witten equation

Gang Tian and Guangbo Xu

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2020-0022

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Abstract

In this paper, we construct virtual cycles on moduli spaces of solutions to the perturbed gauged Witten equation over a fixed smooth r-spin curve, under the framework of [G. Tian and G. Xu, Analysis of gauged Witten equation, J. reine angew. Math. 740 (2018), 187–274]. Together with the wall-crossing formula proved in the companion paper [G. Tian and G. Xu, A wall-crossing formula for the correlation function of gauged linear σ-model, preprint], this paper completes the construction of the correlation function for the gauged linear σ-model announced in [G. Tian and G. Xu, Correlation functions in gauged linear σ-model, Sci. China Math. 59 (2016), 823–838] as well as the proof of its invariance.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1607091

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: NSFC-11890661

Funding statement: Gang Tian is supported by DMS-1607091 and NSFC-11890661. Guangbo Xu is supported by AMS-Simons Travel Grant.

A Topological virtual orbifolds and virtual cycles

We recall the framework of constructing virtual fundamental cycles associated to moduli problems. Such constructions, usually called “virtual technique”, has a long history since it first appeared in algebraic Gromov–Witten theory by [24]. The current method is based on the topological approach of [25].

A.1 Topological manifolds and transversality

In this subsection we review the classical theory about topological manifolds and (microbundle) transversality.

Definition A.1 (Topological manifolds and embeddings).

A topological manifold is a second countable Hausdorff space M which is locally homeomorphic to an open subset of ℝ n .
A subset S ⊂ M is a submanifold if S equipped with the subspace topology is a topological manifold.
A map f : N → M between two topological manifold is called a topological embedding if f is a homeomorphism onto its image.
A topological embedding f : N → M is called locally flat if for any p ∈ f ⁢ ( N ) , there is a local coordinate φ p : U p → ℝ m , where U p ⊂ M is an open neighborhood of p such that φ p ⁢ ( f ⁢ ( N ) ∩ U p ) ⊂ ℝ n × { 0 } .

In this paper, without further clarification, all embeddings of topological manifolds are assumed to be locally flat. In fact, we will always assume (or prove) the existence of a normal microbundle which implies local flatness.

A.1.1 Microbundles

The discussion of topological transversality needs the concept of microbundles, which was introduced by Milnor [33].

Definition A.2 (Microbundles).

Let B be a topological space.

A microbundle over a B is a triple ( E , i , p ) , where E is a topological space, i : M → E (the zero section map) and p : E → M (the projection) are continuous maps, satisfying the following conditions:
1. p ∘ i = Id M .
2. For each b ∈ B there exist an open neighborhood U ⊂ B of b and an open neighborhood V ⊂ E of i ⁢ ( b ) with i ⁢ ( U ) ⊂ V , j ⁢ ( V ) ⊂ U , such that there is a homeomorphism V ≃ U × ℝ n which makes the following diagram commutes:
Two microbundles ξ = ( E , i , p ) and ξ ′ = ( E ′ , i ′ , p ′ ) over B are equivalent if there are open neighborhoods of the zero sections W ⊂ E , W ′ ⊂ E ′ and a homeomorphism ρ : W → W ′ which is compatible with the structures of the two microbundles.

Vector bundles and disk bundles are particular examples of microbundles. More generally, an ℝ n - bundle over a topological manifold M is a fibre bundle over M whose fibres are ℝ n and whose structure group is the group of homeomorphisms of ℝ n which fix the origin. Notice that an ℝ n -bundle has a continuous zero section, thus an ℝ n -bundle is naturally a microbundle. A very useful fact, which was proved by Kister [23] and Mazur [27], says that microbundles are essentially ℝ n -bundles.

Theorem A.3 (Kister–Mazur theorem).

Let B be a topological manifold (or a weaker space such as a locally finite simplicial complex) and let ξ = ( E , i , p ) be a microbundle over B. Then ξ is equivalent to an R n -bundle, and the isomorphism class of this R n -bundle is uniquely determined by ξ.

However, ℝ n -bundles are essentially different from vector bundles. For example, vector bundles always contain disk bundles, which is not true for ℝ n -bundles.

Definition A.4 (Normal microbundles).

Let f : S → M be a topological embedding.

A normal microbundle of f is a pair ξ = ( N , ν ) , where N ⊂ M is an open neighborhood of f ⁢ ( S ) and ν : N → S is a continuous map such that together with the natural inclusion S ↪ N they form a microbundle over N. A normal microbundle is also called a tubular neighborhood.
Two normal microbundles ξ 1 = ( N 1 , ν 1 ) and ξ 2 = ( N 2 , ν 2 ) are equivalent if there is another normal microbundle ( N , ν ) with N ⊂ N 1 ∩ N 2 and

ν 1 | N = ν 2 | N = ν .

An equivalence class is called a germ of normal microbundles (or tubular neighborhoods).

For example, for a smooth submanifold S ⊂ M in a smooth manifold, there is always a normal microbundle. Its equivalence class is not unique though, as we need to choose the projection map.

A.1.2 Transversality

We first recall the notion of microbundle transversality. Let Y be a topological manifold, let X ⊂ Y be a submanifold and let ξ = ( N , ν ) be a normal microbundle of X. Let f : M → Y be a continuous map.

Definition A.5 (Microbundle transversality).

Let Y be a topological manifold, X ⊂ Y be a submanifold and ξ X be a normal microbundle of X. Let f : M → Y be a continuous map. We say that f is transverse to ξ if the following conditions are satisfied:

f - 1 ⁢ ( X ) is a submanifold of M.
There is a normal microbundle ξ ′ = ( N ′ , ν ′ ) of f - 1 ⁢ ( X ) ⊂ M such that the following diagram commute:

and the inclusion f : N ′ → N induces an equivalence of microbundles.

More generally, if C ⊂ M is any subset, then we say that f is transverse to X near C if the restriction of f to an open neighborhood of C is transverse to X.

It is easy to see that the notion of being transverse to ξ only depends on the germ of ξ.

Remark A.6.

The notion of microbundle transversality looks too restrictive at the first glance. For example, the line x = y in ℝ 2 intersects transversely with the x-axis in the smooth category, however, the line is not transverse to the x-axis with respect to the natural normal microbundle given by the projection ( x , y ) → ( x , 0 ) .

The following theorem, which is of significant importance in our virtual cycle construction, shows that one can achieve transversality by arbitrary small perturbations.

Theorem A.7 (Topological transversality theorem).

Let Y be a topological manifold and let X ⊂ Y be a proper submanifold. Let ξ be a normal microbundle of X. Let C ⊂ D ⊂ Y be closed sets. Suppose f : M → Y is a continuous map which is microbundle transverse to ξ near C. Then there exists a homotopic map g : M → Y which is transverse to ξ over D such that the homotopy between f and g is supported in a given open neighborhood of f - 1 ⁢ ( ( D ∖ C ) ∩ X ) .

Remark A.8.

The theorem was proved by Kirby and Siebenmann [22] with a restriction on the dimensions of M, X and Y. Then Quinn [38, 39, 10] completed the proof of the remaining cases. Notice that in [39], the transversality theorem is stated for an embedding i : M → Y and the perturbation can be made through an isotopy. This implies the above transversality result for maps as we can identify a map f : M → Y with its graph f ~ : M → M × Y , and an isotopic embedding of f ~ , written as g ~ ⁢ ( x ) = ( g 1 ⁢ ( x ) , g 2 ⁢ ( x ) ) , can be made transverse to the submanifold X ~ = M × X ⊂ M × Y with respect to the induced normal microbundle ξ ~ . Then it is easy to see that it is equivalent to g 2 : M → Y being transverse to ξ.

In most of the situations of this paper, the notion of transversality is about sections of vector bundles or ℝ n -bundles. Suppose f : M → ℝ n is a continuous map. The origin 0 ∈ ℝ n has a canonical normal microbundle. Therefore, one can define the notion of transversality for f as a special case of Definition A.5. Now suppose E → M is an ℝ n -bundle and

φ U : E | U → U × ℝ n

is a local trivialization. Each section s : M → E induces a map s U : U → ℝ n . Then we say that s is transverse over U if s U is transverse to the origin of ℝ n . This notion is clearly independent of the choice of local trivializations. Then s is said to be transverse if it is transverse over a sufficiently small neighborhood of every point of M.

Notice that the zero section of E has a canonical normal microbundle in the total space, and this notion of transversality for sections never agrees with the notion of transversality for graphs of the sections with respect to this canonical normal microbundle. Hence there is an issue about whether this transversality notion for sections behaves as well as the microbundle transversality.

Theorem A.9.

Let M be a topological manifold and let E → M be an R n -bundle. Let C ⊂ D ⊂ M be closed subsets. Let s : M → E be a continuous section which is transverse near C. Then there exists another continuous section s ′ of E which is transverse near D and which agrees with s over a small neighborhood of C. Moreover, if E is a vector bundle equipped with a continuous norm, then for any ϵ > 0 , one may require that

(A.1) ∥ s ′ - s ∥ C 0 ≤ ϵ .

Proof.

We cannot directly apply the topological transversality theorem (Theorem A.7) as a transverse section is not microbundle transverse to the zero section a priori. Hence we need to use local trivializations to view the section locally as a map to ℝ n . For each p ∈ D , choose a precompact open neighborhood U p ⊂ M of p and a local trivialization

φ p : E | U p ≃ U p × ℝ n .

All U p form an open cover of D. Since M is paracompact, so is D. Hence there exists a locally finite refinement. Moreover, D is Lindelöf, hence this refinement has a countable subcover, denoted by { U i } i = 1 ∞ . Over each U i there is an induced trivialization of E.

We claim that there exists precompact open subsets V i ⊏ U i such that { V i } i = 1 ∞ still cover D. We construct V i inductively. Indeed, a topological manifold satisfies the T 4 -axiom, hence we can use open sets to separate the two closed subsets

D ∖ ⊔ i ≠ 1 U i , D ∖ U 1 .

This provides a precompact V i ⊏ U i such that replacing U 1 by V 1 one still has an open cover of D. Suppose we can find V 1 , … , V k so that replacing U 1 , … , U k by V 1 , … , V k still gives an open cover of D. Then one can obtain V k + 1 ⊏ U k + 1 to continue the induction. We see that { V i } i = 1 ∞ is an open cover of D because every p ∈ D is contained in at most finitely many U i .

Now take an open neighborhood U C ⊂ M of C over which s is transverse. Since M is a manifold, one can separate the two closed subsets C and M ∖ U C by a cut-off function

ρ C : M → [ - 1 , 1 ]

such that

ρ C - 1 ⁢ ( - 1 ) = C , ρ C - 1 ⁢ ( 1 ) = M ∖ U C .

Define a sequence of open sets

C k = ρ C - 1 ⁢ ( [ - 1 , 1 k + 1 ) ) .

Similarly, we can choose a sequence of shrinkings

V i ⊏ ⋯ ⊏ V i k + 1 ⊏ V i k ⊏ ⋯ ⊏ U i .

