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I am slowly creating a bunch of entries on basic concepts of equivariant stable homotopy theory, such as
At the moment I am mostly just indexing Stefan Schwede’s
I can’t see that the Tambara functor literature has linked up with the equivariant stable homotopy theory literature very much. Isn’t this surprising, given the former is concerned with equivariant ring spectra? As Neil Strickland puts it
Tambara functors are to rings what Mackey functors are to abelian groups.
Rather than correspondences, they seem to deal with polynomials.
Mike Hill discusses them in his notes for Derived Equivariant Algebraic Geometry, so I’ve added that as a reference at Tambara functor.
At equivariant homotopy group, in the section ’Definition: Via G-equivariant maps’ it talks of a subgroup $H$ of $G$ but then doesn’t use it in the definition. Should mention of $H$ be removed?
Thanks for catching this. So I had started out saying this in one of the two possible ways and thne switched to the other half-way. Have made it consistent now.
I have again been touching a bunch of entries related to equivariant stable homotopy theory, such as equivariant homotopy group and Mackey functor and others. Mainly I have been adding statement of and pointers to various bits of May 96. In particular at Mackey functor – cohomology I have finally started to spell out the definition of the “ordinary” cohomology of a topological G-space with coefficients in a Mackey functor.
This was to set myself straight, for I wanted to properly convince myself of what I had previously thought was true only to then begin to worry about, namely that, for instance for $G \subset Aut_{\mathbb{R}}(\mathbb{H})$ we have equivalences such as
$H^{(4 - \mathbb{H}) + 1}(S^1, \pi_4(\Sigma^\infty_G S^{\mathbb{H}})) \simeq Hom_{Mackey}( \pi^{st}_5(S^1 \wedge S^{\mathbb{H}}), \pi_4(\Sigma^\infty_G S^{\mathbb{H}}) ) \simeq Hom_{Mackey}( \pi_4(\Sigma^\infty S^{\mathbb{H}}), \pi_4(\Sigma^\infty S^{\mathbb{H}}))$I want to eventually further clean this all up, but for tonight I do need to call it quits.
made a start at equivariant stable homotopy category and tried to improve a bit more at equivariant homotopy group and tom Dieck splitting.
Looking around related entries I guess coordinate-free spectrum should link up with G-spectrum and G-universe, no? Isn’t the ’universe’ of the former a $G$-universe for the trivial group?
That’s right, coordinate-free spectra are equivalently $G$-spectra indexed on a $G$-universe for $G$ the trivial group. Yes, these entries should be interlinked better. I don’t have time to do so right now, but if you have time, please feel invited to edit!
just noticed that the pdf link in this reference is broken now:
Might anyone know a working pointer, or have the file available, that we can upload it?
Might anyone know a working pointer, or have the file available, that we can upload it?
Its in the wayback machine
though you probably shouldn’t link there but instead rehost it.
added pointer to:
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