A geometric approach to equivariant factorization homology and nonabelian Poincaré duality

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an EV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{E}_V$$\end{document}-algebra using a two-sided bar construction, generalizing (Andrade, From manifolds to invariants of En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_n$$\end{document}-algebras. PhD thesis, Massachusetts Institute of Technology, 2010; Kupers and Miller, Math Ann 370(1–2):209–269, 2018). This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincaré duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds, when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} in them, the scanning map gives group completions. This generalizes the recognition principle for V-fold loop spaces in Guillou and May (Algebr Geom Topol 17(6):3259–3339, 2017).

Non-equivariantly, factorization homology has multiple origins.The most well-known approach started in Beilinson-Drinfeld's study of an algebraic geometric approach to conformal field theory [BD04] under the name of chiral homology.Lurie [Lur,5.5]and Ayala-Francis [AF15] introduced and extensively studied the algebraic topology analogue, named as factorization homology.This route relies heavily on ∞-categorical foundations.An alternative geometric model is Salvatore's configuration spaces with summable labels [Sal01].This construction is close to the geometric intuition, but is not homotopical.Yet another model, using the bar construction and developed by Andrade [And10], Miller [Mil15] and Kupers-Miller [KM18], is homotopically wellbehaved while staying close to the geometric intuition of configuration spaces.It is this last approach that we will generalize equivariantly.
To give context, we first give an introduction to this approach to non-equivariant factorization homology.It is a classical theorem by Dold-Thom [DT58] that the ordinary integral homology groups of a connected space M are exactly the homotopy groups the configuration space on M with summable labels in N, the commutative monoid of natural numbers.Salvatore [Sal01] observed that one can form the configuration space on M with summable labels in an E n -algebra A, which has less structure than a commutative monoid, if the space M has the structure of a framed smooth manifold of dimension n, because the local Euclidean chart of M offers the way to sum the labels in the E n -algebra A. In [And10,KM18], the authors used this idea and defined the factorization homology of a framed smooth manifold M with coefficients in an E n -algebra A to be the two sided bar construction (1.1) where D n is the monad associated to the little n-disks operad and D M is a certain functor associated to embeddings of disks in M.
This bar construction definition (1.1) is a concrete point-set level model of the ∞categorical definition of [Lur,AF15].One can construct a topological category Mfld fr n of framed smooth n-dimensional manifolds and framed embeddings, which is a common ground for both definitions.It is a symmetric monoidal category under disjoint unions.Let Disk fr n be the full subcategory spanned by objects equivalent to ∐ k R n for some k ≥ 0. An E n -algebra A can be viewed as a symmetric monoidal topological functor out of Disk fr n .The ∞-categorical factorization homology [AF15, definition 3.2] is the derived symmetric monoidal topological left Kan extension of A along the inclusion: (1.2) Horel [Hor17,7.7]showed the equivalence of (1.1) and (1.2).
1.2.The definition of equivariant factorization homology.We fix an integer n and a finite group G throughout.An equivariant version of an E n -algebra is an E Valgebra, where E V is a monad associated to a G-operad that is equivalent to the little Vdisks operad D V (see Section 3.4).The E V -algebras give the correct concrete coefficient input of equivariant factorization homology on V -framed smooth G-manifolds.Here, a smooth G-manifold M is V -framed if there is a trivialization of its tangent bundle.
In line with (1.1), we define the equivariant factorization homology of a V -framed smooth G-manifold M with coefficients in an E V -algebra A to be (Definition 3.14): (1.4) Remark 1.5.As will be made clear in [KMZ], where . This bar construction is a cofibrant replacing of A in D fr V V -algebra, and thus the equivariant factorization homology could be understood as first taking a cofibrant replacement, and then extending from local to global by tensoring with D fr V M over D fr V V .
We explain the definition (1.4) in a conveniently generalized context.A tangential structure is a G-map θ : B → B G O(n) for some well-chosen G-space B1 .A morphism of two tangential structures is a G-map over B G O(n).All tangential structures form a category T S, which is simply the over category GTop/ B G O(n) .
Denote by ζ n the universal G-n-vector bundle over B G O(n). Pulling back along θ gives a bundle θ * ζ n over B. A θ-framing on a smooth G-manifold M is an equivariant bundle map φ M : TM → θ * ζ n .The G-manifold M has a θ-framing if and only if the classifying map of its tangent bundle τ : M → B G O(n) factors up to G-homotopy through θ : B → B G O(n).Indeed, a θ-framing on M is the same data as a map τ B : M → B plus a homotopy between the two classifying maps τ and θ • τ B from M to B G O(n) (see Corollary B.10 with Definition B.4).The V -framing (1.3) is a special case: it is fr V -framing for a particular tangential structure fr V : * → B G O(n).
In Section 3.1, we construct a GTop-enriched category Mfld θ G,n , the category of smooth n-dimensional θ-framed G-manifolds and θ-framed embeddings.In particular, there is the category of V -framed smooth G-manifold Mfld fr V G,n .It takes some effort to define the morphisms in the category.For example, V -framed embeddings between little V -disks should be just the linear embeddings in the definition of the little V -disks operad.However, we do not have the notion of linear embeddings between general V -framed manifolds.The solution is to allow all embeddings and to add in path data to correct the homotopy type, so that we do not see the unwanted rotations.This idea goes back to Steiner [Ste79] and was used non-equivariantly by Andrade [And10] and Kupers-Miller [KM18].Using paths in the framing space, we define the θ-framed embedding space of θ-framed manifolds (Definition 3.6).This construction is covariant as a functor of θ.
In Section 3.2, we use the GTop-enriched category Mfld fr V G,n to build the V -framed factorization homology by the bar construction (1.4).The representation V can be viewed as a G-manifold with a canonical V -framing, so each ∐ k V also has a canonical V -framing.Let Λ be the category of based finite sets k = {0, 1, 2, • • • , k} with base point 0 and based injections.For any ) gives a functor Λ op → GTop.Such functors E : Λ op → GTop and their associated functors E : GTop * → GTop * (Construction 2.4) give a convenient context for reduced operads and monads, which we explain in Section 2.1.
Taking M = V , compositions in Mfld fr V G,n equip the sequence D fr V V with the structure of a reduced G-operad.It is the endomorphism operad of the object V .Moreover, it is equivalent to the little V -disks operad D V (Proposition 3.33), so it is an E V -operad.The functors associated to M , and thus (1.4) makes sense for a D fr V V -algebra A. For a tangential structure θ so that V is θ-framed (possible under the conditions on θ prescribed in Proposition 3.10), one can define the θ-framed equivariant factorization homology with coefficient in a D θ V -algebra A as Specializing to θ = fr V , (1.6) gives (1.4).This construction is homotopically wellbehaved.
Proposition 1.7.(Proposition 3.15).The functor 1.3.Main results.In Section 3.3, we prove that the embedding space in Mfld fr V G,n has a close connection to the configuration space.
Proposition 1.8.(Proposition 3.30) Evaluating at 0 of the embedding gives a (G×Σ k )homotopy equivalence: Here, F M (k) is the ordered configuration space of k points in M.This is used to justify that We also prove an invariance result in the equivariant setting.Such a result is known non-equivariantly [AF15, Proposition 3.9] and expected equivariantly.
Theorem 1.9.(Theorem 3.20) Let q : θ 1 → θ 2 be a morphism of tangential structures and V be θ 1 -framed.We also write V for the θ 2 -framed G-manifold q * V .Then for a θ 1 -framed G-manifold M and a D θ 2 V -algebra A, there is a G-equivalence Due to the invariance, we may drop the θ from the notation θ when the context is clear.