Define

W k = C k ∪ ⋃ i = 1 k V i k ,

which is a sequence of open subsets of M.

Now we start an inductive construction. First, over U 1 , the section can be identified with a map s 1 : U 1 → ℝ n . By our assumption, s 1 is transverse over U C ∩ U 1 . Then apply the theorem for the pair of closed subsets C 1 ¯ ∩ U 1 ⊂ ( C 1 ¯ ∩ U 1 ) ∪ V 1 1 ¯ of U 1 . Then one can modify it so that it becomes transverse near ( C 1 ¯ ∩ U 1 ) ∪ V 1 1 ¯ , and the change is only supported in a small neighborhood of V 1 1 ¯ ∖ C 1 ¯ . This modified section still agrees with the original section near the boundary of U 1 , hence still defines a section of E. It also agrees with the original section over a neighborhood of C 2 ¯ . Moreover, the modified section is transverse near

W 1 ¯ := C 1 ¯ ∪ V 1 1 ¯ .

Now suppose we have modified the section so that it is transverse near

W k ¯ := C k ¯ ∪ ⋃ i = 1 k V i k ¯ ,

and such that s agrees with the original section over an open neighborhood of C k + 1 ¯ . Then by similar method, one can modify s (via the local trivialization over U k + 1 ) to a section which agrees with s over a neighborhood of

C k + 1 ¯ ∪ ⋃ i = 1 k V i k + 1 ¯

and is transverse near W k + 1 ¯ . In particular, this section still agrees with the very original section over C k + 1 .

We claim that this induction process provides a section s of E which satisfies the requirement. Indeed, since the open cover { U i } i = 1 ∞ of D is locally finite, the value of the section becomes stabilized after finitely many steps of the induction, hence defines a continuous section. Moreover, in each step the value of the section remains unchanged over the open set ρ C - 1 ⁢ ( [ - 1 , 0 ) ) . Transversality also holds by construction. Lastly, when E is a vector bundle equipped with a continuous norm, the requirement (A.1) can be satisfied as one can make the perturbation be supported in any small neighborhood of the zero section. ∎

A.2 Topological orbifolds and orbibundles

We use Satake’s notion of V-manifolds [43] instead of groupoids to treat orbifolds, and only discuss it in the topological category. In this paper we only consider effective orbifolds.

Definition A.10.

Let M be a second countable Hausdorff topological space.

Let x ∈ M be a point. A topological orbifold chart (with boundary) of x consists of a triple ( U ~ x , Γ x , φ x ) , where U ~ x is a topological manifold with possibly empty boundary ∂ ⁡ U ~ x , Γ x is a finite group acting continuously on ( U ~ x , ∂ ⁡ U ~ x ) and

φ x : U ~ x / Γ x → M

is a continuous map which is a homeomorphism onto an open neighborhood of x. Denote the image U x = φ x ⁢ ( U ~ x / Γ x ) ⊂ M and denote the composition

φ ~ x : U ~ x → U ~ x / Γ x → φ x M .
If p ∈ U ~ x , take Γ p = ( Γ x ) p ⊂ Γ x the stabilizer of p. Let U ~ p ⊂ U ~ x be a Γ p -invariant neighborhood of p. Then there is an induced chart (which we will call a subchart) ( U ~ p , Γ p , φ p ) , where φ p is the composition

φ p : U ~ p / Γ p ↪ U ~ x / Γ x → φ p M .
Two charts ( U ~ x , Γ x , φ x ) and ( U ~ y , Γ y , φ y ) are compatible if for any p ∈ U ~ x and q ∈ U ~ y with φ x ⁢ ( p ) = φ y ⁢ ( q ) ∈ M , there exist an isomorphism Γ p → Γ q , subcharts U ~ p ∋ p , U ~ q ∋ q and an equivariant homeomorphism φ q ⁢ p : ( U ~ p , ∂ ⁡ U ~ p ) ≃ ( U ~ q , ∂ ⁡ U ~ q ) .
A topological orbifold atlas of M is a set { ( U ~ α , Γ α , φ α ) ∣ α ∈ I } of topological orbifold charts of M such that M is covered by all U α and for each pair α , β ∈ I , ( U ~ α , Γ α , φ α ) and ( U ~ β , Γ β , φ β ) are compatible. Two atlases are equivalent if the union of them is still an atlas. A structure of topological orbifold (with boundary) is an equivalence class of atlases. A topological orbifold (with boundary) is a second countable Hausdorff space with a structure of topological orbifold (with boundary).

We will often skip the term “topological” in the rest of this paper.

Now consider bundles. Let E, B be orbifolds and π : E → B be a continuous map.

Definition A.11.

A vector bundle chart (resp. disk bundle chart) of π : E → B is a tuple ( U ~ , F n , Γ , φ ^ , φ ) , where F n = ℝ n (resp. F n = 𝔻 n ), ( U ~ , Γ , φ ) is a chart of B and ( U ~ × F n , Γ , φ ^ ) is a chart of E, where Γ acts on F n via a representation Γ → Gl ⁡ ( ℝ n ) (resp. Γ → O ⁢ ( n ) ). The compatibility condition is required, namely, the following diagram commutes:

If ( U ~ p , Γ p , φ p ) is a subchart of ( U ~ , Γ , φ ) , then one can restrict the bundle chart to π ~ - 1 ⁢ ( U ~ p ) .

We can define the notion of compatibility between bundle charts, the notion of orbifold bundle structures and the notion of orbifold bundles in a similar fashion as in the case of orbifolds. We skip the details.

A.2.1 Embeddings

Now we consider embeddings for orbifolds and orbifold vector bundles. First we consider the case of manifolds. Let S and M be topological manifolds and E → S , F → M be continuous vector bundles. Let ϕ : S → M be a topological embedding. A bundle embedding covering ϕ is a continuous map ϕ ^ : E → F which makes the diagram

commute and which is fibrewise a linear injective map. Since ϕ ^ determines ϕ, we also call ϕ ^ : E → F a bundle embedding.

Definition A.12 (Orbifold embedding).

Let S, M be orbifolds and let f : S → M be a continuous map which is a homeomorphism onto its image. Then ϕ is called an embedding if for any pair of orbifold charts, ( U ~ , Γ , φ ) of S and ( V ~ , Π , ψ ) of M, any pair of points p ∈ U ~ , q ∈ V ~ with ϕ ⁢ ( φ ⁢ ( p ) ) = ψ ⁢ ( q ) , there are subcharts

( U ~ p , Γ p , φ p ) ⊂ ( U ~ , Γ , φ ) and ( V ~ q , Π q , ψ q ) ⊂ ( V ~ , Π , ψ ) ,

an isomorphism Γ p ≃ Π q and an equivariant locally flat embedding ϕ ~ p ⁢ q : U ~ p → V ~ q such that the following diagram commutes:

A.2.2 Multisections and perturbations

The equivariant feature of the problem implies that transversality can only be achieved by multi-valued perturbations. Here we review basic notions and facts about multisections. Our discussion mainly follows the treatment of [15].

Definition A.13 (Multimaps).

Let A, B be sets, l ∈ ℕ , and let 𝒮 l ⁢ ( B ) be the l-fold symmetric product of B.

An l - multimap f from A to B is a map f : A → 𝒮 l ⁢ ( B ) . For another a ∈ ℕ , there is a natural map

m a : 𝒮 l ⁢ ( B ) → 𝒮 a ⁢ l ⁢ ( B )

by repeating each component a times.
If both A and B are acted by a group Γ, then we say that an l-multimap f : A → 𝒮 l ⁢ ( B ) is Γ-equivariant if it is equivariant with respect to the Γ-action on A and the induced Γ-action on 𝒮 l ⁢ ( B ) .
If A and B are both topological spaces, then an l-multimap f : A → 𝒮 l ⁢ ( B ) is called continuous if it is continuous with respect to the topology on 𝒮 l ⁢ ( B ) induced as a quotient of B l .
A continuous l-multimap f : A → 𝒮 l ⁢ ( B ) is said to be liftable if there are continuous maps f 1 , … , f l : A → B such that

f ⁢ ( x ) = [ f 1 ⁢ ( x ) , … , f l ⁢ ( x ) ] ∈ 𝒮 l ⁢ ( B ) for all ⁢ x ∈ A .

The maps f 1 , … , f l are called branches of f.
An l 1 -multimap f 1 : A → 𝒮 l 1 ⁢ ( B ) and an l 2 -multimap f 2 : A → 𝒮 l 2 ⁢ ( B ) are said to be equivalent if there exists a common multiple l = a 1 ⁢ l 1 = a 2 ⁢ l 2 of l 1 and l 2 such that

m a 1 ∘ f 1 = m a 2 ∘ f 2

as l-multimaps from A to B.
Being equivalent is clearly reflexive, symmetric and transitive. A multimap from A to B, denoted by f : A ⁢ → 𝑚 ⁢ B , is an element of

( ⊔ l ≥ 1 Map ( A , 𝒮 l ( B ) ) ) / ∼ ,

where ∼ is the above equivalence relation.

In the discussions in this paper, we often identify an l-multimap with its equivalence class as a multimap.

Definition A.14 (Multisections).

Let M be a topological orbifold and let E → M be a vector bundle.

A representative of a (continuous) multisection of E is a collection

{ ( U ~ α , ℝ k , Γ α , φ ^ α , φ α ; s α , l α ) ∣ α ∈ I } ,

where { ( U ~ α , ℝ k , Γ α , φ ^ α , φ α ) ∣ α ∈ I } is a bundle atlas of E and s α : U ~ α → 𝒮 l α ⁢ ( ℝ k ) is a Γ α -equivariant continuous l α -multimap, satisfying the following compatibility condition:
1. For any p ∈ U ~ α and q ∈ U ~ β with φ α ⁢ ( p ) = φ β ⁢ ( q ) ∈ M , there exist two subcharts U ~ p ⊂ U ~ α , U ~ q ⊂ U ~ β , an isomorphism ( φ ^ p ⁢ q , φ p ⁢ q ) of subcharts, a common multiple l = a α ⁢ l α = a β ⁢ l β of l α and l β such that
  
  φ ^ p ⁢ q ∘ m a β ∘ s β | U ~ q = m a α ∘ s α | U ~ p ∘ φ p ⁢ q .
Two representatives are equivalent if their union is also a representative. An equivalence class is called a multisection of E, denoted by

s : M ⁢ → 𝑚 ⁢ E .
A multisection s : M ⁢ → 𝑚 ⁢ E is called locally liftable if for any p ∈ M , there exists a local representative ( U ~ p × ℝ k , Γ p , φ ^ p , φ p ; s p , l p ) such that s p : U ~ p → 𝒮 l p ⁢ ( ℝ k ) which is a liftable continuous l p -multimap.
A multisection s : M ⁢ → 𝑚 ⁢ E is called transverse if it is locally liftable and for any liftable local representative s p : U ~ p → 𝒮 l ⁢ ( ℝ k ) , all branches are transverse to the origin of ℝ k .

