Equivariant properties of the space ℤ (X) for a stratifiable space X

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Introduction

In the category of stratifiable spaces and continuous images, we include one construction belonging to the test space [1; 2] that defines the covariant functor in this category. This construction defines the functor that allows each stratified space X to be immersed in a closed manner into some other space Z(X), which is the stratified space with "good"   functorial, geometric and topological properties.

A stratifiable spaces can be defined of as topological space that is divided into smooth manifolds. Then a stratification in this context [3] is a structure associated with the emergence of closed sets, which is locally a decomposed space. This is what [4] refer to as a “germinal stratification". Each decomposed space causes a bundle of germs-stratifications, hence the concepts are consistent. Another notion of stratification can be found in [4], where the boundary conditions are slightly different. Stratifiable spaces always admit tangent bundle. They are relevant because they are really singular, while the usual vector bundle is not [5]. Families of examples and applications [6; 7] arise from smooth equivariant vector bundles. The next main motivation for developing stratifiable vector bundles is to use them for quantization purposes. In particular, in the Kostant–Suriot–Weyl quantization picture, three components of the initial data are required: a symplectic manifold, a complex linear bundle with a connection, and polarization, all of which satisfy various compatibility conditions [8]. Therefore, it is relevant to consider and define theorems for equivariant properties of spaces over a stratifiable space.

Let X be a stratified (briefly, S - space) space. For each open subset U - of the space X and any point x ∈ U of the set U we put:

1.  n(U, x) = min{m : x ∈ U_m}, where $U={\cup }_{k=1}^{\infty }{U}_k$
2.  $U_x=U_{n\left(U\mathrm{,}x\right)}\backslash {\overline{X\backslash \left\{x\right\}}}_{n\left(U\mathrm{,}x\right)}$

Obviously, the set U_x is an open neighborhood of the point x and U_x ⊂ U. The set U_x has the following properties:

1°. U_x is an open neighborhood of the point x;

2°. If  U_x ∩ V_y ≠ ∅ and n(U, x) ≤ n(V, y) then y ∈ U.

3°. If U_x ∩ V_y ≠ ∅ then x ∈ V or y ∈ U.

Let X be the topological space, |F(X)| be a complete simplicial complex whose vertices are points in the space X, i.e. |F(X)⁰| = X. The space |F(X)|  has a weak topology. Now we define the topology on the space |F(X)|, the bases of open sets of which we denote by Z(X) consists of W open in F(X), satisfies following conditions:

o1. W ∩ X is an open in X;
o2. |F (W ∩ X)| ⊂ W;
i. e. τ_Z(X) = {W ∈ τ_|F(X)| : W satisfies the conditions o1-o2}.

Condition o2 means that every simplex σ ∈ F(X), is contained in W if all vertices of this simplex σ lie entirely in W ∩ X.

1  Main results

For the subset A⊂ X, the set F(A) is a subcomplex of the full complex  F(X) and Z(A)  is a subspace of the space Z(X). Obviously, Z(A) is closed in Z(A) if A is closed in X.

For each n ∈ ℤ_₊ = N ∪{0} we put Z_n (X) = |F (X)ⁿ| Z_n (X) is a subspace of Z(X). Then Z₀ (X) ≅ X  and  $Z\left(X\right){\cup }_{n=0 }^{\infty }{Z}_n\left(X\right)$.

It is easy to see that for any n ∈ Z_₊ the subspace Z_n (X) is closed in Z(X).