The bar construction definition (1.6) stays close to the geometric origin, which readily leads to proofs of the following results using classical techniques.

A.
In Section 4, we prove that our definition satisfies the following theorem.
Theorem 1.11.(Theorem 4.7 and Theorem 4.41) Let M be a V -framed manifold and A be a D fr V V -algebra in GTop.There is a G-map: Here, M + is the one-point compactification of M; B V A is a model for the V -fold deloop of A defined in Section 4.2.
In Theorem 1.11, part (1) is an equivariant version of the nonabelian Poincaré duality theorem due to several authors, including [Sal01, Theorem 6.6] and [Lur,5.5.6.6];specializing to M = V in Theorem 1.11, it recovers the equivariant recognition principle of [GM17, Theorem 1.14].In particular, if the E V -algebra A is grouplike, then A ≃ Ω V B V A. This justifies the definition of B V A.
Corollary 1.12.Let M and A be as in Theorem 1.11 and A be G-connected.

Then we have
The map p M in the eNPD theorem is induced by a scanning map, a natural transformation of right D fr V V -functors: The scanning map has been studied in various forms in [McD75,BM88,MT14].In particular, Rourke-Sanderson [RS00] proved that McDuff's scanning map is a weak G-equivalence on G-connected objects.Classically, given a configuration of k points in M, regarded as an embedding of k to M, the Pontryagin-Thom collapse gives an element of Map * (M + , ∨ k S n ).Note that the i-th wedge component S n is in fact the fiber at the image of i ∈ k of the sphere bundle Sph(TM).The scanning map pushes the target further to the codomian Section c (M, Sph(TM)) independent of k, so that the individual Pontryagin-Thom maps vary continuously for the configurations.To do this, one needs an identification of the normal bundle of the embedded points with the tangent bundle of the manifold.There are conceptually two ways to do this: to use geodesics to generate a canonical local vector field ( [McD75]), or to fatten the configuration space to include the data of a tubular neighborhood ( [MT14]).
In the V -framed case, we can give an easy definition of the scanning map (4.2).In Appendix A, we compare our scanning map to the scanning maps in the literature.In particular, we prove in Proposition A.10 that equivariant versions of the scanning maps in [McD75] and [MT14] are homotopic, which is claimed without proof in [MT14, Remark 3.2].
Our proof of eNPD has two steps.We sketch it out when A is G-connected.The first step is to use the scanning map (1.13).It assembles to a simplicial map Using the Rourke-Sanderson result, the induced map on the geometric realization is a weak G-equivalence The second step is to pull the M + out of the geometric realization.The map is a G-equivalence only when K • satisfies some connectivity conditions.Non-equivariantly, for M = R so that M + = S 1 , a sufficient connectivity condition is given in [May72,Theorem 12.3].Let ν be a function from the conjugacy classes of subgroups of G to Z ≥−1 .We say a finite-dimensional based G-CW complex X has cell dimension ν if its cells in the form of G/H × D n have highest dimension ν(H).We define the function dim(X) to be dim(X)(H) = max H⊂L ν(L).
Combining the non-equivariant result with induction shows: When A is G-connected, the K • constructed out of it satisfies this connectivity condition, so the eNPD theorem follows.
1.4.Comparison to other work.In this paper we give a homotopical point set definition of equivariant factorization homology generalizing [And10]. 2 There are axiomatic approaches to ∞-categorical equivariant factorization homology [Hor19, Wee20] using G-∞-categories and ∞-G-categories respectively.Our definition and [Wee20], being generalizations of (1.1) and (1.2) respectively, are equivalent. 3The definition of equivariant factorization homology in [Hor19] is called "genuine", meaning that it considers H-manifolds for all subgroups H ⊂ G. Restricted to G-manifolds, a theory of [Hor19] gives a theory of [Wee20].
In joint work with Horev and Klang [HHK + 20], the author studies equivariant factorization homology of Thom G-spectra in the context of [Hor19].There, a very different proof of the eNPD theorem adapted to the ∞-categorical context is given, generalizing Corollary 4.6 of [AF15].The alternative proof is an axiomatic one, based on equivariant handle-body decompositions of the G-manifold M. In contrast, we provide a geometrically-seen scanning map that gives the equivalence in this paper.The scanning map was used to prove homological stability properties of non-equivariant configuration spaces and factorization homology in [McD75,Mil15,KM18].The approach in our paper should lead to equivariant stability results.
Another advantage of our approach to the equivariant factorization homology and the eNPD theorem is that it gives a simplicial filtration on the mapping space Map * (M + , Y ) (taking A = Ω V Y ), thus offering a spectral sequence.It could be useful for obtaining 2 Note that [And10] is non-equivariant: their G in Emb G is a subgroup of GL n (R) and therefore refers to a tangential structure θ : BG → BGL n (R).
3 In [Wee20], their G is our θ : BG → BO(n); their Γ is our G; their ρ is our V ; their Γ ρ Orb G n is our Mfld frV G,n with the adjustment that the morphisms are replaced by the G-fixed points of the morphisms; their Γ ρ Disk G n -algebra is defined in a symmetric monoidal category C whose objects do not necessarily have G-actions, and a D frV equivariant generalizations of [CT88].However, as computations of equivariant homology of the free E V -algebra on A, H G ⋆ (D fr V V A), and in general, H G ⋆ (D fr V M A), remains open for any coefficients, this computational tool has not yet been explored.
Our definition of Mfld θ G,n in Section 3.1 is closely related to Ayala-Francis [AF15], which we compare in Appendix B. For the trivial tangential structure id : induced by the map tangential structure θ → id.We also identify the automorphism G-space Emb θ (V, V ) in Theorem B.15. 1.5.Notations.
• GTop is the Top-enriched category of G-spaces and G-equivariant maps.
• Top G is the GTop-enriched category of G-spaces and non-equivariant maps where G acts by conjugation on the mapping space.For a space M and a fiber bundle E → M, • F M (k) is the ordered configuration space of k points in M.
• F E↓M (k) is the ordered configuration space of k points in E whose images are k distinct points in M.

Preliminaries on operads and equivariant bundles
2.1.Λ-sequences and operads.To streamline the monadic bar construction in the main body, we use an elementary categorical framework of Λ-objects.This framework is studied in more detail in a paper with May and Zhang [MZZ20].This subsection is a summary of the relevant content towards Example 2.10 and Proposition 2.11, which are used in later sections.
Let Λ be the category of based finite sets k = {0, 1, 2, • • • , k} with base point 0 and based injections.The morphisms of Λ are generated by permutations and the ordered injections s k i : k − 1 → k that skip i for 1 ≤ i ≤ k.It is a symmetric monoidal category with wedge sum as the symmetric monoidal product.
For a symmetric monoidal category (V , ⊗, I), let V I be the category under the unit.In [MZZ20], V is more general, but here we will work only with the Cartesian monoidal category (GTop, ×, * ).The empty G-space ∅ is an initial object.
where morphisms are natural transformations of functors.The category of all unital Λ-sequences in GTop is denoted Λ op * [GTop], where morphisms are natural transformations of functors that are identity at level zero.
The category Λ op [GTop] admits a symmetric monoidal structure (Λ op [GTop], ⊠, I 0 ).Here, ⊠ is the Day convolution of functors on the closed symmetric monoidal category Λ op .The unit is given by The symmetric monoidal product ⊠ on Λ op [GTop] induces a symmetric monoidal product on Λ op [GTop] I 0 and Λ op * [GTop], which we still denote by ⊠.