The space of continuous multisections of E → M , denoted by C m 0 ⁢ ( M , E ) , is acted by the space of continuous functions C 0 ⁢ ( M ) on M by pointwise multiplication. The space C m 0 ⁢ ( M , E ) also has the structure of a commutative monoid, but not an abelian group. The additive structure is defined as follows. If s 1 , s 2 : M ⁢ → 𝑚 ⁢ E are multisections, then for liftable local representatives with branches s 1 a : U ~ → ℝ n , 1 ≤ a ≤ l , s 2 b : U ~ → ℝ n , 1 ≤ b ≤ k , define

( s 1 + s 2 ) a ⁢ b = [ s 1 a + s 2 b ] 1 ≤ a ≤ l 1 ≤ b ≤ k .

However, there is no inverse to this addition: one can only invert the operation of adding a single-valued section. It is enough, though, since we have the notion of being transverse to a single-valued section which is not necessarily the zero section.

We also want to measure the size of multisections. A continuous norm on an orbifold vector bundle E → M is a continuous function ∥ ⋅ ∥ : E → [ 0 , + ∞ ) which only vanishes on the zero section such that over each local chart, it lifts to an equivariant norm on the fibres. It is easy to construct norms in the relative sense, as one can extend continuous functions defined on closed sets.

The following lemma, which is a generalization of Theorem A.9, shows one can achieve transversality for multisections by perturbation relative to a region where transversality already holds.

Lemma A.15.

Let M be an orbifold (with boundary) and let E → M be an orbifold vector bundle. Let C ⊂ D ⊂ M be closed subsets. Let S : M → E be a single-valued continuous section and let t C : M ⁢ → 𝑚 ⁢ E be a multisection such that S + t C is transverse over a neighborhood of C. Then there exists a multisection t D : M ⁢ → 𝑚 ⁢ E satisfying the following condition:

t C = t D over a neighborhood of C.
S + t D is transverse over a neighborhood of D.

Moreover, if E has a continuous norm ∥ ⋅ ∥ , then for any ϵ > 0 , one can require that

∥ t D ∥ C 0 ≤ ∥ t C ∥ C 0 + ϵ .

Proof.

Similar to the proof of Theorem A.9, one can choose a countable locally finite open cover { U i } i = 1 ∞ of D satisfying the following conditions:

Each U i ¯ is compact.
There are a collection of precompact open subsets V i ⊏ U i such that { V i } i = 1 ∞ is still an open cover of D.
Over each U i there is a local representative of t C , written as

( U ~ i × ℝ k , Γ i , φ ^ i , φ i ; t i , l i ) ,

where t i : U ~ i → 𝒮 l i ⁢ ( ℝ k ) is a Γ i -equivariant l i -multimap which is liftable. Write

t i ⁢ ( x ) = [ t i 1 ⁢ ( x ) , … , t i l i ⁢ ( x ) ] .

In this chart also write S as a map S i : U ~ i → ℝ k .

The transversality assumption implies that there is an open neighborhood U C ⊂ M of C such that for each i, each a ∈ { 1 , … , l i } , s i a is transverse to the origin over

U ~ i , C := φ i - 1 ⁢ ( U C ) ⊂ U ~ C .

Using an inductive construction which is very similar to that in the proof of Theorem A.9, one can construct a valid perturbation. We only sketch the construction for the first chart. Indeed, one can perturb each t 1 a over U ~ 1 to a function t ˙ 1 a : U ~ 1 → ℝ k such that S 1 + t ˙ 1 a is transverse over a neighborhood of the closure of V ~ 1 := φ 1 - 1 ⁢ ( V 1 ) inside U ~ 1 , but t ˙ 1 a = t 1 a over a neighborhood of U ~ 1 , C and the near the boundary of U ~ 1 . Moreover, given ϵ > 0 we may require that

(A.2) sup x ∈ U ~ 1 ⁡ ∥ t ˙ 1 a ⁢ ( x ) ∥ ≤ sup x ∈ U ~ 1 ⁡ ∥ t 1 a ⁢ ( x ) ∥ + ϵ 2 .

Then we obtain a continuous l 1 -multimap

t ˙ 1 ⁢ ( x ) = [ t ˙ 1 1 ⁢ ( x ) , … , t ˙ 1 l 1 ⁢ ( x ) ] .

This multimap may not be Γ 1 -equivariant. We reset

t ˙ 1 ⁢ ( x ) := [ t 1 a ⁢ b ] 1 ≤ a ≤ l 1 1 ≤ b ≤ n 1 := [ g b - 1 ⁢ t ˙ 1 a ⁢ ( g b ⁢ x ) ] 1 ≤ a ≤ l 1 1 ≤ b ≤ n 1 ,

where Γ 1 = { g 1 , … , g n 1 } . It is easy to verify that this is Γ 1 -invariant, and agrees with t 1 over a neighborhood of U ~ 1 , C and near the boundary of U ~ 1 . There still holds

sup a , b ⁡ sup x ∈ U ~ 1 ⁡ ∥ t ˙ 1 a ⁢ b ⁢ ( x ) ∥ ≤ sup a ⁡ sup x ∈ U ~ 1 ⁡ ∥ t 1 a ⁢ ( x ) ∥ + ϵ 2 .

Therefore, together with the original multisection over the complement of U 1 , t ˙ 1 defines a continuous multisection of E. Moreover, it agrees with the original one over a neighborhood of C and is transverse near C ∪ V 1 ¯ . In this way we can continue the induction to perturb over all U ~ i . At the k-th step of the induction, we replace ϵ 2 by ϵ 2 k in the C 0 bound (A.2). Since U i is locally finite, near each point, the value of the perturbation stabilizes after finitely many steps of this induction. This results in a multisection t D which satisfies our requirement. ∎

A.3 Virtual orbifold atlases

Now we introduce the notion of virtual orbifold atlases. This notion plays a role as a general structure of moduli spaces we are interested in, and is a type of intermediate objects in concrete constructions. The eventual objects we would like to construct are good coordinate systems, which are special types of virtual orbifold atlases.

Definition A.16.

Let X be a compact and Hausdorff space.

A virtual orbifold chart (chart for short) is a tuple C := ( U , E , S , ψ , F ) , where
1. U is a topological orbifold (with boundary),
2. E → U is a continuous orbifold vector bundle,
3. S : U → E is a continuous section,
4. F ⊂ X is an open subset,
5. ψ : S - 1 ⁢ ( 0 ) → F is a homeomorphism;
F is called the footprint of this chart C, and the integer dim ⁡ U - rank ⁡ E is called the virtual dimension of C.
Let C = ( U , E , S , ψ , F ) be a chart and let U ′ ⊂ U be an open subset. The restriction of C to U ′ is the chart

C ′ = C | U ′ = ( U ′ , E ′ , S ′ , ψ ′ , F ′ ) ,

where E ′ = E | U ′ , S ′ = S | U ′ , ψ ′ = ψ | ( S ′ ) - 1 ⁢ ( 0 ) , and F ′ = 𝐈𝐦𝐚𝐠𝐞 ⁡ ψ ′ . Any such chart C ′ induced from an open subset U ′ ⊂ U is called a shrinking of C. A shrinking C ′ = C | U ′ is called a precompact shrinking if U ′ ⊏ U , denoted by C ′ ⊏ C .

A very useful lemma about shrinkings is the following, whose proof is left to the reader.

Lemma A.17.

Suppose C = ( U , E , S , ψ , F ) is a virtual orbifold atlas and let F ′ ⊂ F be an open subset. Then there exists a shrinking C ′ of C whose footprints is F ′ . Moreover, if F ′ ⊏ F , then C ′ can be chosen to be a precompact shrinking.

Definition A.18.

Let C i := ( U i , E i , S i , ψ i , F i ) for i = 1 , 2 be two charts of X. An embedding of C 1 into C 2 consists of a bundle embedding ϕ ^ 21 satisfying the following conditions:

The following diagrams commute:
(Tangent Bundle Condition) There exist an open neighborhood N 21 ⊂ U 2 of ϕ 21 ⁢ ( U 1 ) and a subbundle E 1 ; 2 ⊂ E 2 | N 21 which extends ϕ ^ 21 ⁢ ( E 1 ) such that S 2 | N 2 : N 2 → E 2 | N 2 is transverse to E 1 ; 2 and S 2 - 1 ⁢ ( E 1 ; 2 ) = ϕ 21 ⁢ ( U 1 ) .

The following lemma is left to the reader.

Lemma A.19.

The composition of two embeddings is still an embedding.

Definition A.20.

Let C i = ( U i , E i , S i , ψ i , F i ) , i = 1 , 2 , be two charts. A coordinate change from C 1 to C 2 is a triple T 21 = ( U 21 , ϕ 21 , ϕ ^ 21 ) , where U 21 ⊂ U 1 is an open subset and ( ϕ 21 , ϕ ^ 21 ) is an embedding from C 1 | U 21 to C 2 . They should satisfy the following conditions:

ψ 1 ⁢ ( U 21 ∩ S 1 - 1 ⁢ ( 0 ) ) = F 1 ∩ F 2 .
If x k ∈ U 21 converges to x ∞ ∈ U 1 and y k = ϕ 21 ⁢ ( x k ) converges to y ∞ ∈ U 2 , then x ∞ ∈ U 21 and y ∞ = ϕ 21 ⁢ ( y ∞ ) .

Lemma A.21.

Given two charts C i = ( U i , E i , S i , ψ i , F i ) , i = 1 , 2 , and a coordinate change T 21 = ( U 21 , ϕ 21 , ϕ ^ 21 ) from C 1 to C 2 . Suppose C i ′ = C i | U i ′ be a shrinking of C i . Then the restriction T 21 ′ := T 21 | U 1 ′ ∩ ϕ 21 - 1 ⁢ ( U 2 ′ ) is a coordinate change from C 1 ′ to C 2 ′ .

Proof.

Left to the reader. ∎

We call T 21 ′ in the above lemma the induced coordinate change from the shrinking.

Now we introduce the notion of atlases.

Definition A.22.

Let X be a compact metrizable space. A virtual orbifold atlas of virtual dimension d on X is a collection

𝔄 := ( { C I := ( U I , E I , S I , ψ I , F I ) ∣ I ∈ ℐ } , { T J ⁢ I = ( U J ⁢ I , ϕ J ⁢ I , ϕ ^ J ⁢ I ) ∣ I ≼ J } ) ,

where

( ℐ , ≼ ) is a finite, partially ordered set,
for each I ∈ ℐ , C I is a virtual orbifold chart of virtual dimension d on X,
for I ≼ J , T J ⁢ I is a coordinate change from C I to C J .