Let us introduce the following notation:

T(A) = {σ ∈ F(X)\F(A) : σ ∩ A ≠ ∅};
M(A) = {x ∈ Z(X) : exists σ ∈ F(A) such that x($\hat{\mathrm{\sigma }}$) > 0};
T_n(A) = T (A) ∩ (F(X)ⁿ \ F(X)ⁿ⁻¹);
M_n(A) = Z (A) ∪ (M (A) ∩ Z_n(X);

For each ε ∈ (0,1)^T(A) and for each n ∈ N, define the set:

M (A, ε) = ∪_n∈ℤ₊+M_n(A, ε), 

where M₀ (A, ε) = Z (A) = |F (A)| and M_n (A, ε) = Z (A) ∪ {σ(ε(σ)) ∩ ${\mathrm{\pi }}_{\mathrm{\sigma }}^{-1}$(M_n−1(A, ε)) : σ ∈ T_n(A)}.

Then thee quality M(A,ε) ∩ X = A holds.

For each open set $\mathcal{U}$ of the space X, the set M ($\mathcal{U}$, ε) is open in Z(X).

In this case, the family $\mathcal{B}$(M) = {($\mathcal{U}$, ε) : $\mathcal{U}$ is open in ε ϵ ${(0,1)}^{\left(\mathcal{U}\right)}$} is an open base of the space Z(X).

Therefore, if for every n ∈ N and every ε ∈ (0, 1)^{T_₁(A) ∪T_₂(A)∪...∪T_n(A)} the set M_n(A, ε) is defined, thenthe family

$\mathcal{B}$(M) = {M₁, ($\mathcal{U}$, ε)} : $\mathcal{U}$ is open in X and ε ∈ (0, 1)^{T_₁($\mathcal{U}$)}} is an open base for Z₁(X), i.e. the following holds.

In the work [2] R. Cauty claimed that for the space Z(X) the following are true:

Lemma [9]. Families{M ($\mathcal{U}$, ε) : $\mathcal{U}$  is open in X and  and ε ∈ (0,1)^{T($\mathcal{U}$)}} and {M₁ ($\mathcal{U}$, ε) : $\mathcal{U}$  is open the base in X  of the space and ε ∈ (0,1)^{T($\mathcal{U}$)}} is the base of the space Z(X), (respectively, the space Z₁(X)).

In the work [2] R. Cauty claimed that for the space Z(X) the following are true:

1. Each continuous map f : A → Y, where A is a closed subset of the stratified space X, has a continuous extension to all X with values in Z(X): that is, the following diagram holds  , 
   

         $A\stackrel{f}{\to }Y$  

        $\stackrel{\curvearrowright }{X}\stackrel{\overline{f}}{\to \stackrel{\curvearrowright }{Z\left(Y\right)}}$  

       ${\widetilde{f\mathit{ }|}}_A=f\mathrm{.}X\mathrm{,} Y\in S\mathrm{.}$   

2. The stratifiable space X is AR(S)  (ANR(S))  if and only if X is a retract (respectively, a neighborhood retract) of the space Z(X).

Definition [6]. A topological space L is called hyper-connected (respectively, m -hyper-connected) if for each i ∈ N, there is a mapping hi : Lⁱ × σ^i-1 → L satisfying a, b and c (respectively, a, b and d):

1. t ∈ σ^n-1and t_i = 0 implies h_n (x, t) = h_n-1 (δ_i · x, δ_i · t) for each x ∈ Lⁿ and n = 2, 3, . . .
2. For each x ∈ Lⁿ the mapping t → L_n (x, t) maps the sets σ^n-1 to L continuously;
3. For each  x ∈ L and a neighborhood  U of x, there is a neighborhood  V of x such that ${\cup }_{i=1}^n$h_i(Vⁱ × σ^i-1) ⊂ U and V ⊂ U ;
4. For each  x ∈ L and a neighborhood  U of the point x, there is a neighborhood V of the point x such that   ${\cup }_{i=1}^n$h_i(Vⁱ × σ^i-1) ⊂ U  and V ⊂ U,   where ${\sigma }^{n-1}=\{t\in R^n{\sum }_{i=1}^n{t}_i=t_i\ge 0\}$ - (n-1) is  a dimensional simplex    a  δ_i : Aⁿ → A^n-1      mapping defined by the formula      δ_i(a₁, a₂, . . . a_n ) = ( a₁, a₂, . . . , a_i−1, a_i+1, . . . , a_n) $i\overline{\mathrm{1,}n}$, i.e.     δ_i – ”forgetting” i - th coordinate of the product.