The categories Λ op [GTop] I 0 and Λ op * [GTop] admit a second (nonsymmetric) monoidal product ⊙ in addition to ⊠, called the circle product.It is analogous to Kelly's circle product on symmetric sequences [Kel05].The unit for ⊙ is given by where the only non-trivial morphism I 1 (1) → I 1 (0) is the identity.For a brief definition of ⊙, see Construction 2.6 (2).An operad in GTop, as defined in [May97], gives an example of a symmetric sequence in GTop.If the operad is unital, meaning the 0-space of the operad is the unit, it has the structure of a Λ-sequence in GTop.A unital operad in Top or GTop, is also called a reduced operad in [May97].In fact, we have the unital variant of Kelly's observation [Kel05]: We give a construction which will be used in the definition of equivariant factorization homology: the associated functor of a unital Λ-sequence.This construction specializes to the monad associated to a reduced operad of [May97]; it also appears in the definition of the circle product ⊙.Assume that (W , ⊗, J ) is a cocomplete symmetric monoidal category tensored over GTop.
Construction 2.3.Let X ∈ W J be an object under the unit.Define X * : Λ → W to be the covariant functor that sends n to X ⊗n .On morphisms, it sends the permutations to permutations of the X's and sends the injection s k where η : J → X is the unit map of X.By convention, X ⊗0 = J .
This defines a functor (−) * : W J → Fun(Λ, W ). Then one can form the categorical tensor product over Λ of the contravariant functor E and the covariant functor Remark 2.5.It is sometimes useful to take the quotient in two steps and use the following alternative formula for E: x ∈ X ⊗k−1 .We will use ≈ or ∼ for the equivalence relation to be clear which formula we are using and refer to ∼ as the base point identification.
Construction 2.6.We focus on the following context of Construction 2.4.
(2) Let W = (Λ op [GTop], ⊠, I 0 ) with the Day monoidal structure.Then W is tensored over GTop in the obvious way by levelwise tensoring.One gets the circle product for These two cases are further related: the 0-th level functor gives an inclusion of a full symmetric monoidal subcategory, so we have In words, the reduced monad construction is what happens at the 0-space of the circle product.Using this, one can show: A monad is a monoid in the functor category.Using the associativity of the circle product and (2.7), it is easy to prove that when C is an operad, the associated functor C is a monad; and that when F is a left/right module over the monoid C in (Λ op * [GTop], ⊙), the associated functor F is a left/right module over C. The following construction gives examples.Construction 2.9.([MZZ20, Section 8]) Suppose that we have a GTop-enriched symmetric monoidal category (W , ⊗, J ) such that W (J , Y ) ∼ = * for all objects Y of W . Then we can construct a Λ op * [GTop]-enriched category H W .The objects are the same as those of W , while the enrichment is given by The definition of the composition in H W is similar to the structure maps of an endomorphism operad.So, for any objects V is a monad and D θ M is a right module over D θ V .We will use that the circle product is strong symmetric monoidal in the first variable: That is, the circle product distributes over the Day convolution: for any D, D ′ ∈ Λ op (GTop) I 0 , we have 2.2.Equivariant bundles.As pointed out in the introduction, we define θ-framed embeddings using maps between equivariant bundles.In this subsection, we list some preliminary results on equivariant vector bundles for the reader's reference.The proofs of the results as well as a clarification of different notions of equivariant fiber bundles can be found in [Zou21].
Let G and Π be compact Lie groups, where G is the ambient action group and Π is the structure group.
Definition 2.12.A G-n-vector bundle a map p : E → B such that the following statements hold: (1) The map p is a non-equivariant n-dimensional vector bundle; (2) Both E and B are G-spaces and p is G-equivariant; (3) The G-action is linear on fibers.
Definition 2.13.A principal G-Π-bundle is a map p : P → B such that the following statements hold: (1) The map p is a non-equivariant principal Π-bundle; (2) Both P and B are G-spaces and p is G-equivariant; (3) The actions of G and Π commute on P .
Theorem 2.15.There is an equivalence of categories between {G-n-vector bundles over B} and {principal G-O(n)-bundles over B}.
The classical procedure of passing from n-vector bundles to principal O(n)-bundles is called taking the space of admissible maps.The equivariant bundles mentioned are both just non-equivariant bundles with G-actions, and the classical procedure is compatible with the G-actions.
or V -framing of the bundle.This is analogous to the case of nonequivariant vector bundles, except that equivariance adds in the representation V that's part of the data.However, the representation V in the equivariant trivialization of a fixed vector bundle may not be unique.(1) Let G = C 2 , σ be the sign representation.The unit sphere, S(2σ), is S 1 with the 180 degree rotation action.As C 2 -vector bundles, (2) Take V and W to be any two representation of G that are of the same dimension and take B to have free G-action.
We do have the uniqueness of V in the following case ([Zou21, Corollary 3.2]).
Equivariantly, G-representations serve the role of R n .So it is natural to consider the V -framing bundle Fr V (E) for an orthogonal n-dimensonal representation V .
Definition 2.18.Let p : E → B be a G-n-vector bundle.Let Fr V (E) be the space of admissible maps with the G-action g(ψ) = gψρ(g) −1 .
In other words, Fr V (E) has the same underlying space as Fr R n (E), but we think of admissible maps as mapping out of V instead of R n .
Let H ⊂ G be a subgroup and Rep(H, Π) be the set: Denote the centralizer of the image of ρ in Π by Z Π (ρ).It is a closed subgroup of Π, and we define Furthermore, for any fixed representative ρ, p −1 (B 0 ) This is essentially [LM86, Theorem 12] and is explained in [Zou21, Section 2.6].Note that a principal G-Π-bundle morphism preserves the associated homomorphism There is a notion of the universal G-Π-bundle E G Π → B G Π, so that principal G-Πbundles over a base G-space B are classified by G-homotopy classes of maps from B to B G Π. We denote the universal G-n-vector bundle by The G-homotopy type of the universal base can be obtained from information about the fixed-point spaces of total space.We have Theorem 2.20.([Las82, Theorem 2.17]) Here, O(V ) is the space of isometric self maps of V with G acting by conjugation.
One can make explicit the classifying maps of V -trivial bundles as follows.A G-map θ : * → B G O(n) gives the following data: it lands in one of the G-fixed components of Proposition 2.22.The pullback of the universal bundle along this map is exactly , is a G-space with the pointwise G-action on the loops.Via concatenation of loops, it is an A ∞ -algebra in G-spaces.Using the Moore loop space Definition 2.23.A G-monoid is a monoid in G-spaces, that is, an underlying monoid such that the multiplication is G-equivariant.A morphism of G-monoids is an equivalence if it is a weak G-equivalence.
Theorem 2.24.([Zou21, Theorem 3.12]) Let b be a fixed point in the (1) There is a G-homotopy equivalence The equivalence of G-monoids is explicitly given by a zigzag (see Remark B.17).Theorem 2.24 is used in Theorem B.15 to understand the automorphism space of a framed disk V .Recall that ζ n is the universal G-n-vector bundle over B G O(n). Pulling back along the tangential structure θ : B → B G O(n) gives a bundle θ * ζ n over B. This is meant to be the universal θ-framed vector bundle.For an n-dimensional smooth G-manifold M, the tangent bundle of M is a G-n-vector bundle.It is classified by a G-map up to G-homotopy: A θ-framing on a smooth G-manifold M is a θ-framing φ M on its tangent bundle.We abuse notations and refer to the map on the base spaces as φ M as well.