They are subject to the following conditions:

(Covering Condition) X is covered by all the footprints F I .
(Cocycle Condition) For I ≼ J ≼ K ∈ ℐ , denote U K ⁢ J ⁢ I = U K ⁢ I ∩ ϕ J ⁢ I - 1 ⁢ ( U K ⁢ J ) ⊂ U I . Then we require that

ϕ ^ K ⁢ I | U K ⁢ J ⁢ I = ϕ ^ K ⁢ J ∘ ϕ ^ J ⁢ I | U K ⁢ J ⁢ I

as bundle embeddings.
(Overlapping Condition) For I , J ∈ ℐ , we have

F I ¯ ∩ F J ¯ ≠ ∅ ⟹ I ≼ J ⁢ or ⁢ J ≼ I .

All virtual orbifold atlases considered in this paper have definite virtual dimensions, although sometimes we do not explicitly mention it.

A.3.1 Orientations

Now we discuss orientation. When M is a topological manifold, there is an orientation bundle Orient M → M which is a double cover of M (or a ℤ 2 -principal bundle). The manifold M is orientable if and only if Orient M is trivial. If E → M is a continuous vector bundle, then E also has an orientation bundle Orient E → M as a double cover. Since ℤ 2 -principal bundles over a base B are classified by H 1 ⁢ ( B ; ℤ 2 ) , the orientation bundles can be multiplied. We use ⊗ to denote this multiplication.

Definition A.23 (Orientability).

A virtual orbifold chart C = ( U , E , S , ψ , F ) is locally orientable if for any bundle chart ( U ~ , ℝ n , Γ , φ ^ , φ ) of E, if we denote E ~ = U ~ × ℝ n , then for any γ ∈ Γ , the map

γ : Orient U ~ ⊗ Orient E ~ * → Orient U ~ ⊗ Orient E ~ *

is the identity over all fixed points of γ.
If C is locally orientable, then Orient U ~ ⊗ Orient E ~ * for all local charts glue together a double cover Orient C → U . If Orient C is trivial (resp. trivialized), then we say that C is orientable (resp. oriented).
A coordinate change T 21 = ( U 21 , ϕ 21 , ϕ ^ 21 ) between two oriented charts C 1 and C 2 is called oriented if the embeddings ϕ 21 and ϕ ^ 21 are compatible with the orientations on Orient C 1 and Orient C 2 .
An atlas 𝔄 is oriented if all charts are oriented and all coordinate changes are oriented.

A.4 Good coordinate systems

Now we introduce the notion of shrinkings of virtual orbifold atlases.

Definition A.24.

Let 𝔄 = ( { C I ∣ I ∈ ℐ } , { T J ⁢ I ∣ I ≼ J } ) be a virtual orbifold atlas on X.

A shrinking of 𝔄 is another virtual orbifold atlas 𝔄 ′ = ( { C I ′ ∣ I ∈ ℐ } , { T J ⁢ I ′ ∣ I ≼ J } ) indexed by elements of the same partially ordered set ℐ such that for each I ∈ ℐ , C I ′ is a shrinking C I | U I ′ of C I and for each I ≼ J , T J ⁢ I ′ is the induced shrinking of T J ⁢ I given by Lemma A.21.
If for every I ∈ ℐ , U I ′ is a precompact subset of U I , then we say that 𝔄 ′ is a precompact shrinking of 𝔄 and denote 𝔄 ′ ⊏ 𝔄 .

Given a virtual orbifold atlas 𝔄 = ( { C I ∣ I ∈ ℐ } , { T J ⁢ I ∣ I ≼ J } ) , we define a relation ⋎ on the disjoint union ⊔ I ∈ ℐ U I as follows. We have U I ∋ x ⋎ y ∈ U J if one of the following holds:

I = J and x = y ,
I ≼ J , x ∈ U J ⁢ I and y = ϕ J ⁢ I ⁢ ( x ) ,
J ≼ I , y ∈ U I ⁢ J and x = ϕ I ⁢ J ⁢ ( y ) .

If 𝔄 ′ is a shrinking of 𝔄 , then it is easy to see that the relation ⋎ ′ on ⊔ I ∈ ℐ U I ′ defined as above is induced from the relation ⋎ for 𝔄 via restriction.

For an atlas 𝔄 , if ⋎ is an equivalence relation, we can form the quotient space

| 𝔄 | := ( ⊔ I ∈ ℐ U I ) / ⋎

with the quotient topology. There is a natural injective map

X ↪ | 𝔄 | .

We call | 𝔄 | the virtual neighborhood of X associated to the atlas 𝔄 . Denote the quotient map by

(A.3) π 𝔄 : ⊔ I ∈ ℐ U I → | 𝔄 | ,

which induces continuous injections

π I : U I ↪ | 𝔄 | .

A point in | 𝔄 | is denoted by | x | , which has a certain representative x ∈ U I for some I.

Definition A.25.

A virtual orbifold atlas 𝔄 on X is called a good coordinate system if the following conditions are satisfied:

⋎ is an equivalence relation.
The virtual neighborhood | 𝔄 | is a Hausdorff space.
For all I ∈ ℐ , the natural maps π I : U I → | 𝔄 | are homeomorphisms onto their images.

The conditions for good coordinate systems are very useful for later constructions (this is the same as in the Kuranishi approach, see [13]), for example, the construction of suitable multisection perturbations. In these constructions, the above conditions are often implicitly used without explicit reference. Therefore, an important step is to construct good coordinate systems.

Theorem A.26 (Constructing good coordinate system).

Let A be a virtual orbifold atlas on X with the collection of footprints F I indexed by I ∈ I . Let F I □ ⊏ F I for all I ∈ I be a collection of precompact open subsets such that

X = ⋃ I ∈ ℐ F I □ .

Then there exists a shrinking A ′ of A such that the collection of shrunk footprints F I ′ contains F I □ ¯ for all I ∈ I and A ′ is a good coordinate system.

Moreover, if A is already a good coordinate system, then any shrinking of A remains a good coordinate system.

We give a proof of Theorem A.26 in the next subsection. A similar result is used in the Kuranishi approach while our argument potentially differs from that of [13].

Remark A.27.

If 𝔄 is a good coordinate system, and 𝔄 ′ is a shrinking of 𝔄 , then the shrinking induces a natural map

| 𝔄 ′ | ↪ | 𝔄 | .

If we equip both | 𝔄 ′ | and | 𝔄 | with the quotient topologies, then the natural map is continuous. However, there is another topology on | 𝔄 ′ | by viewing it as a subset of | 𝔄 | . We denote this topology by ∥ 𝔄 ′ ∥ and call it the subspace topology. In most cases, the quotient topology is strictly stronger than the subspace topology. Hence it is necessary to distinguish the two different topologies.

A.5 Shrinking good coordinate systems

In this subsection we prove Theorem A.26. First we show that by precompact shrinkings one can make the relation ⋎ an equivalence relation.

Lemma A.28.

Let A be a virtual orbifold atlas on X with the collection of footprints { F I ∣ I ∈ I } . Let F I □ ⊏ F I be precompact open subsets such that

X = ⋃ I ∈ ℐ F I □ .

Then there exists a precompact shrinking A ′ of A whose collection of footprints F I ′ contains F I □ ¯ for all I ∈ I such that the relation ⋎ on A ′ is an equivalence relation.

Proof.

By definition, the relation ⋎ is reflexive and symmetric. By the comments above, any shrinking of 𝔄 will preserve reflexiveness and symmetry. Hence we only need to shrink the atlas to make the induced relation transitive.

For any subset ℐ ′ ∈ ℐ , we say that ⋎ is transitive in ℐ ′ if for x , y , z ∈ ⊔ I ∈ ℐ ′ U I , x ⋎ y , y ⋎ z imply x ⋎ z . Being transitive in any subset ℐ ′ is a condition that is preserved under shrinking. Hence it suffices to construct shrinkings such that ⋎ is transitive in { I , J , K } for any three distinct elements I , J , K ∈ ℐ . Let x ∈ U I , y ∈ U J , z ∈ U K be general elements.

Since U I (resp. U J , U K ) is an orbifold and hence metrizable, we can choose a sequence of precompact open subsets U I n ⊏ U I (resp. U J n ⊏ U J , U K n ⊏ U K ) containing F I □ ¯ (resp. F J □ ¯ , F K □ ¯ ) such that

U I n + 1 ⊏ U I n , U J n + 1 ⊏ U J n , U K n + 1 ⊏ U K n

and

⋂ n U I n = ψ I - 1 ⁢ ( F I □ ¯ ) , ⋂ n U J n = ψ J - 1 ⁢ ( F J □ ¯ ) , ⋂ n U K n = ψ K - 1 ⁢ ( F K □ ¯ ) .

Then for each n, U I n , U J n , U K n induce a shrinking of the atlas 𝔄 , denoted by 𝔄 n . Let the induced binary relation on U I n ⊔ U J n ⊔ U K n still by ⋎ . We claim that for n large enough, ⋎ is an equivalence relation on this triple disjoint union. Denote the domains of the shrunk coordinate changes by U J ⁢ I n , U K ⁢ J n and U K ⁢ I n , respectively.

If this is not true, then without loss of generality, we may assume that for all large n, there exist points x n ∈ U I n , y n ∈ U J n , z n ∈ U K n such that

(A.4) x n ⋎ y n , y n ⋎ z n , but ⁢ ( x n , z n ) ∉ ⋎ .

Then for some subsequence (still indexed by n),

x n converges to x ∞ ∈ ψ I - 1 ⁢ ( F I □ ¯ ) ⊂ U I ,
y n converges to y ∞ ∈ ψ J - 1 ⁢ ( F J □ ¯ ) ⊂ U J ,
z n converges to z ∞ ∈ ψ K - 1 ⁢ ( F K □ ¯ ) ⊂ U K .

By the definition of coordinate changes (Definition A.20), one has

x ∞ ⋎ y ∞ , y ∞ ⋎ z ∞ ⟹ ψ I ⁢ ( x ∞ ) = ψ J ⁢ ( y ∞ ) = ψ K ⁢ ( z ∞ ) ∈ F I □ ¯ ∩ F J □ ¯ ∩ F K □ ¯ .

By the Overlapping Condition of Definition A.22, { I , J , K } is totally ordered. Since the roles of K and I are symmetric, we may assume that I ≼ K . Then since

x ∞ ∈ ψ I - 1 ⁢ ( F I □ ¯ ∩ F K □ ¯ ) ⊂ ψ I - 1 ⁢ ( F I ∩ F K ) = ψ I - 1 ⁢ ( F K ⁢ I ) ⊂ U K ⁢ I

and U K ⁢ I ⊂ U I is an open set, for n large enough one has

x n ∈ U K ⁢ I .

We have the following:

If I ≼ J ≼ K , then by the Cocycle Condition of Definition A.22,

ϕ K ⁢ I ⁢ ( x n ) = ϕ K ⁢ J ⁢ ( ϕ J ⁢ I ⁢ ( x n ) ) = ϕ K ⁢ J ⁢ ( y n ) = z n .

So x n ⋎ z n , which contradicts (A.4).
If J ≼ I ≼ K , then the Cocycle Condition of Definition A.22,

z n = ϕ K ⁢ J ⁢ ( y n ) = ϕ K ⁢ I ⁢ ( ϕ I ⁢ J ⁢ ( y n ) ) = ϕ K ⁢ I ⁢ ( x n ) .