A space L is said to be a locally hyper - connected if for each point x  L, there exists a neighborhood V of the point x such that V is hyper-connected.

In the paper [2] R. Cauty proved that X ∈ A(N)R(S) if and only if X is hyperconnected (respectively, locally hyperconnected).

Theorem 1. For an arbitrary S - space X, the space Z(X)\X  is a AR(S) space.

Proof. Let n ∈ N. We construct mapping h_n(z₁, . . . , z_n,t) : (ZX\X)ⁿ × σⁿ⁻¹ → ZX\X   

assuming h_n(z₁, z₂,. . . . , z_n)(t₁, t₂, . . . , t_n) = ${\sum }_i^n{z}_i{t}_i$,

where (z₁, z₂,. . . . , z_n ) ∈ (ZX\X)ⁿ, (t₁, t₂, . . . , t_n) ∈  σⁿ⁻¹, ${\sum }_{i\mathit{=}\mathit{1}}^n{t}_i\mathit{=}\mathit{ }\mathit{1}$, t_i ≥ 0.

It is easy to show that h_n((z₁, . . . , z_n) × t) ∈ Z(X)\X.

Now we show that the space Z(X)\X is hyperconnected.

1. Let  t ∈ σⁿ⁻¹, t = (t₁, t₂, . . . , t_i−1, 0, t_i+1, t_i+2, . . . , t_n). Then h_n(z, t) =
   

    = h_n((z₁, z₂, . . . , z_n)(t₁, t_2, . . . , t_i−1, 0, t_i+1, t_i+2, . . . , t_n)  =  (t₁z₁+ t₂z₂+. . .+t_i−1z_i−1+ t_i+1z_i+1+ . . . t_nz_n) = 

   = h_n−1((z₁, z₂, . . . , z_i−1, z_i, z_i+1, . . . , z_n)(t₁, t₂, . . . , t_n)) = h_n−1(δ_iz, δ_it).

2. We fix z_₀∈ (Z(X)\X)ⁿ ,$z_0\mathit{ }\mathit{=}\mathit{ }(z_1^0\mathrm{,}{z}_2^0\mathrm{,}\dots \mathrm{,}{z}_n^0)$, $z_1^0\in ZX\backslash X$. 
   

   Hence $z_{\mathit{1}}^{\mathit{0}}\mathit{=}{\sum }_{k\mathit{=}\mathit{1}}^{l_i}{m}_k^i\mathit{,}\mathit{ }\overline{\mathit{ }{x}_k^l}\mathit{,}\mathit{ }\mathit{ }{\sum }_{\mathit{i}\mathit{=}\mathit{1}}^{{\mathit{l}}_{\mathit{i}}}{m}_{\mathit{k}}^{\mathit{i}}\mathit{=}\mathit{1}\mathit{,}\mathit{ }\mathit{ }{m}_{\mathit{k}}^{\mathit{i}}\ge 0$.  Let t ∈ σⁿ⁻¹, then  $t\to h_n\left(z_0\mathrm{,}t\right)={\sum }_{i=1}^n{t}_i{z}_i^0 =$$=t_1{\sum }_{k=1}^{l_1}{\mu }_k\overline{x_k^1}+t_2{\sum }_{k=1}^{l_2}{\mu }_k\overline{x_k^2}+\dots +t_n{\sum }_{k=1}^{l_n}{\mu }_k\overline{x_k^n}$;   

   Let us put t_iμ_j = aij, t_i ≥ 0, µ_j ≥ 0, a_ij ≥ 0, i = 1, n, j = l₁, . . . , ln, ∑_ij a_ij = 1. 