Note that for a manifold M to be θ-framed, it must be of dimension n.We consider the empty set to be uniquely θ-framed for any n and any θ : A bundle has a θ-framing if and only if its classifying map τ : Example 3.3.As seen in Proposition 2.22, the tangential structure fr V : * → B G O(n) characterizes V -trivializations.We call it the V -framing tangential structure, and emphasize that is not only a space B = * but also a map fr V .
Definition 3.4.Given two θ-framed bundles E 1 , E 2 with framings φ 1 , φ 2 , the space of θ-framed bundle maps between them is defined as: We use the following model for the homotopy fiber in (3.5): Here, the function l is the length of the Moore paths and locally constant means being constant on path components.The θ-framed bundle maps have unital and associative composition, with the unit in Hom θ (E, E) given by (id E , φ const , 0 const ).Treating the path data l as 1 const , the composition is defined up to homotopy as: Note that in the definition of Hom θ (E 1 , E 2 ), everything is taken non-equivariantly.The spaces Hom(E 1 , E 2 ) and Hom(E 1 , θ * ζ n ) have G-actions by conjugation.Since φ 1 and φ 2 are G-maps, the homotopy fiber Hom θ (E 1 , E 2 ) inherits the conjugation G-action.So we have built a GTop-enriched category Vec θ G,n of θ-framed bundles and θ-framed bundle maps.
Definition 3.6.The space of θ-framed embeddings between two θ-framed manifolds is defined as the pullback displayed in the following diagram of G-spaces: Let ∐ be the disjoint union of θ-framed vector bundles or manifolds and ∅ be the empty bundle or manifold.Both (Vec θ G,n , ∐, ∅) and (Mfld θ G,n , ∐, ∅) are GTop-enriched symmetric monoidal categories.In both categories, ∅ is the initial object.In Vec θ G,n , ∐ is the coproduct, but it is not so in Mfld θ G,n .Remark 3.9.We need the length of the Moore path to be locally constant as opposed to constant for the enrichment to work.Namely, the map ) is given by first post-composing with the obvious θ-framed map then using a homeomorphism, as follows: If the length of the Moore path were constant, the displayed homeomorphism would only be a homotopy equivalence, as the length of a Moore path can be different on the two parts.
To set up factorization homology in Section 3.2, we fix an n-dimensional orthogonal G-representation V ; in addition, we suppose that V is θ-framed and fix a θ-framing on From the proof of the next proposition, we may assume without loss of generality that the base of φ : V → B is the constant map to φ(0) ∈ B G (which is a V -indexed component in the sense of Theorem 2.19).
Proposition 3.10.Write ρ : G → O(n) for a matrix representation of V and So a θ-framing on V exists, if and only if the intersection of θ(B) and the V -indexed component of (B G O(n)) G as introduced in Theorem 2.20 is non-empty.
Proof.Since TV ∼ = V as G-vector bundles, the space of θ-framings on V is We have Proposition 3.10 and Theorem 2.20 give: Corollary 3.11.Let V, W be n-dimensional G-representations.
(1) The G-manifold W can be fr V -framed if and only if W ∼ = V as G-representations.
(2) For a tangential structure θ so that V and W are both θ-framed and H ⊂ G, We also describe the change of tangential structures.Let q be a morphism from The morphism q also induces a map on framed-morphisms.So we have a functor , and similarly q * : Mfld θ 1 G,n → Mfld θ 2 G,n .3.2.Equivariant factorization homology.In this subsection, we use the GTopenriched category Mfld θ G,n developed in Section 3.1 to define the equivariant factorization homology as a monadic bar construction.We have fixed an n-dimensional orthogonal G-representation V and a θ-framing φ : TV → θ * ζ n on the G-manifold V .
Recall that Λ is the category of finite based sets k and based injections.From Example 2.10, we have a Λ-sequence D θ M for a θ-framed manifold M. Explicitly, on objects, we have On morphisms, Σ k acts by permuting the copies of V , and s k V is a reduced G-operad.Using Construction 2.6, we get associated functors of D θ M and D θ V , which we denote by The associated functor D θ V is a monad (with natural transformations η : id The following is a standard definition: Definition 3.13.Let C be a reduced operad in (GTop, ×) and C be the associated monad.An object A ∈ GTop * is a C -algebra, or equivalently a C-algebra, if there is a map γ : CA → A such that the following diagrams commute, where the unlabeled maps are the unit and multiplication map of the monad C: In what follows, let A be a D θ V -algebra in GTop * .It is conceptually a left D θ M -module.We have a simplicial G-space, whose q-th level is The face maps are induced by the above-given structure maps Proof.The proof is a formal argument assembling the literature.We show that the bar construction is Reedy cofibrant in the deferred Corollary 4.19.The claim then follows since geometric realization preserves levelwise weak equivalences between Reedy cofibrant simplicial G-spaces, as quoted in the deferred Theorem 4.14.
We have the following properties of the factorization homology.
Proof.Both follow from the extra degeneracy argument of [May72, Propositions 9.8 and 9.9].For the first equivalence, the extra degeneracy coming from the unit map of the Proposition 3.17.For θ-framed manifolds M and N, Proof.Without loss of generality, we may assume that both M and N are connected.Then This is the formula of the Day convolution of D θ M and D θ N .So we have (3.18) We drop the θ in the rest of the proof.By (3.18) and iterated use of Proposition 2.11, there is an isomorphism in Λ op * (GTop) for each q: Iterated use of (2.7) identifies so evaluating on the 0-th level of (3.19) gives an equivalence of simplical G-spaces: The claim follows from passing to geometric realization and commuting the geometric realization with the product.Theorem 3.20.Let q : θ 1 → θ 2 be a morphism of tangential structures and V = (V, φ 1 ) be θ 1 -framed.We also write V for the θ 2 -framed G-manifold q * V = (V, qφ 1 ).For a θ 1framed G-manifold M and a D θ 2 V -algebra A, there is a G-equivalence The proof is deferred to the end of Section 3.4.3.3.Relation to configuration spaces.Now we restrict our attention to the Vframed case for an orthogonal n-dimensional G-representation V .We give V the canonical V -framing TV ∼ = V × V and let M be a G-manifold of dimension n.When M is V -framed, we denote the V -framing by φ M : TM → V .
In this subsection, we first prove that a smooth embedding of ∐ k V into M is determined by its images and derivatives at the origin up to a contractible choice of homotopy (Proposition 3.26).Then we proceed to prove that a V -framed embedding space of ∐ k V into M as defined in (3.7) is homotopically the same as choosing the center points (Proposition 3.30).
To formulate the result, we first define the suitable equivariant configuration space related to a manifold, which will be "the space of points and derivatives".
We use F E (k) to denote the ordered configuration space of k distinct points in E, topologized as a subspace of E k .When E is a G-space, F E (k) has a G-action by pointwise acting.It commutes with the Σ k -action that permutes the points.Definition 3.22.For a fiber bundle p : E → M, define F E↓M (k) to be configurations of k-ordered distinct points in E with distinct images in M. F E↓M (k) is a subspace of F E (k) and inherits a free Σ k -action.When p is a G-fiber bundle, In general, we have the following pullback diagram: Now, we take E = Fr V (TM).Recall that Fr V (TM) = Hom(V, TM) is a G-bundle over M. For an embedding ∐ k V → M, we take its derivative and evaluate at 0 ∈ V .We will get k-points in Fr V (TM) with different images projecting to M. In other words, the composition Proposition 3.26.The map d 0 in (3.25) is a G-Hurewicz fibration and (G × Σ k )homotopy equivalence.
