The stable-homotopy tag has no usage guidance.

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### Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...

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613 views

### Calabi-Yau cohomology?

My question here is going to be this -- but I'll give a bit of background to explain myself in a moment:
What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ ...

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284 views

### Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...

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383 views

### Refinement of concept of support of a module

My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...

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330 views

### How to see the quaternionic hopf map generates the stable 3-stem?

I am looking for a direct proof that the quaternionic hopf map generates (after suspension) the 3rd stable homotopy group of spheres. There are some related MO questions, for example:
...

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191 views

### “topological” Ochanine genus?

The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus I am aware of a lift to a "spin orientation of Tate K-theory", namely to ...

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105 views

### A tensor product for triangulated categories?

Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category.
Stable $\infty$-categories give ...

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227 views

### stable homotopy of BG_+?

Let $G$ be an abelian group (if this helps, let's say also finite). Let $BG_+$ be the classifying space together with a disjoint base point. What are the stable homotopy groups
$$\pi^s_m (BG_+) := ...

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153 views

### Fibrations of orthogonal G-spectra and fixed points

There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...

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190 views

### May's infinite loop machine for Friedlander's result for Adams conjecture

Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space.
...

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238 views

### Is it possible to define a perverse $t$-structure for a certain triangulated category of sheaves of spectra?

The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it ...

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364 views

### Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem

I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...

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150 views

### Schwede's DB spectra and MU

In Stefan Schewede's paper Formal groups and stable homotopy of commutative rings, he introduces $\Gamma$-rings (ring spectra) $DB$ for any commutative ring $B$ such that the the set of 1-dimensional ...

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152 views

### Moore spectra are not E-infinity (oldest known proof)

Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map
$$ S^0 \overset{p^i}\longrightarrow S^0 $$
where $S^0$ be the sphere spectrum. In the Mathoverflow ...

**5**

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101 views

### Spectral Sequences of Parametrized Spectra

I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up:
Suppose that I have a parametrized spectra ...

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121 views

### Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...

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197 views

### Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time.
What will follow is sort of vernacular but whether it can be ...

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280 views

### Fields in Stable Homotopy Theory

It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...

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123 views

### Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...

**4**

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207 views

### Gluing Local Spectra

Suppose we have four spectra: $E_1$, $E_2$, $X_1$ and $X_2$ where $X_i$ is $E_i$ local and $L_{E_1\wedge E_2}X_1\simeq L_{E_1\wedge E_2}X_2$. Of course what this really means (I think), if we're ...

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88 views

### Maps between equivariant loop spaces

I have an elementary question about equivariant loop spaces that I feel it should be well known.
Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation ...

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149 views

### Twisting of the power functor

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a ...

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151 views

### Are finite (levelwise) homotopy limits of spectra homotopy invariant?

I found an easy proof that the (levelwise) homotopy limit of a pointwise equivalence of finite diagrams of orthogonal spectra is an equivalence, without assuming that the spectra in the diagrams are ...

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164 views

### In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes

Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map
$1+3: \Omega S^{2} \to BG_{3}$
where $1 +3$ is a unit in $3$-adics.
Here $G_{3}$ is the unit component of ...

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335 views

### Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...

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88 views

### Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...

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242 views

### Functor of points of a tensor triangulated category

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra?
Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be ...

**2**

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201 views

### Units of a ring spectrum

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?
A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...

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165 views

### Quillen functors and stable model categories

Are there any books or papers where I can find some general statements and methods for working with Quillen functors that are not equivalences (and not localizations)? In particular, I would like to ...

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136 views

### Formal n-buds from BU(n) rather than SU(n)

It's known, from Ravenel's green book, as well as other sources, that we build formal group laws over a ring from n-buds, where an n-bud is essentially a truncated formal group law (sometimes called a ...

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121 views

### Bousfield Complements of Interesting Spectra

For a spectrum $X$, Bousfield constructs a spectrum (which is only well-defined up to Bousfield equivalence) $aX$, which he shows satisfies some nice properties, like $\langle a^2X\rangle=\langle ...

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282 views

### Cohomology theory associated to the spectrum BG

Hi, I've recently been interested in Stable Homotopy Theory and was reading this text to understand some basics: http://www.maths.ed.ac.uk/~aar/papers/carlmilg.pdf
Near the end of the text (p582) we ...

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### Periodicity Theorem for D(R)

For a derived category of a Noetherian ring (or perhaps more generally), can we talk about a Periodicity Theorem? We have Thick Subcategory Theorems and Nilpotence Theorems (HPS 91) for D(R), and in ...

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188 views

### Whitehead Theorem for Harmonic Spectra

What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite ...

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522 views

### The stable-homotopy-homology-theory

Hi
Is there a way to stabilise relative homotopy groups into giving the stable-homotopy-homology-functor?
The fact that the homotopy excision theorem holds for exactly the same kind of pair that ...