    Hence, we get ${\sum }_{i=1}^n{\sum }_{j=1}^{l_i}{a}_{ij}\overline{x_j^i}$. Consider the set X₀ = {xⁱ_j : i, j}..     In this case, point h(z₀, t) ∈ Z(X₀), i.e. there is a simplex σ lying in Z(X₀) whose vertices consist of points of the set X₀.  On the other hand, if we consider the simplex σⁿ⁻¹ with vertices $z_1^0\mathrm{,}\dots \mathrm{,}{z}_n^0$, i.e. ${\sigma }_0^{n-1}=⟨z_1^0\mathrm{,}{z}_2^0\mathrm{,}\dots \mathrm{,}{z}_n^0⟩$. The mapping h_n(z, t) with continuity in the argument t or the mapping t → h_n(z, t) completely covers the simplex , i.e. the mapping t → h_n(z, t) as a homeomorphism maps σⁿ⁻¹ to z₀ⁿ⁻¹ to . Hence, the mapping t → h_n(z, t) is continuous.
3. Let z_₀ ∈ Z(X)\X and U_z₀ be an arbitrary neighborhood of the point z_₀ in Z(X)\X. Consider suppz_₀ = {x₁, x₂, . . . , x_k}   the support  of  the point z_₀ of the space Z(X)\X. Then z_₀ ∈ ⟨x₁, x₂, . . . , x_k⟩ = σ.       By the definition of topology in the space Z(X)\X , the set $V^1\sigma \cap U_{z_0}$ is open.  Consider a set V of the form $\{z\in \sigma \cap U_{z_0}=V^1:$ segment $\left[z\mathrm{,}{z}_0\right]\subset =V^1\}$. Obviously, the set V is open and convex. By definition, the following takes place: V ⊂ V ⊂ V_z₀. Note that if z ∈ (Z(X)\X)ⁿ, z = (z₁, z₂, . . . , z_n) and suppz_i ⊂ A,  A ⊂ X, then supp h_n(z, t) ⊆ A.   If V is convex, then the maps h_n(z, t) by definition maps Vⁿ × σⁿ⁻¹ to V.  Therefore, the following holds: ${\cup }_{n\mathit{=}\mathit{1}}^{\infty }{h}_n(V^n\times {\sigma }^{n-1})\subset U$.    Hence, the space Z(X)\X is hyperslash. By virtue of R. Cauty’s theorem [1], we obtain that Z(X)\X is AR(S).  Theorem 1 is proved.

Theorem 2. The finite product of Z(N)R(S) spaces is Z(N)R(S) spaces. Theorem 2 is proved in [6].

Let X be a topological space, G is a topological group θ : G × X → X  is a continuous mapping such that

The mapping θ is called the action of the group G on the space X.

The space X with a fixed action θ of the group G is called a G - space.

A set A is called invariant under the action of the group G (or G - invariant) if G(A) = A,  where G(A) = {g(x) : g ∈ G, x ∈ A}.

For g ∈ G, we define the mapping θ_g : X → X  by the formula θ_g(x) = g(x) = θ(g, x). By virtue of (1) or (2) we have θ_g ◦ θ_h = θ_gh and θ_e and  is the identity mapping 1_X of the space X into itself.

Thus, θ_g ◦ θ_g-1 = θ_e = 1_X  = θ_g-1 · θ_g  therefore, for g ∈ G, the mapping θ_g  is a homeomorphism of the space X onto itself.

An action θ is called effective if  Kerθ = e(i.e., the mapping θ is injective), where Kerθ = {g ∈ G : g(x) = x} for any x ∈ X  the kernel of the action of θ, and is almost effective if Kerθ – is a discrete subgroup of the group G. Obviously, the kernel Kerθ is a normal divisor of the group G and is closed in G.

We note, that some of the supporting statements were considered in the articles [10–14].

Definition [15]. Let X and Y be G - spaces.