One can find an equivariant local trivialization.The proof is tedious and can be found in [Zou20, Prop 5.5.5].
A section and homotopy inverse exists uniquely up to homotopy: For k = 1, it is given by the exponential map: Since there is a (contractible) choice of the radius at each point for the exponential map to be homeomorphism, σ is unique only up to homotopy.
Lemma 3.28.For a V -framed manifold M, the projection is a trivial bundle with fiber (Hom(V, V )) k .We call the section that selects (id V ) k in each fiber the zero section z.
Proof.Regarding V as a bundle over a point, we may identify Fr The claim follows from Example 3.24.
We can restrict the exponential map (3.27) to the zero section in Lemma 3.28 to get (3.29) Now we are ready to justify the equivalence of Emb fr V (∐ k V, M) and the configuration spaces of M.Moreover, we show that this equivalence is compatible over Emb(∐ k V, M).This will be used in later sections to compare different scanning maps.
Proposition 3.30.For a V -framed manifold M, we have: (1) Evaluating at 0 of the embedding gives a (G × Σ k )-homotopy equivalence: the sense that the following diagram is (G × Σ k )-homotopy commutative: (1) By Definition 3.6 and (3.12), Emb fr V (∐ k V, M) is the homotopy fiber of the composite: We would like to restrict the composite at and i 0 : V → TV is a G-homotopy equivalence of G-vector bundles, ev 0 : Hom(∐ k TV, TM) is a (G × Σ k )-homotopy equivalence.So in the following commutative diagram, the vertical maps are all (G × Σ k )-homotopy equivalences: We focus on the top composition D and the bottom map proj 2 .The map ev 0 between their codomains is a based map.Indeed, the base point of Hom(∐ k TV, V ) is from the V -framing of ∐ k V and is (G×Σ k )-fixed.It is mapped to id k , the base point of Fr V (V ) k .Consequently, there is a (G × Σ k )-homotopy equivalence between the homotopy fibers of these two maps.
Our desired ev 0 in question is the composite of (3.32) and the following map: It suffices to show that X is a (G × Σ k )-equivalence.Indeed, X is the comparison of the homotopy fiber and the actual fiber of proj 2 .Write temporarily The claim follows from the fact that (2) We examine the following diagram, where z is the zero section in Lemma 3.28: The left column is given by the (homotopy) fibers of the first and second rows of (3.31), so the solid diagram is (G × Σ k )-homotopy commutative.As σ 0 = σ • z and σ is a (G×Σ k )-homotopy inverse of d 0 by Proposition 3.26, the upper triangle with the dotted arrow is homotopy commutative.

Comparison of operads and the invariance theorem.
In this subsection, we study the θ-framed little V -disk operad D θ V .For θ = fr V , D fr V V is equivalent to the little V -disks operad D V .For background, D V is a well-studied notion introduced for recognizing V -fold loop spaces; see [GM17, 1.1].Roughly speaking, D V (k) is the space of non-equivariant embeddings of k copies of the open unit disks D(V ) to D(V ), each of which takes only the form v → av + b for some 0 < a ≤ 1 and b ∈ D(V ), called linear.In particular, the spaces are the same as those of the non-equivariant little n-disks operad, and so are the structure maps.The G-action on D V (k) is by conjugation.It is well-defined, commutes with the Σ k -action and the structure maps are G-equivariant.Proposition 3.33.There is an equivalence of G-operads β : D V → D fr V V .Proof.To construct the map of operads β, we first define β(1) : D V (1) → D fr V V (1).Take e ∈ D V (1); we must give β(1)(e) = (f, l, α) ∈ D fr V V (1).Explicitly, e : D(V ) → D(V ) is e(v) = av + b for some 0 < a ≤ 1 and b ∈ D(V ).

Define
f For α, Hom(TV, V ) ∼ = Map(V, O(V )), I is the unit element of O(V ) and c is the constant map to the indicated element.It can be checked that β(1) as defined is a map of Gmonoids. Restricting We have shown that ev 0 is a levelwise equivalence (Proposition 3.30 (1)).So β is also a levelwise homotopy equivalence.
For general θ, D θ V also allows θ-framed automorphisms of the embedded V -disks.By Theorem B.15, the θ-framed automorphism space of V is equivalent to Λ φ B, the Moore loop space of B based at φ(0).

Proposition 3.34. ([Zou20, B.2.8])
There is a G-monoid ΛB equivalent to Λ φ B which acts on D V .Furthermore, there is an equivalence of G-operads D V ⋊ ΛB → D θ V .Explanation.Without loss of generality we assume V is θ-framed by a constant map.
Note that fr V is initial for such tangantial structures, so we have ) be the sub-G-monoid of embeddings that preserves the origin 0 ∈ V .We claim that the composition map is an equivalence by Proposition 3.30, where the map ev 0 is evaluation at 0 and is a G × Σ k fibration.Its fiber is (Emb θ 0 (V, V )) k .So it follows that (3.35) is an equivalence.Combining Proposition 3.33 with (3.35), there is a G × Σ k -equivalence for each k.In [Zou20, Appendix B], this equivalence is upgraded to an equivalence of G-operads Here, ΛB is a replacement of Λ φ B that acts on D V (k), D V ⋊ ΛB is a G-operad whose k-th space is D V × ( ΛB) k , and the semi-direct product notation is introduced in [SW03] to indicate a twisting in the structure maps.
Proof of Theorem 3.20.Without loss of generality we assume V is θ 1 -framed by a constant map.We omit the q * and q * in the proof.As B(D It suffices to show that natural map of right D θ 2 V -functors M is an equivalence.Using (3.35), one can already construct a retract of (3.36).To construct a deformation retract, we need the full strength of Proposition 3.34.There are equivalences of Goperads fitting in a commutative diagram (3.37) The monad associated to And similarly the associated functors for k → D M (k) × ( ΛB i ) k are given by D Note that ΛB i is a G-monoid, so the functor A → ΛB i × A is a monad, which we still write as ΛB i .We have is an equivalence.Here, the last equivalence is given by a deformation retract using an extra degeneracy argument [May72, Proposition 9.9].Now, in the following commutative diagram whose vertical maps are equivalences induced by the approximation (3.37),

B(D
we see that ǫ is an equivalence.

Nonabelian Poincaré Duality for V -framed manifolds
Configuration spaces have scanning maps out of them.It turns out that equivariantly the scanning map is an equivalence in the case of G-connected labels X.Since the factorization homology is built up simplicially by the configuration spaces, we can upgrade the scanning equivalence to what is known as the nonabelian Poincaré duality theorem.
4.1.Scanning map for V -framed manifolds.In this subsection we construct the scanning map, a natural transformation of right D fr V V -functors: (4.1) Here, Map c (X, Y ) for a based space Y denotes the space of maps f so that the support f −1 (Y \ * ) is compact.In Appendix A, we compare our scanning map to the existing different constructions in the literature.This allows us to utilize known results about equivariant scanning maps to give Theorem 4.5, a key input to the nonabelian Poincaré duality theorem in Section 4.2.