So x n ⋎ z n , which contradicts (A.4).
If I ≼ K ≼ J , then since ϕ K ⁢ I ⁢ ( x ∞ ) ∈ ψ K - 1 ⁢ ( F J ∩ F K ) ⊂ U J ⁢ K , for sufficiently large n, x n ∈ ϕ K ⁢ I - 1 ⁢ ( U J ⁢ K ) . Then by the Cocycle Condition of Definition A.22,

ϕ J ⁢ K ⁢ ( z n ) = y n = ϕ J ⁢ I ⁢ ( x n ) = ϕ J ⁢ K ⁢ ( ϕ K ⁢ I ⁢ ( x n ) ) .

Since ϕ J ⁢ K is an embedding, we have ϕ K ⁢ I ⁢ ( x n ) = z n . Therefore, x n ⋎ z n , which contradicts (A.4).

Therefore, ⋎ n is an equivalence relation on U I n ⊔ U J n ⊔ U K n for large enough n. We can perform the shrinking for any triple of elements of ℐ , which eventually makes ⋎ an equivalence relation. By construction, the shrunk footprints F I ′ still contain F I □ ¯ . ∎

Lemma A.29.

Suppose A is a virtual orbifold atlas on X such that the relation ⋎ is an equivalence relation. Suppose there is a collection of precompact subsets F I □ ⊏ F I of footprints of A such that

X = ⋃ I ∈ ℐ F I □ .

Then there exists a precompact shrinking A ′ ⊏ A satisfying the following conditions:

The shrunk footprints F I ′ all contain F I □ .
The virtual neighborhood | 𝔄 ′ | is a Hausdorff space.

Before proving Lemma A.29, we need some preparations. Order the finite set ℐ as { I 1 , … , I m } such that for k = 1 , … , m ,

I k ≼ J ⟹ J ∈ { I k , I k + 1 , … , I m } .

For each k, ⋎ induces an equivalence relation on ⊔ i ≥ k U I i and denote the quotient space by | 𝔄 k | . Then the map π 𝔄 of (A.3) induces a natural continuous map

π k : ⊔ i ≥ k U I i → | 𝔄 k | .

Lemma A.30.

For k = 1 , … , m , if | A k | is Hausdorff and A ′ is a shrinking of A , then | A k ′ | is also Hausdorff.

Proof.

Left to the reader. A general fact is that the quotient topology is always stronger than (or homeomorphic to) the subspace topology (see Remark A.27). ∎

Lemma A.31.

The natural map

(A.5) | 𝔄 k + 1 | → | 𝔄 k |

is a homeomorphism onto an open subset.

Proof.

The map is clearly continuous and injective. To show that it is a homeomorphism onto an open set, consider any open subset O k + 1 of its domain. Its preimage under the quotient map

U I k + 1 ⊔ ⋯ ⊔ U I m → | 𝔄 k + 1 |

is denoted by

O ~ k + 1 = O I k + 1 ⊔ ⋯ ⊔ O I m ,

where O I k + 1 , … , O I m are open subsets of U I k + 1 , … , U I m , respectively. Define

O I k := ⋃ i ≥ k + 1 , I i ≥ I k ϕ I i ⁢ I k - 1 ⁢ ( O I i ) .

This is an open subset of U I k . Then the image of O k + 1 under the map (A.5), denoted by O k , is the quotient of

O ~ k := O I k ⊔ O I k + 1 ⊔ ⋯ ⊔ O I m ⊂ U I k ⊔ ⋯ ⊔ U I m .

On the other hand, O ~ k is exactly the preimage of O k under the quotient map. Hence by the definition of the quotient topology, O k is open. This show that (A.5) is a homeomorphism onto an open subset. ∎

Proof of Lemma A.29.

For each k, we would like to construct shrinkings U I i ′ ⊏ U I i for all i ≥ k such that | 𝔄 k ′ | is Hausdorff and the shrunk footprints F I i ′ contain F I i □ ¯ for all i ≥ k . Our construction is based on a top-down induction. First, for k = m , | 𝔄 m | ≃ U I m and hence is Hausdorff. Suppose after shrinking | 𝔄 k + 1 | is already Hausdorff.

Choose open subsets F I i ′ ⊏ F I i for all i ≥ k such that

F I i □ ¯ ⊂ F I i ′ , X = ⋃ i ≥ k F I i ′ ∪ ⋃ i ≤ k - 1 F I i .

Choose precompact open subsets U I i ′ ⊏ U I i for all i ≥ k such that

ψ I i ⁢ ( U I i ′ ∩ S I i - 1 ⁢ ( 0 ) ) = F I i ′ , ψ I i ⁢ ( U I i ′ ¯ ∩ S I i - 1 ⁢ ( 0 ) ) = F I i ′ ¯ .

Then U I i ′ for i ≥ k and U I i for i < k provide a shrinking 𝔄 ′ of 𝔄 . We claim that | 𝔄 k ′ | is Hausdorff.

Indeed, pick any two different points | x | , | y | ∈ | 𝔄 k ′ | . If | x | , | y | ∈ | 𝔄 k + 1 ′ | ⊂ | 𝔄 k ′ | , then by the induction hypothesis and Lemma A.30, | x | and | y | can be separated by two open subsets in | 𝔄 k + 1 ′ | . Then by Lemma A.31, these two open sets are also open sets in | 𝔄 k ′ | . Hence we assume that one or both of | x | and | y | are in | 𝔄 k ′ | ∖ | 𝔄 k + 1 ′ | .

Case 1. Suppose | x | and | y | are represented by x , y ∈ U I k ′ . Choose a distance function on U I k which induces the same topology. Let O x ϵ and O y ϵ be the open ϵ-balls in U I k ′ centered at x and y, respectively. Then for ϵ small enough, O x ϵ ¯ ∩ O y ϵ ¯ = ∅ .

Claim.

For ϵ sufficiently small, for all I k ≼ I i and I k ≼ I j , one has

π k + 1 ⁢ ( ϕ I i ⁢ I k ⁢ ( O x ϵ ∩ U I i ⁢ I k ′ ) ¯ ) ∩ π k + 1 ⁢ ( ϕ I j ⁢ I k ⁢ ( O y ϵ ∩ U I j ⁢ I k ′ ) ¯ ) = ∅ .

Here the closures are the closures in U I i and U I j , respectively.

Proof of the claim.

Suppose it is not the case. Then there exist a sequence ϵ n → 0 , I k ≼ I i , I k ≼ I j , and a sequence of points

| z n | ∈ π k + 1 ⁢ ( ϕ I i ⁢ I k ⁢ ( O x ϵ n ∩ U I i ⁢ I k ′ ) ¯ ) ∩ π k + 1 ⁢ ( ϕ I j ⁢ I k ⁢ ( O y ϵ ∩ U I j ⁢ I k ′ ) ¯ ) ⊂ | 𝔄 k + 1 | .

In U I i , | z n | has its representative p n ∈ ϕ I i ⁢ I k ⁢ ( O x ϵ n ∩ U I k ⁢ I i ′ ) ¯ ; and in U I j , | z n | has its representative q n ∈ ϕ I j ⁢ I k ⁢ ( O y ϵ n ∩ U I j ⁢ I k ′ ) ¯ . Then p n ⋎ q n and without loss of generality, assume that I i ≼ I j . Then p n ∈ U I j ⁢ I i and q n = ϕ I j ⁢ I i ⁢ ( p n ) .

Choose distance functions d i on U I i and d j on U I j which induce the same topologies. Then one can choose x n ∈ O x ϵ n ∩ U I i ⁢ I k ′ and y n ∈ O y ϵ n ∩ U I j ⁢ I k ′ such that

(A.6) d i ⁢ ( p n , ϕ I i ⁢ I k ⁢ ( x n ) ) ≤ ϵ n , d ⁢ ( q n , ϕ I i ⁢ I k ⁢ ( y n ) ) ≤ ϵ n .

Since U I i ′ ¯ and U I j ′ ¯ are compact and p n ∈ U I i ′ ¯ , q n ∈ U I j ′ ¯ , for some subsequence (still indexed by n), p n converges to some p ∞ ∈ U I i ′ ¯ and q n converges to some q ∞ ∈ U I j ′ ¯ . Then p ∞ ⋎ q ∞ . Moreover, by (A.6), one has

lim n → ∞ ⁡ ϕ I i ⁢ I k ⁢ ( x n ) = p ∞ , lim n → ∞ ⁡ ϕ I j ⁢ I k ⁢ ( y n ) = q ∞ .

On the other hand, x n converges to x and y n converges to y. By the property of coordinate changes, one has that x ∈ U I i ⁢ I k , y ∈ U I j ⁢ I k and

x ⋎ p ∞ ⋎ q ∞ ⋎ y .

Since ⋎ is an equivalence relation and it remains an equivalence relation after shrinking, x ⋎ y , which contradicts x ≠ y . ∎

Now choose such an ϵ and abbreviate O x = O x ϵ , O y = O y ϵ . Denote

P I i = ϕ I i ⁢ I k ⁢ ( O x ∩ U I i ⁢ I k ′ ) ¯ ⊂ U I i ′ ¯ , Q I i = ϕ I i ⁢ I k ⁢ ( O y ∩ U I i ⁢ I k ′ ) ¯ ⊂ U I i ′ ¯

(which could be empty). They are all compact, hence

P k + 1 := π k + 1 ⁢ ( ⊔ i ≥ k + 1 P I i ) ⊂ | 𝔄 k + 1 | , Q k + 1 := π k + 1 ⁢ ( ⊔ i ≥ k + 1 Q I i ) ⊂ | 𝔄 k + 1 |

are both compact. The above claim implies that P k + 1 ∩ Q k + 1 = ∅ . Then by the induction hypothesis which says that | 𝔄 k + 1 | is Hausdorff, they can be separated by two open sets V k + 1 , W k + 1 ⊂ | 𝔄 k + 1 | . Write

π k + 1 - 1 ⁢ ( V k + 1 ) = ⊔ i ≥ k + 1 V I i , π k + 1 - 1 ⁢ ( W k + 1 ) = ⊔ i ≥ k + 1 W I i .

Define V I i ′ = V I i ∩ U I i ′ , W I i ′ = W I i ∩ U I i ′ and

V I k ′ := O x ∪ ⋃ I k ≼ I i ϕ I i ⁢ I k - 1 ⁢ ( V I i ′ ) ∩ U I k ′ , W I k ′ := O y ∪ ⋃ I k ≼ I i ϕ I i ⁢ I k - 1 ⁢ ( W I i ′ ) ∩ U I k ′

and

V k ′ := π k ⁢ ( ⊔ i ≥ k V I i ′ ) ⊂ | 𝔄 k ′ | , W k ′ := π k ⁢ ( ⊔ i ≥ k W I i ′ ) ⊂ | 𝔄 k ′ | .