The mapping φ : X → Y is called an equivariant mapping (or -mapping) if φ commutes by actions, 

i.e. φ(g(x)) = g(φ(x)) for all g ∈ G and all x ∈ X.

For a fixed group G, the class of G- spaces is a class of objects of a certain category, whose morphisms are called equivariant maps.

An equivariant mapping φ : X → Y that is a homeomorphism is called the G -equivalence of G -spaces.

Note that if we denote by Homeo(X) the group (with respect to, composition) of all homomorphisms of the space X onto itself.

The mapping g → θ_g defines a homeomorphism θ : G → Homeo(X).

Let X be some G space, and let x ∈ X. The set G_x = {g ∈ G : g(x) = x} of elements of the group, for which x is a fixed point, is obviously a closed subgroup of the group G. This subgroup G_x is called the stationary subgroup (or the stabilizer of the point x).

On the other hand, note that kerθ is exactly ${\cap }_{x\in X}{G}_x$, i.e. kerθ = ${\cap }_{x\in X}{G}_x$.

The action of the group G on the space X is called free if for any point x ∈ X the subgroup G_x is trivial. An action is called semi-free if the stationary subgroup G_x of any point x ∈ X is either trivial or is the whole of G.

Take x ∈ X. The subspace G_x = {g(x) : g ∈ G} is called the orbit of the point x (with respect to the action of the group G). Note that G(x) ⊂ X  for any x ∈ X and for points x and y the sets G(x) and G(y) either do not intersect each other or coincide, i.e. G(x)$\cap$G(y) = Ø or G(x) = G(y) for any x, y ∈ X. By X \G ={G(x) : x ∈ X} we denote the orbit set of G - space X.

Let  π = π_X : X → X/G be a natural mapping, associating the point x and the orbit x* = G(x).

Then X\G is endowed with the quotient topology in the usual way (i.e., the set U ⊂ X\G is open if π⁻¹(U) and only if  is open in X ), and the resulting topological space is called the orbit space.

Note that if U ⊂ X is open, then the set  G(U) = is open, since each of the sets g(U) = θ_g(U)(recall that θ_g : X → X is a homeomorphism).

Therefore, for an open U ⊂ X, the set π⁻¹(U) = G(U) is also open, which by definition means that the set π(U) is open in X\G. Hence the projection  is a continuous open map.

Theorem [15]. Let the group G be compact and X is some G - space. Then 

Let X be a stratified  G - space that is the topological group G acts on the space X,    i.e.   there is a continuous mapping (G, X) : G × X → X defined by the formula: (g, x) = gx.

On the test space Z(X), the action of the group  is defined as follows:  (G, Z(X)) : G × Z(X) → Z(X), g ∈ G, z ∈ Z(X), z = ${\sum }_{i=1}^k{m}_i\overline{x_i}$ , ${\sum }_{i=1}^k{m}_i$ = 1, m_i ≥ 0 (g, z) = g · z = ${\sum }_{i=1}^k{m}_i\overline{g{x}_i}$. Thus, the space Z(X) is a G - space.

It is easy to see that the space X  in the G space Z(X) is an invariant G -subset, i. e. if x ∈ X, then g(x) = x ∈ X.

Thus, the following holds.

Take the point $z{\sum }_{i=1}^k{m}_i\overline{x_i}$, m_i ≥ 0, ${\sum }_{i=1}^k{m}_i =$ 1, then

$G_z=$$\{gz : g\in G, gz = g\sum _{i=1}^k{m}_i\overline{x_i}\text{ = }\sum _{i=1}^k{m}_i\overline{g{x}_i}\}$.

Obviously, $G_z=$$G_{m_1\overline{x_1}+\dots +m_k\overline{x_k}}=$$\{m_1\overline{g{x}_1}+\dots +m_k\overline{g{x}_k}\text{\hspace{0.33em}:\hspace{0.33em}}g\in G\}$.