Assume that the scanning map (4.1) has been constructed for a moment.When we take M = V , (4.1) gives a map of monads s : induces the right D fr V V -module structure for the functor Map c (M, Σ V −).Now we construct the scanning map.For any G-space X, recall that where ∼ is the base point identification.Take an element Here, each fi = (f i , α i ) consists of an embedding f i : V → M and a homotopy α i of two bundle maps TV → V , see Definition 3.6.We use only the embeddings f i to define an element s X (P ) ∈ Map c (M, Σ V X): Notice that if x i is the base point, f −1 i (m) ∧ x i is the base point regardless of what f i is.So passing to the quotient, (4.2) yields a well-defined map ≈ in Construction 2.4, but commutes with ∼.This is because the H-action preserves the filtration and ∼ only identifies elements of different filtrations.The single point at filtration k = 0, or equivalently the point at any k with all labels being the base point of X, is the base point of (F V X) H . Since the Σ k -action is free on F V (k) × X k and commutes with the G-action, we have a principal G-Σ k -bundle To get H-fixed points on the base space, we need to consider the Λ α -fixed points on the total space for all the subgroups Λ α ⊂ G × Σ k that are the graphs of some group homomorphisms α : H → Σ k .More precisely, by Theorem 2.19, we have Here, the coproduct is taken over Σ k -conjugacy classes of group homomorphisms and We would like to make the expression coordinate-free for k.A homomorphism α can be identified with an H-action on the set {1, • • • , k}.For an H-set S, write X S = Map(S, X) and F V (S) = Emb(S, V ).Then So we have: If we take care of the base point identification, we end up with: (4.12) (F V X) H = [S]:iso classes of finite H-set Suppose that the H-set S breaks into orbits as S = ∐ i r i (H/K i ) for i = 1, • • • , s, where K i 's are in distinct conjugacy classes of subgroups of H and r i > 0, then we know explicitly each coproduct component is: Since X K i are all connected, so are the spaces i (X K i ) r i .They contain the base point of the labels * = i r i * → i (X K i ) r i .So after the gluing ∼ H , each component in (4.12) is in the same component as the base point of Proof.This is a consequence of Theorem 4.5 and Lemma 4.11.
For geometric realization, we have: Theorem 4.14 (Theorem 1.10 of [MMOar]).A levelwise weak G-equivalence between Reedy cofibrant simplicial objects realizes to a weak G-equivalence.4.4.Cofibrancy.We take care of the cofibrancy issues in this part, following details in [May72].We first show that some functors preserve G-cofibrations.One who is willing to take it as a blackbox may skip directly to Definition 4.17.We uses NDR data, which give a hands-on way to handle cofibrations.Definition 4.15 (Definition A.1 of [May72]).A pair (X, A) of G-spaces with A ⊂ X is an NDR pair if there exists a G-invariant map u : X → I = [0, 1] such that A = u −1 (0) and a homotopy given by a map h : I → Map G (X, X) satisfying • h 0 (x) = x for all x ∈ X; • h t (a) = a for all t ∈ I and a ∈ A; • h 1 (x) ∈ A for all x ∈ u −1 [0, 1).The pair (h, u) is said to a representation of (X, A) as an NDR pair.A pair (X, A) of based G-spaces is an NDR pair if it is an NDR pair of G-spaces with the h t being based maps for all t ∈ I.
An NDR pair gives a G-cofibration A → X.The function u gives an open neighboorhood U of A by taking U = u −1 [0, 1).The function h restricts on I × U to a neighborhood deformation retract of A in X.
We have the following lemma by elaborating the NDR data.Its proof is tedious and omitted here (See [Zou20, Section 6.4]).
Lemma 4.16.Any functor F associated to F ∈ Λ op * [GTop], in particular both D fr V V and D fr V M , sends NDR pairs to NDR pairs.The functors Map c (M, −), Map * (M + , −) and Σ V all send NDR pairs to NDR pairs.Definition 4.17 (Lemma 1.9 of [MMOar]).A simplicial G-space X • is Reedy cofibrant if all degeneracy operators s i are G-cofibrations.
The following lemma shows that monadic bar constructions are Reedy cofibrant.
Lemma 4.18 (adaptation of Proposition A.10 of [May72]).Let C be a reduced operad in G-spaces such that the unit map η : * → C (1) gives a non-degenerate base point.Let C be the reduced monad associated to C .Let A be a C-algebra in GTop * and F : GTop * → GTop * be a right-C-module functor.Suppose that F sends NDR pairs to NDR pairs.Then B • (F, C, A) is Reedy cofibrant.
Proof.We need to show that for any n ≥ 0 and 0 ≤ i ≤ n, the degeneracy map By Lemma 4.16, C sends NDR pairs to NDR pairs.Starting from the NDR pair (A, * ) and applying this functor (n − i) times, we get an NDR pair (C n−i A, * ) = (X, * ).Together with the assumption that C (1) is non-degenerately based, we can show (CX, X) is an NDR pair where X is identified with the image η X : X → CX (see the proof of [May72, A.10]).Applying C another i times and then F , we get the NDR pair Corollary 4.19.Let M, V, A be as in Theorem 4.7.Then the following are Reedy cofibrant simplicial G-spaces: Proof.In Lemma 4.18, we take C = D fr V V and respectively By Lemma 4.16, each F does send NDR pairs to NDR pairs.4.5.Dimension.We start by recalling some facts about G-CW complexes and equivariant dimensions following [May96, I.3].A G-CW complex X is a union of G-spaces X n , where X 0 is a disjoint union of orbits, and X n is obtained by inductively gluing cells G/K × D n for subgroups K ⊂ G via G-maps along their boundaries G/K × S n−1 to the previous skeleton X n−1 .
We shall look at functions from the conjugacy classes of subgroups of G to Z ≥−1 and typically denote such a function by ν.We say that a G-CW complex X has dimension ≤ ν if its cells of orbit type G/H all have dimensions ≤ ν(H), and that a It is worth pointing out that this notion of dimension should be more appropriately called the cell dimension.(It is not the dimension of X H , as we explain shortly.)It gives information on which cells to consider in an induction.For the purpose of induction, we use the following ad hoc definition in this paper: Definition 4.20.A based G-CW complex is a union of G-spaces X n obtained by inductively gluing cells to X 0 , a disjoint union of orbits plus a disjoint base point * .
(The gluing maps are non-based maps.)In a based map out of X, the base point * has no freedom but to be sent to the base point.So we do NOT count it as a cell for a based G-CW complex, excluding it from counting the dimension as well.It then makes sense to write X −1 = * .This is not the same as a based G-CW complex in [May96, Page 18], where the base point is put in the 0-skeleton X 0 .Fix a subgroup H ⊂ G.A function ν from the conjugacy classes of subgroups of G to Z ≥−1 induces a function from the conjugacy classes of subgroups of H to Z ≥−1 , which we still call ν.We have the double coset formula where each for some element g i ∈ G.So a (based) G-CW structure on X restricts to a (based) H-CW structure on the H-space Res G H X. However, for X of cell dimension ≤ ν, Res G H X may not be of cell dimension ≤ ν, as we see in (4.21) that an H/K i -cell can come from a G/K-cell for a larger group K.For a function ν, we define the function d ν to be Then Res G H X is of cell dimension ≤ d ν .Remark 4.23.More specifically, we define the cell dimension of a (based) G-CW complex X to be the minimum ν such that X is of cell dimension ≤ ν.Suppose that X has cell dimension ν.From (4.21), we get: (i) The (based) H-CW complex Res G H X has cell dimension ν H , where ν H (K) = max K⊂L K=L∩H ν(L).
