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For reasonable spaces $X$, the Dold-Thom Theorem states that $\pi_i(SP(X)) \cong \tilde{H}_i(X)$ where $SP(X) = \bigsqcup_i \mathrm{Sym}^i(X)$. There is a purely algebro-geometric realization of this Theorem, as explain here. I have two (unrelated) questions here:

  1. Is the Dold-Thom theorem true if we take the homotopy quotient for $\mathrm{Sym}^i(X) = X^i//S_n$ (or equivalently treat it as a stack)?

  2. To what extent can we use the Dold-Thom theorem as a definition of cohomology? For instance, can one directly prove something like the (Grothendieck) Lefschetz trace formula with respect to $\pi_i(SP(X))$? Or how about the rationality of the zeta function for varieties over a finite field?

The reason I am interested in this is that I would like to define a new cohomology theory where I believe I have the analog of $\mathrm{Sym}^n(X)$ and use a Dold-Thom theorem like statement to define my cohomology theories. If this is to be useful, one must be able to prove things about cohomology in terms of the homotopy theory interpretation...

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    $\begingroup$ Have you looked into Suslin homology, as in this paper of Suslin-Voevodsky link.springer.com/article/10.1007/BF01232367 ? For your question (1), the answer is no if you take torsion coefficients (think about $X$ being a point) but yes rationally. $\endgroup$
    – Phil Tosteson
    Commented Jan 22 at 3:23
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    $\begingroup$ In topology, if you use the homotopy orbit space $ES_n\times _{S_n}X^n$ instead of the orbit space $Sym^n(X)$, then you get stable homotopy groups instead of homology groups (after group completion or plus-construction) $\endgroup$
    – Tom Goodwillie
    Commented Jan 22 at 3:24

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