Note that Z(X/G) ≅ Z(X)/G and is invariant in Z(X/G).

$\begin{array}{l}X\stackrel{{\pi }_{G\left(x\right)}}{\to }X\text{\hspace{0.05em}/\hspace{0.05em}}G x\to G_x\\ \downarrow i \downarrow i\\ Z\left(X\right)\overleftarrow{{\pi }_{Gz}}Z\left(X\right)G/ z\to G_z\end{array}$

In his paper [9] Cauty proved the following

Lemma 1.2 [1]. Let X be a topological stratified space. If Y ∈ S and A ⊂ Y - is closed and f : A → X  is continuous, then the mapping $\tilde{f}$ has a continuous extension $\tilde{f}$ : Y → Z(X).

Lemma 1. Let X be a topological stratified G - space, A ⊂ Y a closed G  invariant subset, f : A → X an equivariant continuous mapping, when the mapping  f has a continuous equivariant extension  $\tilde{f}$ : Y → Z(X)..

Proof. Let A be a closed subset of the space X. We put  W = X\A. W′ = {x ∈ W : x ∈ U_y, y ∈ A and U is open in X} and m(x) = max{n(U, y) : y ∈ A and x ∈ U_y}.

Obviously, W′ ⊂ W and for every x ∈ W′  and for every  there is m(x) < n(W,x) < ∞.

Let W = Y\A, W′ = {x ∈ W : x ∈ U_y , y ∈ A  and U  is open in X}.

Consider the open covered W^* = {W_x : x ∈ W } subspace W.

Since the subspace W is paracompact, there exists a locally finite G - cover V inscribed in W^*.

For any v ∈ V, we fix a point (vertex) x_v ∈ W such that gx_v = x_gv, where g ∈ G.

If a point x_v ∈ W' we fix such a point (vertex) a_v ∈ A and an open set S_v, a_v ∈ S_v such that  $x_v\in (S{)}_{{\mathrm{a}}_{\mathrm{v}}}$and n(S_v , a_v ) = m(x_v), and ga_v = a_gv.

If $x_v\overline{\in }{W}^{\text{'}}$ we put a_v fixed a₀ ∈ A. Let {P_v : v ∈ V} be partition of unity subordinate to V and P_v(x) = P_gv(gx).

The required continuation F : Y  → Z(X) is defined as follows:

$F\left(x\right) =$$\left\{\begin{array}{ll}f\left(x\right);& ifx\in A\\ \sum P_v\left(x\right)\cdot f\left(a_v\right)\mathrm{,}& ifx\in W\end{array}\right.$

1. Now we show that the mapping F : Y  → Z(X) is equivariant, i.e. gF(x) = F(gx), where g ∈ G.
   
   1)  If the point x ∈ A, due to the invariance of the set, we have, gF(x) = gf(x) = f(gx) = F(gx). 
   2)  Let the point x ∈ W then we have gF(x) = g∑(P_v(x) · f(a_v)) = ∑(P_v(x) · gf(a_v) = ∑P_v(x) · gf(a_v) = ∑P_v(x) f(a_gv).   gF(x) = ∑P_gvg(x) · f(a_gv) = ∑P_v(x) f(a_gv).  
   Hence, gF(x) = F(gx) i.e. the mapping F is equivariant.
2. Due to the - invariance of the closed set A and the simpliciality of a certain mapping F : Y → Z(X), the mapping F(x) is continuous. 

 Lemma 1 is proved.

By Lemma 1 and Theorem 1, we have

Theorem 4. The space X ∈ G  A(N)R(S)  if and only if there is a G - retraction r (neighborhood)  G - space Z(X)  on the G - space X.

Lemma 2. Let X ∈ A(N)R(S) .

Then there is a G -  retraction R_n : O(Xⁿ) → Xⁿ such that G ⊆ Sⁿ 

and n ∈ N such that O(Xⁿ) is a neighborhood Xⁿ to Z(Xⁿ) .