We have ν H (K) ≤ d ν (K), and it can be strictly less.(For a trivial example, take H = G.) (ii) The (based) CW-complex X H has dimension ν H (H) = d ν (H) ≥ ν(H).(In the based case, we also exclude the base point from counting the dimension of X H , so that if X H = * , the dimension of X H is -1.)Definition 4.24.
(1) For a (based) G-CW complex X of cell dimension ν, dim(X) is the function d ν .
From Remark 4.23, we have two observations: First, dim(X)(H) is equal to the dimension of the CW-complex X H .So dim(X) is independent of the G-CW decomposition of the underlying G-space of X.Second, for a unbased G-CW complex X, the based G-CW complex X + = X ∐ * satisfies dim(X + ) = dim(X) because * is excluded from cells in the based case.
We prepare the following results regarding dimension for the next subsection.
Theorem 4.25 (Theorem 3.6 of [Ill78]).For a smooth G-manifold M and a closed smooth G-submanifold N, there exists a smooth G-equivariant triangulation of (M, N).
Lemma 4.26.Let M be a V -framed manifold and A be a G-space, then (1) M + has the homotopy type of a G-CW complex of cell dimension ≤ dim(V ). (2) Proof.(1) Since M is a V -framed, the exponential maps give local coordinate charts of M H as a (possibly empty) manifold of dimension dim(V H ). If M is compact we take W = M, otherwise we take a manifold W with boundary such that M is diffeomorphic to the interior of W .By Theorem 4.25, (W, ∂W ) has a G-equivariant triangulation.It gives a relative G-CW structure on (W, ∂W ) with relative cells of type G/H of dimension ≤ dim(V H ). The quotient W/∂W gives the desired G-CW model for M + .
(2) For any subgroup H ⊂ G, we have 4.6.Commuting mapping space and geometric realization.Let X be a based G-CW complex and K • be a simplicial G-space.Then the levelwise evaluation is a whose adjoint gives a G-map Non-equivariantly, it is one of the key steps in May's recognition principal that (4.27) is a weak equivalence when each K • is dim(X)-connected [May72,Theorem 12.3].The goal of this subsection is to give a sufficent condition for ζ to be a weak G-equivalence.
The strategy is to induce on cells.However, the geometric realization of a levelwise fibration is not necessarily a fibration.Dold-Thom came up with the notion of quasifibrations, which is good enough for handling the homotopy groups.
Definition 4.28.A map p : Y → W of spaces is a quasi-fibration if p is onto and it induces an isomorphism on homotopy groups π * (Y, p −1 (w), y) → π * (W, w) for all w ∈ W and y ∈ p −1 (w).In other words, there is a long exact sequence on homotopy groups of the sequence p −1 (w) → Y → W for any w ∈ W . Theorem 4.30.Let G be a finite group.If X is a finite-dimensional based G-CW complex and K • is a simplicial G-space such that for any n, K n is dim(X)-connected, then the natural map (4.27) The base case k = −1 is obvious.Suppose that (i) and (ii) hold for k.Take the cofiber sequence and map it into K • .We then apply (4.27) and get the following commutative diagram: (4.31) Since maps out of a cofiber sequence form a fiber sequence, we have a fiber sequence in the second row and a realization of the following levelwise fiber sequence in the first row: By the inductive hypothesis (i) and Theorem 4.29, it realizes to a quasi-fibration.We first show the inductive case of (i).We can write where each K i is a subgroup of G.When K i is presented, ν(K i ) ≥ k + 1.From (4.21), we can further write X k+1 /X k ∼ = ∨ i ∨ j (H/K i,j ) + ∧ S k+1 as spaces with H-action, where each K i,j is G-conjugate to a subgroup of K i .Then d ν (K i,j ) ≥ ν(K i ) ≥ k + 1, and the following space is connected by assumption: This space is the fiber in (4.32).The connectedness of the base space given by (i) then implies the connectedness of the total space.We next show the inductive case of (ii).Commuting geometric realization with finite products and with fixed points, the left vertical map of (4.31) is a product of maps Since we have d ν (K i,j ) ≥ k + 1, these maps are weak equivalences by [May72,Theorem 12.3].By (ii), the right vertical map is a weak equivalence.Comparing the long exact sequences of homotopy groups, this implies that the middle vertical map is also a weak equivalence.
Remark 4.33.Non-equivariantly, Miller [Mil15,Cor 2.22] observed that the theorem is also true if K n is only (dim(X) − 1)-connected for all n, since the only thing that fails in the proof is the claim (i) for k = dim(X e ).Equivariantly, one needs (i) to hold for all inductive steps of k < d ν (e).So we can only relax the assumption to the following extent: If K H n is min{d ν (H), d ν (e) − 1}-connected for all n and H, then the natural map (4.27) is a weak G-equivalence.This is an improvement only when Nevertheless, when X = ΣZ and Z is of cell dimension ν, so that X is of cell dimension ν + 1, we can relax the assumption further.
Proof.The cofiber sequence S 0 ∨ S 0 → S 0 → S 1 gives a levelwise fiber sequence By Theorem 4.30 and its proof, (4.35) has a G-connected base and realizes to a quasifibration; the same method will show the claim.
The unbased version of Theorem 4.30 is due to Hauschild and written down by Costenoble-Waner [CW91, Lemma 5.4], stated as: Theorem 4.30 improves Theorem 4.36 slightly in the case when X G = * .On one hand, taking X in Theorem 4.36 to be Y ∐ { * } recovers Theorem 4.30.On the other hand, for a based G-CW complex X we have the levelwise fibration sequence If the cell dimension of X satisfies ν(H) ≥ 0 for all H, then dim(X)(H) = d ν (H) ≥ 0. The assumptions imply that K n is G-connected, we can use the quasi-fibration technique The commutative diagram (A.8) is central in this section.In the first row, fatten and φ ǫ are the two steps in McDuff's scanning map.The map γ + is from Section A.1.We will define the undefined spaces and maps as we go along.
Define Cǫ (M) 1 ≡ {exp m 0 : T m 0 M → M such that it is a diffeomorphism on the ǫ-ball}; images of e i on the δ-balls are disjoint in M}.
For preparation, we write down an explicit homeomorphism Here, D ǫ (R n ) is the disk of radius ǫ in R n .Then, abusively we also have Define E M to be the bundle over M whose fiber over m is D 1 (T m M)/∂D 1 (T m M), which is identified with Sph(T m M) through η 1 .This is the right vertical map in (A.8).
We give the vertical map in the middle of (A.8).For an element exp m 0 ∈ exp m 0 , the composite exp m 0 • η −1 ǫ is an embedding R n → M, so we can identify Cǫ (M) 1 with a subspace of Emb(R n , M).Similarly, we can include as subspace: let us define φ ǫ and compare it to the map γ + locally.Put a Riemannian metric on M. The input for φ ǫ are the exponential maps in Cǫ (M) 1 .Define Here, the values are vectors in D 1 (T m M); t(m, m 0 ) is the unit tangent at m of the minimal geodesic from m 0 to m; dist(m, m 0 ) is the distance between m and m 0 .Now, it can be easily verified that . We can work the same way to extend φ ǫ to Cǫ (M) and we have the commutativity part of (A.8): . In McDuff's second step, we describe the fattening map in (A.8).We can take a continuous positive function ǫ on M such that for any m 0 ∈ M, the exponential map exp m 0 : T m 0 M → M is always a diffeomorphism on the ǫ(m 0 )-ball.(We note the change Definition B.4.For φ M : M → B and φ N : N → B, the space Hom GTop h / B (M, N) and the G-space Hom Top h G / B (M, N) are given by: Hom ) and a homotopy from φ M to φ N • f ; the map f is not necessarily a G-map, but we do require φ M and φ N to be G-maps.And we have The category Top h G / B models θ-framed bundles: Proposition B.6.For i = 1, 2, let E i → B i be G-n-vector bundles with θ-framings φ i : E i → θ * ζ n .We have the following equivalences of G-spaces that are natural with respect to the two variables as well as the tangential structure: Hom(E 1 , E 2 ) Map(B 1 , B 2 ) We claim that the bottom square is a pullback.Since each column is a homotopy fiber sequence, this implies immediately that β is a G-equivalence.
