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Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in Cohen-Cohen-Mann-Milgram's paper that

"there is an equivariant embedding $f: F(S',k)\to F(S',k+1)$."

Why? Does this mean the following commutative diagram? What is the map $f$?

enter image description here

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I would label the right hand vertical arrow with $\Sigma_k$ instead of $\Sigma_{k+1}$, but yes.

You can let $f$ send the configuration $(p_1,\dots ,p_k)$ to $(p_1,\dots ,p_k,q)$, where $q$ is a point outside the configuration that depends continuously on it. Roughly, $q$ can be taken to be very near $p_k$, say, and can be specified by using a nonzero tangent vector of $S$ at $p_k$ that depends continuously on $p_k$. Such a vector field exists because the tangent bundle of $S'$ is trivial (unlike that of $S$ itself).

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    $\begingroup$ A common choice that is $\Sigma_k$-equivariant is to fix a homeomorphism $\phi$ between $S'$ and $S\backslash \mathrm{disk}$ and a fixed point $q$ in $\mathrm{disk}\backslash \mathrm{point}$ and then take the configuration $(p_1,\cdots p_k)$ to $(\phi(p_1),\cdots, \phi(p_k),q)$. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Nov 12, 2015 at 12:28
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If you are interested in an unified accounting of the various maps between ordered configuration spaces and what happens when you compose them, you could check out the paper FI-modules and stability for representations of symmetric groups by Church–Ellenberg–Farb.

  1. The forgetful maps from $F(M,k+1)\to F(M,k)$ always fit together into a contravariant functor $F(M,-)$ from FI to spaces, where FI denotes the category of finite sets and injections.
  2. When $M$ is an open manifold, the maps from $F(M,k)\to F(M,k+1)$ that you're talking about fit together into a covariant functor from FI to spaces up to homotopy.
  3. These two structures together interact nicely and define a functor from FI# to spaces up to homotopy, where FI# denotes the category of finite sets and "partially defined injections".

Instead of talking about equivariance of these various maps and their compositions, here that is all encoded in the functoriality; personally I find it a lot easier to keep track of that way. (The point of the paper isn't the existence of this structure, of course, but rather the surprisingly powerful consequences it has for the algebraic invariants of the configuration spaces.)

The applications to configuration spaces are in Chapter 6, and the maps on open manifolds you're talking about in Chapter 6.4. The category FI# is defined at the beginning of Chapter 4.

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