It is easy to check that V k ′ and W k ′ are disjoint open subsets containing | x | and | y | , respectively. Therefore | x | and | y | are separated in | 𝔄 k ′ | .

Case 2. Suppose | x | is represented by x ∈ U I k ′ and | y | ∈ | 𝔄 k + 1 ′ | . Similar to Case 1, we claim that for ϵ sufficiently small, for all I k ≼ I i , one has

| y | ∉ π k + 1 ( ϕ I i ⁢ I k ⁢ ( O x ϵ ) ∩ U I i ⁢ I k ′ ¯ ) = : P I i ⊂ U I i ′ ¯ .

The proof is similar and is omitted. Then choose such an ϵ and abbreviate O x = O x ϵ . P I i are compact sets and so is

P k + 1 := π k + 1 ⁢ ( ⊔ i ≥ k + 1 P I i ) ⊂ | 𝔄 k + 1 | .

The above claim implies that | y | ∉ P k + 1 . Then by the induction hypothesis, | y | and P k + 1 can be separated by open sets V k + 1 and W k + 1 of | 𝔄 k + 1 | . By similar procedure as in Case 1 above, one can produce two open subsets of | 𝔄 k ′ | which separate | x | and | y | .

Therefore, we can finish the induction and construct a shrinking such that | 𝔄 ′ | is Hausdorff. Eventually the shrunk footprints still contain F I □ ¯ . ∎

Now we can finish proving Theorem A.26. Suppose | 𝔄 | is Hausdorff. By the definition of the quotient topology, the natural map U I ↪ | 𝔄 | is continuous. Since U I is locally compact and | 𝔄 | is Hausdorff, a further shrinking can make this map a homeomorphism onto its image. This uses the fact that a continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism. Hence the third condition for a good coordinate system is satisfied by a precompact shrinking of 𝔄 , and this condition is preserved for any further shrinking. This establishes Theorem A.26.

A.6 Perturbations

Now we define the notion of perturbations.

Definition A.32.

Let 𝔄 be a good coordinate system on X.

A multi-valued perturbation of 𝔄 , or called a multisection perturbation or simply a perturbation, denoted by 𝔱 , consists of a collection of multi-valued continuous sections

t I : U I ⁢ → 𝑚 ⁢ E I

satisfying (as multisections)

t J ∘ ϕ J ⁢ I = ϕ ^ J ⁢ I ∘ t I | U J ⁢ I for all ⁢ I ≼ J .
Given a multi-valued perturbation 𝔱 , the object

𝔰 = ( S I + t I : U I → 𝑚 E I )

satisfies the same compatibility condition with respect to coordinate changes. The perturbation 𝔱 is called transverse if every S I + t I is a transverse multisection.
The zero locus of a perturbed 𝔰 gives objects in various different categories. Denote

𝔰 - 1 ⁢ ( 0 ) = ⊔ I ∈ ℐ ( S I + t I ) - 1 ⁢ ( 0 ) .

It is naturally equipped with the topology induced from the disjoint union of U I . Denote by

| 𝔰 - 1 ( 0 ) | := 𝔰 - 1 ( 0 ) / ⋎

the quotient of 𝔰 - 1 ⁢ ( 0 ) , which is equipped with the quotient topology. Furthermore, there is a natural continuous injection | 𝔰 - 1 ⁢ ( 0 ) | ↪ | 𝔄 | . Denote by ∥ 𝔰 - 1 ⁢ ( 0 ) ∥ the same set as | 𝔰 - 1 ⁢ ( 0 ) | but equipped with the topology as a subspace of | 𝔄 | .

In order to construct suitable perturbations of a good coordinate system, we need certain tubular neighborhood structures with respect to coordinate changes. In our topological situation, it is sufficient to have some weaker structure near the embedding images.

Definition A.33.

Let 𝔄 be a good coordinate system with charts indexed by elements in a finite partially ordered set ( ℐ , ≼ ) and coordinate changes indexed by pairs I ≼ J ∈ ℐ . A thickening of 𝔄 is a collection of objects

𝒩 = { ( N J ⁢ I , E I ; J ) ∣ I ≼ J } ,

where N J ⁢ I ⊂ U J is an open neighborhood of ϕ J ⁢ I ⁢ ( U J ⁢ I ) and E I ; J is a subbundle of E J | N J ⁢ I . They are required to satisfy the following conditions:

If I ≼ K , J ≼ K but there is no partial order relation between I and J, then

N K ⁢ I ∩ N K ⁢ J = ∅ .
For all triples I ≼ J ≼ K ,

E I ; J | ϕ K ⁢ J - 1 ⁢ ( N K ⁢ I ) ∩ N J ⁢ I = ϕ ^ K ⁢ J - 1 ⁢ ( E I ; K ) | ϕ K ⁢ J - 1 ⁢ ( N K ⁢ I ) ∩ N J ⁢ I .
For all triples I ≼ J ≼ K , one has

E I ; K | N K ⁢ I ∩ N K ⁢ J ⊂ E J ; K | N K ⁢ I ∩ N K ⁢ J .
Each ( N J ⁢ I , E I ; J ) satisfies the Tangent Bundle Condition of Definition A.18.

Definition A.34.

Suppose 𝔄 is thickened by 𝒩 = { ( N J ⁢ I , E I ; J ) ∣ I ≼ J } . We say that a multi-valued perturbation 𝔱 is 𝒩 - normal if for all I ≼ J , one has

t J ⁢ ( N J ⁢ I ) ⊂ E I ; J | N J ⁢ I .

Theorem A.35.

Let X be a compact Hausdorff space and have a good coordinate system

𝔄 = ( { C I = ( U I , E I , S I , ψ I , F I ) ∣ I ∈ ℐ } , { T J ⁢ I ∣ I ≼ J } ) .

Let A ′ ⊏ A be any precompact shrinking. Let N = { ( N J ⁢ I , E I ; J ) } be a thickening of A and let N J ⁢ I ′ ⊂ U J ′ be a collection of open neighborhoods of ϕ J ⁢ I ⁢ ( U J ⁢ I ′ ) such that N J ⁢ I ′ ¯ ⊂ N J ⁢ I . Then they induce a thickening N ′ of A ′ by restriction. Let d I : U I × U I → [ 0 , + ∞ ) be a distance function on U I which induces the same topology as U I . Let ϵ > 0 be a constant. Then there exist a collection of multisections t I : U I ⁢ → 𝑚 ⁢ E I satisfying the following conditions:

For each I ∈ ℐ , S I + t I is transverse.
For each I ∈ ℐ ,

(A.7) d I ⁢ ( ( S I + t I ) - 1 ⁢ ( 0 ) ∩ U I ′ ¯ , S I - 1 ⁢ ( 0 ) ∩ U I ′ ¯ ) ≤ ϵ .
For each pair I ≼ J , we have

ϕ ^ J ⁢ I ∘ t I | U J ⁢ I ′ = t J ∘ ϕ J ⁢ I | U J ⁢ I ′ .

Hence the collection of restrictions t I ′ := t I | U I ′ defines a perturbation 𝔱 ′ of 𝔄 ′ .
𝔱 ′ is 𝒩 ′ - normal.

Proof.

To simplify the proof, we assume that for all I ≼ J , subbundles E I ; J ⊂ E J | N J ⁢ I are defined over U J and they satisfy

E I ; K ⊂ E J ; K for all ⁢ I ≼ J ≼ K .

This is the situation of the current application. The proof in the general case requires minor modifications in a few places.

We use the inductive construction. Order the set ℐ as I 1 , … , I m such that

I k ≼ I l ⟹ k ≤ l .

For each k , l with k < l , define open sets N I l , k - by

N I l , k - = ⋃ a ≤ k , I a ≺ I l N I l ⁢ I a .

Define open sets N I l , k + inductively. First N I m , k + = ∅ . Then

N I l , k + := ⋃ I l ≺ I b ϕ I b ⁢ I l - 1 ⁢ ( N I b , k ) , N I l , k = N I l , k - ∪ N I l , k + .

Replacing N J ⁢ I by N J ⁢ I ′ in the above definitions, we obtain N I l , k ′ ⊂ U I l ′ with

N I l , k ′ ¯ ⊂ N I l , k .

If k ≥ l , define

N I l , k = U I l , N I l , k ′ = U I l ′ .

On the other hand, it is not hard to inductively choose a system of continuous norms on E I such that, for all pairs I a ≼ I b , the bundle embedding ϕ ^ I b ⁢ I a is isometric. Given such a collection of norms, choose δ > 0 such that for all I,

(A.8) d I ⁢ ( x , S I - 1 ⁢ ( 0 ) ∩ U I ′ ¯ ) > ϵ , x ∈ U I ′ ¯ ⟹ ∥ S I ⁢ ( x ) ∥ ≥ ( 2 ⁢ m + 1 ) ⁢ δ .

Now we reformulate the problem in an inductive fashion. We would like to verify the following induction hypothesis.

Induction Hypothesis.

For a , k = 1 , … , m , there exists an open subset O I a , k ⊂ N I a , k which contains N I a , k ′ ¯ and multisections

t I a , k : O I a , k ⁢ → 𝑚 ⁢ E I a .

They satisfy the following conditions:

For all pairs I a ≼ I b , over a neighborhood of the compact subset

N I a , k ′ ¯ ∩ ϕ I b ⁢ I a - 1 ⁢ ( N I b , k ′ ¯ ) ⊂ U I b ⁢ I a ′ ¯ ⊂ U I b ⁢ I a

one has

(A.9) t I b , k ∘ ϕ I b ⁢ I a = ϕ ^ I b ⁢ I a ∘ t I a , k .
In a neighborhood of N I b ⁢ I a ′ ¯ , the value of t I b , k is contained in the subbundle E I a ; I b .
S I a + t I a , k is transverse.
∥ t I a , k ∥ C 0 ≤ 2 ⁢ k ⁢ δ .

It is easy to see that the case k = m implies this theorem. Indeed, (A.7) follows from (A.8) and the bound ∥ t I l , m ∥ ≤ m ⁢ δ . Now we verify the conditions of the induction hypothesis. For the base case, apply Lemma A.15 to

M = U I 1 , C = ∅ , D = U I 1 ,

we can construct a multisection t I 1 , 1 : U I 1 ⁢ → 𝑚 ⁢ E I 1 making S I 1 + t I 1 transverse with ∥ t I 1 ∥ ≤ δ . Now we construct t I a , 1 for a = 2 , … , m via a backward induction. Define

O I a , 1 = N I a , 1 = N I a ⁢ I 1 ∪ ⋃ I a ≺ I b ϕ I b ⁢ I a - 1 ⁢ ( N I b , 1 ) , a = 1 , … , m .

Then (A.9) determines the value of t I m , 1 over the set

ϕ I m ⁢ I 1 ⁢ ( U I m ⁢ I 1 ) ⊂ N I m , 1 = N I m ⁢ I 1 .