We remark that in Proposition B.6, π is not a homotopy equivalence to its image.In other words, a vector bundle map is not just a map on the bases.In contrast, a θ-framed vector bundle map can be seen as a map on the bases as β is an equivalence.Lemma B.8. ([Zou21, Lemma 3.18]) Let p : P → B be any principal G-Π-bundle and Hom(P, E G Π) be the space of (non-equivariant) principal Π-bundle morphisms with G acting by conjugation.Hom(P, E G Π) is G-contractible.
Proof.By definition, Hom id (E 1 , E 2 ) is the homotopy fiber of φ 2 • −, so we have a homotopy fiber sequence of G-spaces:  We can take (B.12) as an alternative definition to (3.7).In practice, (3.7) is easier to deal with.First, the right vertical map in the square is a fibration so the diagram is an actual pullback.Second, the map d is easy to describe.On the other hand, (B.12) has a conceptual advantage.It can be viewed as a comparison of the θ-framing to the trivial framing id : B.2. Automorphism space of (V, φ).With this alternative description of θ-framed mapping spaces in Section B.1, we can identify the automorphism G-space Emb θ (V, V ) of V in Mfld θ G,n by first identifying of the automorphism G-space Hom θ (TV, TV ) of TV in Vec θ G,n .Notation B.14.As φ is an equivariant map, φ(0) for the origin 0 ∈ V is a G-fixed point in B. We denote by Λ φ B the Moore loop space of B at the base point φ(0).

Theorem B.15. We have the following:
(1) There is an equivalence of monoids in G-spaces Hom θ (TV, TV ) which is natural with respect to tangential structures θ : B → B G O(n).Here, the group G acts on both sides by conjugation.(2) The automorphism G-space Emb θ (V, V ) of (V, φ) in Mfld θ G,n fits in the following homotopy pullback diagram of G-spaces: where h t : V → V is any chosen homotopy from h 0 = id to h 1 = proj.Then we have an obvious homotopy: P • I = (id, const φ(0) , 1) ≃ (id, const φ(0) , 0) = id * and using the contraction h t , we can also construct a homotopy: (2) This is an assembly of part (1), Proposition B.11 and Theorem 2.24.However, we note that the map Λ φ B → O(V ) is only a non-canonical G-equivalence.The author does not know how to upgrade it to a map of G-monoids.So although all spaces displayed in the pullback diagram are G-monoids, it is not obvious whether one can write Emb θ (V, V ) as a pullback of G-monoids.
To be more precise, we show how the quoted results assemble.We have the following large commutative diagram (B.16) expanding (B.13).Note that this is a commutative diagram of G-monoids.
The map α is studied in Lemma B.9.The map β and the square 1 ○ are in Proposition B.6.The diagonal unlabeled maps are all induced by the inclusion V → TV and the projection TV → V .In particularly, the parallelogram 2 ○ is in part (1).Naturality of α and β gives the commutativity of 3 ○ and 4 ○.Now, d 0 in the theorem is the composite Emb(V, V ) Hom(TV, TV ) Hom(V, V ).

d ∼
It can be seen that the vertical map in the theorem involves choosing an inverse of the β displayed in the third line.

3.
Tangential structures and factorization homology 3.1.Equivariant tangential structures.In this subsection we fix a tangential structure θ and construct two categories.The first one is Vec θ G,n , the category of ndimensional θ-framed equivariant bundles and θ-framed bundle maps.The second one is Mfld θ G,n , the category of smooth n-dimensional θ-framed G-manifolds and θ-framed embeddings.The category Mfld θ G,n is a subcategory of Vec θ G,n ; both Mfld θ G,n and Vec θ G,n are enriched over GTop.If we let θ vary, both constructions define covariant functors from T S to categories.
the same data as a map τ B : M → B on the base plus a homotopy between the two classifying maps from M to B G O(n).For a detailed proof, see Corollary B.10 with Definition B.4. (3.2) 7) Emb θ (M, N) Hom θ (TM, TN) Emb(M, N) Hom(TM, TN) d Here, Emb(M, N) is the space of smooth embeddings and the map d takes an embedding to its derivative.For the empty manifold, we define Emb θ (∅, N) = * and Emb θ (M, ∅) = ∅ unless M = ∅.The category Mfld θ G,n has objects θ-framed manifolds (including the empty set) and morphism spaces Emb θ .Remark 3.8.Most of the time, we drop the Moore-path-length data and write an element of Emb θ (M, N) as a package of a map f and a homotopy f = (f, α), with f ∈ Emb(M, N) and α : [0, 1] → Hom(TM, TN) satisfying α(0) = φ M and α(1) = φ N • df .There is a functor Mfld θ G,n → Mfld G,n by forgetting the tangential structure.It sends f ∈ Emb θ (M, N) to f ∈ Emb(M, N).
The degeneracy maps are induced by η : id → D θ V .We have the following definition after the non-equivariant work of [And10, IX.1.5]:Definition 3.14.The factorization homology of M with coefficients A is θ M A := B(D θ M , D θ V , A).The category of algebras D θ V [GTop * ] has a transfer model structure via the forgetful functor D θ V [GTop * ] → GTop * ([BM03, 3.2, 4.1]), so that weak equivalences of maps between algebras are just underlying weak equivalences.Proposition 3.15.The functor θ M
In summary, sMD goes in two steps: fatten up the configurations ([McD75, Lemma 2.3]) and use geodesics to give compactly supported vector fields ([McD75, p95]).(A.8) sMD : F M (S 0 ) Cǫ (M) Sect c (M, E M ) Emb M (S 0 ) Sect c (M, Sph(TM)) Top h G / B (B 1 , B 2 ).Proof.One can restrict bundle maps to get maps on the base spaces.We denote this restriction map by π.From our definition of Hom θ in Definition 3.4 and Hom Top h G / B in Definition B.4, π induces the map β and they fit in the following commutative diagram of G-spaces:(B.7)Hom θ (E 1 , E 2 ) Hom Top h G / B (B 1 , B 2 )

Proposition B. 11 .
The G-space Emb θ (M, N) as defined in Definition 3.6 is the homotopy pullback displayed in the following diagram of G-spaces:(B.12)Emb θ (M, N) Hom Top h G / B (M, N) Emb(M, N) Hom Top h G / B G O(n) (M, N)Proof.The lower horizontal map in (B.12) is neither obvious nor canonical.We take it as the composite in the following commutative diagram with a chosen G-homotopy inverse to α.The maps α and β are G-equivalences by Proposition B.6 and Lemma B.9.(B.13)Emb θ (M, N) Hom θ (TM, TN) Hom Top h G / B (M, N) Hom id (TM, TN) Hom Top h G / B G O(n) (M, N)As defined in Definition 3.6, Emb θ (M, N) is the pullback in the left square.It is clear that it is also equivalent to the homotopy pullback of the whole square.