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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 11th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something, on * [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2)) <a href="https://ncatlab.org/nlab/revision/model+structure+on+equivariant+dgc-algebras/1">v1</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+equivariant+dgc-algebras">current</a>

   starting something, on

   • Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexfinally added the actual definition of the model structure <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+equivariant+dgc-algebras/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+equivariant+dgc-algebras/4">v4</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+equivariant+dgc-algebras">current</a>

   finally added the actual definition of the model structure

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 2nd 2020
   • (edited Oct 2nd 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Marek Golasiński]], _Equivariant rational homotopy theory as a closed model category_, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (<a href="https://doi.org/10.1016/S0022-4049(97)00127-8">doi:10.1016/S0022-4049(97)00127-8</a>) <a href="https://ncatlab.org/nlab/revision/diff/model+structure+on+equivariant+dgc-algebras/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+on+equivariant+dgc-algebras/6">v6</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+equivariant+dgc-algebras">current</a>

   added pointer to:

   • Marek Golasiński, Equivariant rational homotopy theory as a closed model category, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 24th 2020
   • (edited Oct 24th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLet $G$ be a finite group acting smoothly on a smooth manifold $X$. Then the fixed loci $X^H$, for $H \subset G$ are smooth manifolds, and hence forming fixed-locus wise vector spaces of smooth differential $n$-forms (for any $n \in \mathbb{N}$) yields a functor $$ \array{ G Orbits &\longrightarrow& VectorSpaces \\ G/H &\mapsto& \Omega^n_{dR}\big( X^H \big) } $$ **Question**: In which generality are these functors injective objects, and what's the proof? I can show this to be so for the case that $G$ is: of order 4 or cyclic of prime order. From the proof of these special cases it is pretty clear how the general case will work, and I suppose I can prove any number of further special cases by a case-by-case analysis; but it remains unclear to me how to formulate the fully general proof. A first inkling of how to approach this issue may be gleaned from the text offered as proof to Prop. 4.3 in [Triantafillou 82](https://ncatlab.org/nlab/show/equivariant+rational+homotopy+theory#Triantafillou82) (where it's PL dR forms instead of smooth forms, but the combinatorial part of the argument is the same), but I don't see how these hints are more than that.

   Let $G$ be a finite group acting smoothly on a smooth manifold $X$. Then the fixed loci $X^H$, for $H \subset G$ are smooth manifolds, and hence forming fixed-locus wise vector spaces of smooth differential $n$-forms (for any $n \in \mathbb{N}$) yields a functor

   $$\array{ G Orbits &\longrightarrow& VectorSpaces \\ G/H &\mapsto& \Omega^n_{dR}\big( X^H \big) }$$

   Question: In which generality are these functors injective objects, and what’s the proof?

   I can show this to be so for the case that $G$ is: of order 4 or cyclic of prime order.

   From the proof of these special cases it is pretty clear how the general case will work, and I suppose I can prove any number of further special cases by a case-by-case analysis; but it remains unclear to me how to formulate the fully general proof.

   A first inkling of how to approach this issue may be gleaned from the text offered as proof to Prop. 4.3 in Triantafillou 82 (where it’s PL dR forms instead of smooth forms, but the combinatorial part of the argument is the same), but I don’t see how these hints are more than that.