Proof. Let X be ANR(S) - space. It follows from the results of R.

Cauty that there is a retraction  r : U → X, where U is a neighborhood of the space X in Z(X).

We put V = (rⁿ)⁻¹(Uⁿ), where V ⊂ (Z(X))ⁿ, rⁿ : Uⁿ → Xⁿ, Uⁿ ⊂ (Z(X))ⁿ.

Now we define the mapping φ : Z(Xⁿ) → (Z(X))ⁿ  as follows: z ∈ Z(Xⁿ), z = ${\sum }_{i=1}^k{m}_i\overline{x_i}$,

$x_i{x}_1^i\mathrm{,}{x}_2^i\mathrm{,}\dots \mathrm{,}{x}_n^i$. We put  φ(z) = $\sum m_i\overline{x_1^i}\sum m_i\overline{x_2^i}\mathrm{,}\dots \mathrm{,}\sum m_i\overline{x_n^i}$. Obviously, φ(z) ∈ (Z(X))ⁿ. It is easy to check that the mapping φ : Z(Xⁿ) → (Z(X))ⁿ is continuous. We put  R_n = rⁿ ◦φ and φ⁻¹(V ) = O( Z(Xⁿ)).

Hence R_n : Z(Xn) φ : Zⁿ. Now we show that R_n is an equivariant mapping, i.e. the equality R_n(gz) = gR_n(z) holds.

R_n(gz) = rⁿ(φ(gz)) = rⁿ(φ(∑m_i$\overline{g{x}_i}$)) = rⁿ(φ(∑m_i$\overline{g(x_1^i . . . , x_n^i )}$)) =

= rⁿ(φ(∑mi$\overline{g(x_{\left(1\right)}^i . . . , x_{g\left(n\right)}^i )}$ )) = rⁿ((∑m_i$\overline{x_{g\left(j\right)}^i}$)) = rⁿ((∑m_i $\overline{x_j^i}$)) = g(rⁿ ∑m_i $\overline{x_j^i}$) = gR_n(z).

Hence, the mapping R_n is equivariant. It is easy to check that R_n is a continuous retraction.

Lemma 2 is proved.

 

Рис. Иллюстрации к теоремам: a — z₀(x) — x; b — z₁(x) — отрезки, вершины — точки X (одномерные симплексы); c — z₂(x) — треугольники, вершины — точки X (двумерные симплексы; d — z₃(x) — тетраэдры, вершины — точки X (трехмерные симплексы)

Иллюстрации к теоремам: a — z₀(x) — x; b — z₁(x) — отрезки, вершины — точки X (одномерные симплексы); c — z₂(x) — треугольники, вершины — точки X (двумерные симплексы; d — z₃(x) — тетраэдры, вершины — точки X (трехмерные симплексы)

 

Theorem 5. Let X ∈ A(N)R(S). Then Xⁿ ∈ G − A(N)R(S), where G ⊆ Sⁿ –  is a subgroup of the group of all permutations.

Proof. Let X ∈ A(N)R(S) and  be a stratified G − space, A its closed invariant G − subset, f : A → Xⁿ  is an arbitrary continuous G − mapping. Let O = φ⁻¹(O(Z(Xⁿ))). 

We put F_g = F◦rⁿ, rⁿ = $R_n^n$− the Cartesian product of retraction  defined by Lemma 2, F : Y → Z(Xn) mapping defined in Lemma 1.

Then the mapping F_g is a G extension, since F_g is the composition of two G mappings F and r^U. Obviously, F_g is an extension of the mapping f.The theorem is proved.

This theorem implies

Corollary 1. If X is a G - space and  X ∈ A(N)R(S) then X ∈ G − A(N)R(S).

By virtue of Lemma 1, we can also assert.

