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edited Feb 2, 2022 at 23:21

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Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. However, in "Restricting the transformation group in equivariant CW complexes"Restricting the transformation group in equivariant CW complexes," Sören Illman showed that the restriction of a $G$-CW complex to an $H$-space does not generally give an $H$-CW complex. He then givesgave a construction of an $H$-homotopy equivalent $H$-CW complex with nice properties.

Related, and related to your question (2), is Illman's paper "Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion"Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion." One consequence of this paper is that $G/H\times G/K$ has a $G$-triangulation, so a $G$-CW structure. The problem that arises when you look at a product of two $G$-CW complexes is that the triangulations of the products of orbits don't have to interact well with the original CW structures.

Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. However, in "Restricting the transformation group in equivariant CW complexes," Sören Illman showed that the restriction of a $G$-CW complex to an $H$-space does not generally give an $H$-CW complex. He then gives a construction of an $H$-homotopy equivalent $H$-CW complex with nice properties.

Related, and related to your question (2), is Illman's paper "Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion." One consequence of this paper is that $G/H\times G/K$ has a $G$-triangulation, so a $G$-CW structure. The problem that arises when you look at a product of two $G$-CW complexes is that the triangulations of the products of orbits don't have to interact well with the original CW structures.

Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. However, in "Restricting the transformation group in equivariant CW complexes," Sören Illman showed that the restriction of a $G$-CW complex to an $H$-space does not generally give an $H$-CW complex. He then gave a construction of an $H$-homotopy equivalent $H$-CW complex with nice properties.

Related, and related to your question (2), is Illman's paper "Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion." One consequence of this paper is that $G/H\times G/K$ has a $G$-triangulation, so a $G$-CW structure. The problem that arises when you look at a product of two $G$-CW complexes is that the triangulations of the products of orbits don't have to interact well with the original CW structures.

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created Feb 2, 2022 at 23:16

Steve Costenoble

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Your question (1) may not be true as stated. One approach that I like is to think of the product of two $G$-CW complexes first as a $(G\times G)$-CW complex, which works because $G/H\times G/K \approx (G\times G)/(H\times K)$. Then you can consider it as a $G$-space by restricting along the diagonal $G\to G\times G$. However, in "Restricting the transformation group in equivariant CW complexes," Sören Illman showed that the restriction of a $G$-CW complex to an $H$-space does not generally give an $H$-CW complex. He then gives a construction of an $H$-homotopy equivalent $H$-CW complex with nice properties.

Related, and related to your question (2), is Illman's paper "Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion." One consequence of this paper is that $G/H\times G/K$ has a $G$-triangulation, so a $G$-CW structure. The problem that arises when you look at a product of two $G$-CW complexes is that the triangulations of the products of orbits don't have to interact well with the original CW structures.