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Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all continuous maps and homotopies preserve the chosen base points). We define $π'_1(X) $ as the set of homotopy classes of products of injective loops in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$. We say that $\mathrm{D} \subset \mathrm{C}$ is essentially wide if for every $X \in \mathrm{C}$ there exists $Y \in \mathrm{D}$ such that $X$ is homotopy equivalent to $Y$.

Is there a $ D $ homotopically wide subclass of $ C $ such that, for $ X \in D $

1. The operation is well defined, i.e. there always exist an injective loop homotopic to $a \cdot b$
2. The natural embedding of $π'_1 (X) \to π_1 (X)$ is an isomorphism.

We define $π''_1 (X)$ similarly, but with homotopies in the class of injective loops.

Is there a $ D $ homotopically wide subclass of $ C $ such that, for $X \in D$

3. The operation is well defined, i.e. among the classes of injective homotopy there exists and is uniquely a class of loops freely homotopic $a \cdot b$
4. The natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism.

Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all continuous maps and homotopies preserve the chosen base points). We define $π'_1(X) $ as the set of homotopy classes of products of injective loops in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$. We say that $\mathrm{D} \subset \mathrm{C}$ is essentially wide if for every $X \in \mathrm{C}$ there exists $Y \in \mathrm{D}$ such that $X$ is homotopy equivalent to $Y$.

Is there a $ D $ essentially wide subclass of $ C $ such that, for $ X \in D $

1. The operation is well defined, i.e. there always exist an injective loop homotopic to $a \cdot b$
2. The natural embedding of $π'_1 (X) \to π_1 (X)$ is an isomorphism.

We define $π''_1 (X)$ similarly, but with homotopies in the class of injective loops.

Is there a $ D $ essentially wide subclass of $ C $ such that, for $X \in D$

3. The operation is well defined, i.e. among the classes of injective homotopy there exists and is uniquely a class of loops freely homotopic $a \cdot b$
4. The natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism.

Let $\mathrm{NiceTop}$ be the class of topological spaces that includes at least all subspaces $\mathbb{R}^n$. Further we are in the category $\mathrm{NiceTop}_{*}$ (category of pointed spaces; all continuous maps and homotopies preserve the chosen basepoints). We define $π'_1 (X)$ as the set of homotopy classes of products of injective loops in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$.

1. Is the operation defined correctly, i.e. does there always exist an injective loop homotopic to $a \cdot b$?
2. Is it true, the natural embedding $π'_1 (X) \to π_1(X)$ is an isomorphism?

We define $π'' _ 1 (X)$ similarly, but with homotopies in the class of injective loops.

3. Is the operation defined correctly, i.e. is it true that among the classes of injective homotopy there is also a unique class of loops freely homotopic $a \cdot b$?
4. Is it true that the natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism?

Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all continuous maps and homotopies preserve the chosen base points). We define $π'_1(X) $ as the set of homotopy classes of products of injective loops in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$. We say that $\mathrm{D} \subset \mathrm{C}$ is essentially wide if for every $X \in \mathrm{C}$ there exists $Y \in \mathrm{D}$ such that $X$ is homotopy equivalent to $Y$.

Is there a $ D $ homotopically wide subclass of $ C $ such that, for $ X \in D $

1. The operation is well defined, i.e. there always exist an injective loop homotopic to $a \cdot b$
2. The natural embedding of $π'_1 (X) \to π_1 (X)$ is an isomorphism.

We define $π''_1 (X)$ similarly, but with homotopies in the class of injective loops.

Is there a $ D $ homotopically wide subclass of $ C $ such that, for $X \in D$

3. The operation is well defined, i.e. among the classes of injective homotopy there exists and is uniquely a class of loops freely homotopic $a \cdot b$
4. The natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism.

Let $\mathrm{NiceTop}$ be the class of topological spaces that includes at least all subspaces $\mathbb{R}^n$. Further we are in the category $\mathrm{NiceTop}_{*}$ (category of pointed spaces; all continuous maps and homotopies preserve the chosen basepoints). We define $π'_1 (X)$ as the set of homotopy classes of injective loops and their exponentiations in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$.

1. Is the operation defined correctly, i.e. does there always exist an injective loop homotopic to $a \cdot b$?
2. Is it true, the natural embedding $π'_1 (X) \to π_1(X)$ is an isomorphism?

We define $π'' _ 1 (X)$ similarly, but with homotopies in the class of injective loops.

3. Is the operation defined correctly, i.e. is it true that among the classes of injective homotopy there is also a unique class of loops freely homotopic $a \cdot b$?
4. Is it true that the natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism?

Let $\mathrm{NiceTop}$ be the class of topological spaces that includes at least all subspaces $\mathbb{R}^n$. Further we are in the category $\mathrm{NiceTop}_{*}$ (category of pointed spaces; all continuous maps and homotopies preserve the chosen basepoints). We define $π'_1 (X)$ as the set of homotopy classes of products of injective loops in $X$ with the operation $[a] \cdot [b] = [a \cdot b]$.

1. Is the operation defined correctly, i.e. does there always exist an injective loop homotopic to $a \cdot b$?
2. Is it true, the natural embedding $π'_1 (X) \to π_1(X)$ is an isomorphism?

We define $π'' _ 1 (X)$ similarly, but with homotopies in the class of injective loops.

3. Is the operation defined correctly, i.e. is it true that among the classes of injective homotopy there is also a unique class of loops freely homotopic $a \cdot b$?
4. Is it true that the natural embedding $π''_ 1(X) \to π_1 (X)$ is an isomorphism?