Corollary 2. Let X be a topological stratifiable G - space. If Y is a stratified G - space A is an invariant G - space, f : A → X\G. LXet|GX is an equivariant mapping. Then  f  has an equivariant continuous extension F : Y → ZX\G.

Corollary 1 implies.

Corollary 3. If X ∈ G − A(N)R(S), then X/G ∈ G − A(N)R(S).

Definition [13]. The set A ⊂ X  is called homotopically dense in X if there exists a homotopy h(x,t) : X ×D[0, 1]→X  such that h(x,0) = id_X  and h (X × (0,1]) ⊂ A.

Theorem 6. For any stratified space X and for any n ∈ N⁺ subspace Z(X)\Z_n(X)  is homotopically dense in Z(X).

Proof. Let X is the stratified space and n ∈ N⁺.

Fixing the point z₀ ∈ Z(X)\Z_n(X), where

z₀ ∈ <  ̅x₁, ̅x₂,..., ̅x_n+1  >  and

$z_0 =m_1^0{\overline{x}}_1+m_2^0{\overline{x}}_2+\mathrm{...}+m_{n+1}^0{\overline{x}}_{n+1}\mathrm{,}$

thus suppz₀ = { ̅x₁, ̅x₂,..., ̅x_n+1}.

We will construct the homotopy h(x,t) : Z(X) ×[0, 1] → ZX, assuming h(z, t) = tz₀ + (1 − t)z.

By virtue of the convexity of the space Z(X) for any z ∈ Z(X) and t ∈ [0, 1] the point h(µ, t) belongs to Z(X), that is h(µ, t) ∈ Z(X), ∀z ∈ Z and ∀t ∈ [0, 1].

If t = 0, then h(µ,0), that is h(µ,0) = id_Z(X).

If t > 0 and t ≤ 1, then h(µ, t) = tz₀ + (1 − t)z belongs to Z(X)\Z_n(X) because the carriers supph(µ, t)  of point h(µ, t)  consist of at least n + 1 points, that is

h(µ, t) = tz₀ + (1 − t)z =

$t(m_1^0{\overline{x}}_1+m_2^0{\overline{x}}_2+\mathrm{...}+m_{n+1}^0{\overline{x}}_{n+1})+$$(1-t)$$(m_1^0{\overline{x^{\text{'}}}}_1+m_2^0{\overline{x^{\text{'}}}}_2+\mathrm{...}+m_k^0{\overline{x^{\text{'}}}}_k)$$\in Z\left(X\right)\backslash Z_n\left(X\right)$,

the point h(µ, t) carrier consists of points z and z₀ carriers, and it consists of different (n+1) points.

So the point h(z, (0,1]) $\overline{\subset }$ Z_n(X) and h(z, (0,1]) ⊂  Z(X)\Z_n(X), which was required to be proved.

The theorem isproved.

Conclusion

In this paper we consider that the functor Z is open, normal and monadic in the category of stratified spaces and continuous maps to itself. The dimensional properties of the space Z(X) for the stratified space X are also studied, the subfunctor of the functor Z with the corresponding nested dimension is determined for each n.

Information about the conflict of interests: authors and reviewers declare no conflict of interests.

Информация о конфликте интересов: авторы и рецензенты заявляют об отсутствии конфликта 

About the authors

Tursunboy F. Zhuraev

Tashkent State Pedagogical University named after Nizami

Author for correspondence.
Email: tursunzhuraev@mail.ru
ORCID iD: 0009-0005-5379-3862

Doctor of Physical and Mathematical Sciences, associate professor of the Department of Mathematics

Uzbekistan, 27, Bunyodkor Street, Tashkent, 700100

Mikhail V. Dolgopolov

Samara State Technical University

Email: mikhaildolgopolov68@gmail.com
ORCID iD: 0000-0002-8725-7831

associate professor, Candidate of Physical and Mathematical Sciences, Department of Higher Mathematics

Russian Federation, 244, Molodogvardeyskaya Street, Samara, 443100

References

Supplementary files