It is a closed subset of N I m , 1 , hence one can extend it to a continuous section of E I 1 ; I m which can be made satisfy the bound

∥ t I m , 1 ∥ ≤ ( 1 + 1 m ) ⁢ δ .

Suppose we have constructed t I a , 1 : O I a , 1 ⁢ → 𝑚 ⁢ E I a for all a ≥ l + 1 such that together with t I 1 , 1 they satisfy the induction hypothesis for k = 1 with the bound

∥ t I a , 1 ∥ ≤ ( 1 + m - l m ) ⁢ δ .

Then we construct t I l , 1 : O I l , 1 ⁢ → 𝑚 ⁢ E I l as follows. Given

z I l ∈ ϕ I l ⁢ I 1 ⁢ ( U I l ⁢ I 1 ) ∪ N I l , 1 + = ϕ I l ⁢ I 1 ⁢ ( U I l ⁢ I 1 ) ∪ ⋃ I l ≺ I b ϕ I b ⁢ I l - 1 ⁢ ( N I b , 1 ) ,

if z I l is in the first component, then define t I l , 1 ⁢ ( z I l ) by formula (A.9) for b = l , a = 1 . If z I l ∈ N I b ⁢ I l ∩ ϕ I b ⁢ I l - 1 ⁢ ( N I b , 1 ) for some b, then define

t I l , 1 ⁢ ( z I l ) = ϕ ^ I b ⁢ I l - 1 ⁢ ( t I b , 1 ⁢ ( ϕ I b ⁢ I l ⁢ ( z I l ) ) ) .

It is easy to verify using the Cocycle Condition that these definitions agree over some closed neighborhood of

ϕ I l ⁢ I 1 ⁢ ( U I l ⁢ I 1 ′ ¯ ) ∪ ⋃ I l ≺ I b ϕ I b ⁢ I l - 1 ⁢ ( N I b ⁢ I l ′ ¯ ) .

Then one can extend it to a continuous multisection of E I 1 ; I l satisfying the bound

∥ t I l , 1 ∥ ≤ ( 1 + m - l + 1 m ) ⁢ δ .

Then the induction can be carried on and stops until l = 2 , for which one has the bound

∥ t I 2 , 1 ∥ ≤ ( 1 + m - 1 m ) ⁢ δ ≤ 2 ⁢ δ .

The transversality of S I a + t I a , 1 for a ≥ 2 follows from the fact that t I a , 1 takes value in E I 1 ; I a , the fact that S I a | N I a ⁢ I 1 intersects with E I 1 ; I a transversely along ϕ I a ⁢ I 1 ⁢ ( U I a ⁢ I 1 ) , and the fact that S I 1 + t I 1 , 1 is transverse. Hence we have verified the k = 1 case of the induction hypothesis.

Suppose we have verified the induction hypothesis for k - 1 . For all a ≤ k - 1 , define

O I a , k = O I a , k - 1 , t I a , k = t I a , k - 1 .

The induction hypothesis implies that we have a multisection

t I k , k - 1 : O I k , k - 1 ⁢ → 𝑚 ⁢ E I k

such that S I k + t I k , k - 1 is transverse and ∥ t I k , k - 1 ∥ ≤ ( 2 ⁢ k - 2 ) ⁢ δ . Then apply Lemma A.15, one can obtain a multisection t I k , k defined over a neighborhood of U I k ′ ¯ = N I k , k ′ ¯ contained in U I k = N I k , k such that S I k + t I k , k is transverse, ∥ t I k , k ∥ ≤ ( 2 ⁢ k - 1 ) ⁢ δ , and t I k , k = t I k , k - 1 over a neighborhood of N I k , k - 1 ′ ¯ which is smaller than O I k , k - 1 . Then by the similar backward induction as before, using the extension property of continuous multi-valued functions, one can construct perturbations with desired properties. The remaining details are left to the reader. ∎

In our argument, condition (A.7) is crucial in establishing the compactness of the perturbed zero locus. In the situation of Theorem A.35, suppose a perturbation 𝔱 ′ is constructed over the shrinking 𝔄 ′ ⊏ 𝔄 ′′ . Then for a further precompact shrinking 𝔄 ′′ ⊏ 𝔄 ′ , (A.7) remains true (with U I ′ ¯ replaced by U I ′′ ¯ ).

Proposition A.36.

Let A be a good coordinate system on X and let A ′ ⊏ A be a precompact shrinking. Let N be a thickening of A . Equip each chart U I a distance function d I which induces the same topology. Then there exists ϵ > 0 satisfying the following conditions. Let t be a multi-valued perturbation of s which is N -normal. Suppose

(A.10) d I ⁢ ( ( S I + t I ) - 1 ⁢ ( 0 ) ∩ U I ′ ¯ , S I - 1 ⁢ ( 0 ) ∩ U I ′ ¯ ) ≤ ϵ for all ⁢ I ∈ ℐ .

Let s ′ be the induced multisection perturbation on A ′ . Then the zero locus ∥ ( s ′ ) - 1 ⁢ ( 0 ) ∥ is sequentially compact with respect to the subspace topology induced from | A ′ | .

Proof.

For each x ∈ | 𝔄 ′ | , consider the subset of indices I ∈ ℐ such that x is contained in π I ⁢ ( U I ′ ) . Because of the overlapping condition, there is a unique minimal in this subset, denoted by I x .

Claim.

Given I, there exists ϵ > 0 such that, for any perturbation s which satisfies condition (A.10), any sequence x i ∈ ∥ ( s ′ ) - 1 ⁢ ( 0 ) ∥ with I x i = I has a convergence subsequence.

Proof of the claim.

Suppose this is not true, then there exist a sequence ϵ k > 0 that converge to zero, a sequence of multi-valued perturbations 𝔱 k satisfying

d J ⁢ ( ( S J + t k , J ) - 1 ⁢ ( 0 ) ∩ U J ′ ¯ , S J - 1 ⁢ ( 0 ) ∩ U J ′ ¯ ) ≤ ϵ k for all ⁢ J ∈ ℐ ,

and sequences of points x ~ k , i ∈ ( S I + t k , J ) - 1 ⁢ ( 0 ) ∩ U I ′ such that the sequence x k , i = π I ⁢ ( x ~ k , i ) does not have a convergent subsequence. Since x ~ k , i ∈ U I ′ ¯ which is a compact subset of U I , for all k one can replace the sequence by a subsequence with a limit x ~ k ∈ ( S I + t k , I ) - 1 ⁢ ( 0 ) ∩ U I ′ ¯ . Then since ϵ k → 0 , the sequence x ~ k has a subsequential limit x ~ ∞ ∈ U I ′ ¯ ∩ S I - 1 ⁢ ( 0 ) . Denote

x ∞ = ψ I ⁢ ( x ~ ∞ ) ∈ ψ I ⁢ ( U I ′ ¯ ∩ S I - 1 ⁢ ( 0 ) ) = F I ′ ¯ ⊂ F I ⊂ X .

We claim that x ∞ ∉ F I ′ . Indeed, if x ∞ ∈ F I ′ , then x ~ ∞ = ψ I - 1 ⁢ ( x ∞ ) ∈ U I ′ which is an open subset of U I . Then for k sufficiently large, x ~ k ∈ U I ′ . Since x ~ k is the limit of x ~ k , i , this contradicts the assumption that x k , i has no convergent subsequence. Therefore x ∞ ∉ F I ′ .

On the other hand, since all F J ′ cover X, there exists J ∈ ℐ such that x ∞ ∈ F J ′ . Then by the Overlapping Condition of the atlas 𝔄 ′ , we have either I ≼ J or J ≼ I but J ≠ I . We claim that the latter is impossible. Indeed, if x ∞ = ψ J ⁢ ( y ~ ∞ ) with y ~ ∞ ∈ U J ′ , then we have y ~ ∞ ∈ U I ⁢ J ′ and x ~ ∞ ∈ φ I ⁢ J ′ ⁢ ( U I ⁢ J ′ ) . Then for k sufficiently large, we have x ~ k in N J ⁢ I . Fix such a large k, then for i sufficiently large, we have x ~ k , i ∈ N J ⁢ I . However, since the perturbation 𝔱 is 𝒩 -normal, it follows that x ~ k , i ∈ φ I ⁢ J ′ ⁢ ( U I ⁢ J ′ ) . This contradicts the assumption that I x k , i = I . Therefore I ≼ J . Then there is a unique y ~ ∞ ∈ U J ′ ∩ S J - 1 ⁢ ( 0 ) such that ψ J ′ ⁢ ( y ~ ∞ ) = x ∞ and y ~ ∞ = φ J ⁢ I ⁢ ( x ~ ∞ ) . Then x ~ ∞ ∈ U J ⁢ I . Therefore, for k sufficiently large, we have x ~ k ∈ U J ⁢ I and we have the convergence

y ~ k := φ J ⁢ I ⁢ ( x ~ k ) → y ~ ∞

since φ J ⁢ I is continuous. Since y ~ ∞ ∈ U J ′ which is an open subset of U J , for k sufficiently large, we have y ~ k ∈ U J ′ . Fix such a large k. Then for i sufficiently large, we have

x ~ k , i ∈ U J ⁢ I ∩ φ J ⁢ I - 1 ⁢ ( U J ′ ) ∩ U I ′ .

Hence we have y ~ k , i := φ J ⁢ I ′ ⁢ ( x ~ k , i ) ∈ U J ′ and

lim i → ∞ ⁡ y ~ k , i = y ~ k .

Since the map U J ′ → | 𝔄 ′ | is continuous, we have the convergence

lim i → ∞ ⁡ x k , i = lim i → ∞ ⁡ π 𝔄 ′ ⁢ ( y ~ k , i ) = π 𝔄 ′ ⁢ ( y ~ ∞ ) .

This contradicts the assumption that x k , i does not converge for all k. Hence the claim is proves. ∎

Now for all I ∈ ℐ , choose the smallest ϵ such that the condition of the above claim hold. We claim this ϵ satisfies the condition of this proposition. Indeed, let 𝔱 be such a perturbation and let x k be a sequence of points in ∥ ( 𝔰 ′ ) - 1 ⁢ ( 0 ) ∥ . Then there exists an I ∈ ℐ and a subsequence (still indexed by k) with I x k = I . Then by the above claim, x k has a subsequential limit. Therefore ∥ ( 𝔰 ′ ) - 1 ⁢ ( 0 ) ∥ is sequentially compact. ∎

A.7 The virtual cardinality

In the application of the virtual technique in this paper, we only consider the situation when the index of the problem is either zero or one. This is the situation when the virtual cardinality is defined and the independence of perturbation is proved. In the companion paper [49], we will further restrict to the situation when all charts are manifolds.

