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David Roberts ♦
David Roberts ♦
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I'll attempt an answer to the mathematical question, without discussing the motivation. As I understand it, $M$ is a manifold with $S_n$-action, and we are asking whether there exists an $S_n$-equivariant map $f:F_n(M)\to M$, where the action on $F_n(M)$ is by permutation of coordinates (and in particular has nothing to do with the action on $M$).

As a first observation, note that if $x\in M$ is a fixed point $x$, then such an $f$ exists; just map everything to $x$.

A general necessary condition for the existence of a $G$-map $f:X\to Y$ between $G$-spaces is given by the Faddell-Husseini index, as decribed for example in this paper.this paper:

Pavle V. M. Blagojević, Wolfgang Lück, Günter M. Ziegler, Equivariant Topology of Configuration Spaces, J. Topology 8 (2015) pp 377–413, doi:10.1112/jtopol/jtu029, arXiv:1207.2852.

Given a $G$-space $X$ and a commutative ring $R$, the Faddell-Husseini index of $X$ is the ideal in $H^*(BG;R)$ defined by $$ \operatorname{Index}_G(X;R):=\ker(p^*:H^*(BG;R)\to H^*(EG\times_G X;R)), $$ where $p:EG\times_G X\to BG$ is the projection of the Borel fibration. Then it is easy to see that if there exists a $G$ map $f:X\to Y$ then $$ \operatorname{Index}_G(Y;R)\subseteq \operatorname{Index}_G(X;R) $$ must hold. You can sometimes rule out existence of equivariant maps using this property. The index of the configuration space $F_n(M)$ is probably difficult to compute in general, but is done in the linked paper for $M=\mathbb{R}^d$.

I'll attempt an answer to the mathematical question, without discussing the motivation. As I understand it, $M$ is a manifold with $S_n$-action, and we are asking whether there exists an $S_n$-equivariant map $f:F_n(M)\to M$, where the action on $F_n(M)$ is by permutation of coordinates (and in particular has nothing to do with the action on $M$).

As a first observation, note that if $x\in M$ is a fixed point $x$, then such an $f$ exists; just map everything to $x$.

A general necessary condition for the existence of a $G$-map $f:X\to Y$ between $G$-spaces is given by the Faddell-Husseini index, as decribed for example in this paper. Given a $G$-space $X$ and a commutative ring $R$, the Faddell-Husseini index of $X$ is the ideal in $H^*(BG;R)$ defined by $$ \operatorname{Index}_G(X;R):=\ker(p^*:H^*(BG;R)\to H^*(EG\times_G X;R)), $$ where $p:EG\times_G X\to BG$ is the projection of the Borel fibration. Then it is easy to see that if there exists a $G$ map $f:X\to Y$ then $$ \operatorname{Index}_G(Y;R)\subseteq \operatorname{Index}_G(X;R) $$ must hold. You can sometimes rule out existence of equivariant maps using this property. The index of the configuration space $F_n(M)$ is probably difficult to compute in general, but is done in the linked paper for $M=\mathbb{R}^d$.

I'll attempt an answer to the mathematical question, without discussing the motivation. As I understand it, $M$ is a manifold with $S_n$-action, and we are asking whether there exists an $S_n$-equivariant map $f:F_n(M)\to M$, where the action on $F_n(M)$ is by permutation of coordinates (and in particular has nothing to do with the action on $M$).

As a first observation, note that if $x\in M$ is a fixed point $x$, then such an $f$ exists; just map everything to $x$.

A general necessary condition for the existence of a $G$-map $f:X\to Y$ between $G$-spaces is given by the Faddell-Husseini index, as decribed for example in this paper:

Pavle V. M. Blagojević, Wolfgang Lück, Günter M. Ziegler, Equivariant Topology of Configuration Spaces, J. Topology 8 (2015) pp 377–413, doi:10.1112/jtopol/jtu029, arXiv:1207.2852.

Given a $G$-space $X$ and a commutative ring $R$, the Faddell-Husseini index of $X$ is the ideal in $H^*(BG;R)$ defined by $$ \operatorname{Index}_G(X;R):=\ker(p^*:H^*(BG;R)\to H^*(EG\times_G X;R)), $$ where $p:EG\times_G X\to BG$ is the projection of the Borel fibration. Then it is easy to see that if there exists a $G$ map $f:X\to Y$ then $$ \operatorname{Index}_G(Y;R)\subseteq \operatorname{Index}_G(X;R) $$ must hold. You can sometimes rule out existence of equivariant maps using this property. The index of the configuration space $F_n(M)$ is probably difficult to compute in general, but is done in the linked paper for $M=\mathbb{R}^d$.

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Mark Grant
Mark Grant
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I'll attempt an answer to the mathematical question, without discussing the motivation. As I understand it, $M$ is a manifold with $S_n$-action, and we are asking whether there exists an $S_n$-equivariant map $f:F_n(M)\to M$, where the action on $F_n(M)$ is by permutation of coordinates (and in particular has nothing to do with the action on $M$).

As a first observation, note that if $x\in M$ is a fixed point $x$, then such an $f$ exists; just map everything to $x$.

A general necessary condition for the existence of a $G$-map $f:X\to Y$ between $G$-spaces is given by the Faddell-Husseini index, as decribed for example in this paper. Given a $G$-space $X$ and a commutative ring $R$, the Faddell-Husseini index of $X$ is the ideal in $H^*(BG;R)$ defined by $$ \operatorname{Index}_G(X;R):=\ker(p^*:H^*(BG;R)\to H^*(EG\times_G X;R)), $$ where $p:EG\times_G X\to BG$ is the projection of the Borel fibration. Then it is easy to see that if there exists a $G$ map $f:X\to Y$ then $$ \operatorname{Index}_G(Y;R)\subseteq \operatorname{Index}_G(X;R) $$ must hold. You can sometimes rule out existence of equivariant maps using this property. The index of the configuration space $F_n(M)$ is probably difficult to compute in general, but is done in the linked paper for $M=\mathbb{R}^d$.