We first define the virtual cardinality. Let 𝔄 be a good coordinate system of virtual dimension zero and let 𝔱 be a transverse multi-valued perturbation for which the perturbed zero locus is sequentially compact (and hence finite). Let x ∈ | 𝔄 | be a point in the perturbed zero locus. Choose a representative x ~ I ∈ U I of x. Over U I one has a perturbed multisection s ~ I : U I ⁢ → 𝑚 ⁢ E I . By Definition A.14, there exist a local bundle chart ( U ~ , ℝ n , Γ , φ ^ , φ ) of E I → U I near x ~ I and an integer l such that with respect to the bundle chart, the multisection S I + t I restricted to this chart is identified with a liftable l-multimap

( S I + t I ) ⁢ ( y ) = [ f I 1 ⁢ ( y ) , … , f I l ⁢ ( y ) ] for all ⁢ y ∈ U ~ .

We may assume that U ~ is an open ball of ℝ n containing the origin and x ~ I ∈ U I is identified with the orbit of the origin. We would like to associate to each i a number ϵ i ∈ { 1 , - 1 , 0 } using the orientation. If f I i ⁢ ( 0 ) ≠ 0 , then define ϵ i = 0 . If f I i ⁢ ( 0 ) = 0 , then transversality implies that f I i is a local homeomorphism between open balls of ℝ n . We define ϵ i = ± 1 as the degree of this local homeomorphism with respect to the orientation of the bundle E I → U I . Then define the multiplicity of this zero x by

m ⁢ ( x ) = 1 l ⁢ ∑ i = 1 l ϵ i .

One can verify that m ⁢ ( x ) is independent of the local representative of s ~ I : U I ⁢ → 𝑚 ⁢ E I . Moreover, because the coordinate changes are orientation preserving, m ⁢ ( x ) is also independent of the representative x ~ I ∈ U I of x. Then we define the virtual cardinality of 𝔄 by

# ⁢ 𝔄 = ∑ x ∈ | 𝔰 - 1 ⁢ ( 0 ) | m ⁢ ( x ) ∈ ℚ .

The compactness of ∥ 𝔰 - 1 ⁢ ( 0 ) ∥ implies that this sum is finite and hence # ⁢ 𝔄 is a finite number. Moreover, if the charts are all manifolds, then transversality can be achieved by single-valued perturbations. Therefore, in that case, the virtual cardinality is an integer.

A.8 Invariance of the virtual cardinality

We expect that the virtual cardinality is independent of various choices, especially the multisection perturbation. To prove this kind of result we need to consider the virtual dimension one case and a certain cobordism argument. The cobordism argument is essentially the same as the arguments used in other versions of the virtual technique, and the relaxation to the topological category does not create more complexity. Here we construct a singular chain supported in the perturbed zero locus and show that its boundary agrees with the perturbed zero locus on the boundary.

We first specify more notations. Let 𝔄 be a virtual orbifold atlas with boundary of dimension 1 on the space X. Then for each chart C I = ( U I , E I , S I , ψ I , F I ) , U I is an orbifold with boundary. The coordinate changes map boundaries to boundaries. Hence there is a well-defined closed subset ∂ ⁡ X ⊂ X and a virtual orbifold atlas (without boundary) of dimension zero on ∂ ⁡ X , denoted by ∂ ⁡ 𝔄 , whose charts are restrictions of C I to ∂ ⁡ U I . Moreover, ∂ ⁡ 𝔄 is a good coordinate system. Suppose we are given a transverse multisection perturbation ∂ ⁡ 𝔰 on ∂ ⁡ 𝔄 with a compact zero locus | ( ∂ ⁡ 𝔰 ) - 1 ⁢ ( 0 ) | so the virtual cardinality can be defined. In fact, we essentially constructed a singular 0-chain (cycle) Z ∂ ⁡ 𝔰 ∈ C 0 ⁢ ( | ( ∂ ⁡ 𝔰 ) - 1 ⁢ ( 0 ) | ; ℚ ) with rational coefficients. We would like to show that this chain is homologous to zero.

Lemma A.37.

There exist a transverse multisection perturbation s on A which extends ∂ ⁡ s and a singular 1-chain Z s with rational coefficients in | s - 1 ⁢ ( 0 ) | such that

(A.11) ∂ ⁡ Z 𝔰 = Z ∂ ⁡ 𝔰 ∈ C 0 ⁢ ( | 𝔰 - 1 ⁢ ( 0 ) | ; ℚ ) .

Proof.

We first prove the following claim.

Claim.

For each I ∈ I , there exist a finite collection of bundle charts

( U ~ I , α , ℝ n I , Γ I , α , φ ^ I , α , φ I , α ) , α = 1 , … , k I ,

of the bundle E I → U I satisfying the following conditions:

Denote U I , α = U ~ I , α / Γ I , α . Then U I , α is precompact in U I and U I , α ¯ is a compact orbifold with boundary and codimension two corner.
As subsets of | 𝔄 | , both X and | ( ∂ ⁡ 𝔰 ) - 1 ⁢ ( 0 ) | are contained in the union of the images of U I , α → U I → | 𝔄 | .

Moreover, one can construct a transverse multisection perturbation s on A which extends ∂ ⁡ s and which satisfies the following conditions:

| 𝔰 - 1 ⁢ ( 0 ) | is contained in the union of the images of U I , α → U I → | 𝔄 | .
The multimap s ~ I , α : U ~ I , α ⁢ → 𝑚 ⁢ ℝ n I is liftable. The induced multisection of E I | U I , α ¯ is transverse to the boundary and the corner and in particular non-vanishing on the corner.

Proof of the claim.

The existence of the finite collection of bundle charts follows from the compactness of X and the perturbed zero locus in | ∂ ⁡ 𝔄 | . Moreover, when we construct 𝔰 inductively, within each bundle chart one can make the perturbation liftable and transverse to the boundary and corner. ∎

Now we can define the singular 1-chain Z 𝔰 . Without loss of generality, we assume that for each chart C I , one only needs to choose one bundle chart ( U ~ I ′ , ℝ n I , Γ I , φ ^ I , φ I ) in the above claim such that U I ′ := U ~ I / Γ I is precompact in U I and its closure is an orbifold with boundary and codimension two corner. The construction in the general case can be carried out similarly. Start from an element I ∈ ℐ . Then the perturbed multisection restricted to U I ¯ can be represented by a liftable transverse multimap

s ~ I = [ f I 1 , … , f I l ] .

Then since each f I i is transverse both in the interior and on the boundary, one get compact oriented 1-manifolds with boundary

Z I i := ( f I i ) - 1 ⁢ ( 0 ) ⊂ U ~ I ′ ¯ .

Via the map U ~ I ′ ¯ → U I → | 𝔄 | , these oriented 1-manifolds with boundary define a singular 1-chain in | 𝔰 - 1 ⁢ ( 0 ) | . By abuse of notation still denote these chains by Z I i . We define

Z I = 1 l ⁢ ∑ i = 1 l Z I i .

Now we consider the compact set

| 𝔰 - 1 ⁢ ( 0 ) | ∖ π I ⁢ ( ( S I + t I ) - 1 ⁢ ( 0 ) ∩ U I ′ ) ⊂ | 𝔰 - 1 ⁢ ( 0 ) | .

This subset can be covered by one fewer charts. Choose another J ≠ I and consider the preimage of | 𝔰 - 1 ⁢ ( 0 ) | ∖ π I ⁢ ( ( S I + t I ) - 1 ⁢ ( 0 ) ∩ U I ′ ) under the map π J : U J ′ ¯ → | 𝔄 | in the U J ′ ¯ . Then there exists compact oriented 1-manifold with boundary

Z J j ⊂ U ~ J ′ ¯ , j = 1 , … , k ,

such that

⋃ j = 1 k Z J j / Γ J = π J - 1 ⁢ ( | 𝔰 - 1 ⁢ ( 0 ) | ∖ π I ⁢ ( ( S I + t I ) - 1 ⁢ ( 0 ) ∩ U I ′ ) ) ⊂ U J ′ ¯ .

Define Z J to be a suitable weighted sum of Z J j . Then inductively one can define

Z 𝔰 = ∑ I Z I ∈ C 0 ⁢ ( | 𝔰 - 1 ⁢ ( 0 ) | ; ℚ ) .

We verify (A.11). Consider a 0-simplex that can appear in ∂ ⁡ Z 𝔰 which is a point either in | ( ∂ ⁡ 𝔰 ) - 1 ⁢ ( 0 ) | or in the interior. If this 0-simplex is in the interior, then the matching condition for multisection perturbations with respect to coordinate changes implies that the coefficient of this 0-simplex is zero. On the other hand, the coefficients of a 0-simplex in | ( ∂ ⁡ 𝔰 ) - 1 ⁢ ( 0 ) | obviously match the coefficients in Z ∂ ⁡ 𝔰 . Hence (A.11) is verified. ∎

Lemma A.37 implies the following corollary.

Corollary A.38.

Let A be an oriented virtual orbifold atlas (without boundary) of dimension zero on X. Then the virtual cardinality is independent of the choice of transverse multisection perturbations.

Acknowledgements

The second named author would like to thank Huai-Liang Chang, Wei Gu, Alexander Kupers, Mauricio Romo, and Mohammad Tehrani for helpful discussions.

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Received: 2017-03-14

Revised: 2020-01-20

Published Online: 2020-07-11

Published in Print: 2021-02-01

Virtual cycles of gauged Witten equation

Abstract

A Topological virtual orbifolds and virtual cycles

A.1 Topological manifolds and transversality

Definition A.1 (Topological manifolds and embeddings).

A.1.1 Microbundles

Definition A.2 (Microbundles).

Theorem A.3 (Kister–Mazur theorem).

Definition A.4 (Normal microbundles).

A.1.2 Transversality

Definition A.5 (Microbundle transversality).

Remark A.6.

Theorem A.7 (Topological transversality theorem).

Remark A.8.

Theorem A.9.

Proof.

A.2 Topological orbifolds and orbibundles

Definition A.10.

Definition A.11.

A.2.1 Embeddings

Definition A.12 (Orbifold embedding).

A.2.2 Multisections and perturbations

Definition A.13 (Multimaps).

Definition A.14 (Multisections).

Lemma A.15.

Proof.

A.3 Virtual orbifold atlases

Definition A.16.

Lemma A.17.

Definition A.18.

Lemma A.19.

Definition A.20.

Lemma A.21.

Proof.

Definition A.22.

A.3.1 Orientations

Definition A.23 (Orientability).

A.4 Good coordinate systems

Definition A.24.

Definition A.25.

Theorem A.26 (Constructing good coordinate system).

Remark A.27.

A.5 Shrinking good coordinate systems

Lemma A.28.

Proof.

Lemma A.29.

Lemma A.30.

Proof.

Lemma A.31.

Proof.

Proof of Lemma A.29.

Claim.

Proof of the claim.

A.6 Perturbations

Definition A.32.

Definition A.33.

Definition A.34.

Theorem A.35.

Proof.

Induction Hypothesis.

Proposition A.36.

Proof.

Claim.

Proof of the claim.

A.7 The virtual cardinality

A.8 Invariance of the virtual cardinality

Lemma A.37.

Proof.

Claim.

Proof of the claim.

Corollary A.38.

Acknowledgements

References

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