Questions tagged [stable-homotopy]

Stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.

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5 votes
1 answer
276 views

Morita equivalence and connectivity

Let $A, B$ be Morita equivalent $\mathbb{E}_1$-ring spectra. Fix an an $(A, B)$-bimodule $P$ and a $(B, A)$-bimodule $Q$ such that $P \otimes_B Q \cong A$ and $Q \otimes_A P \cong B$. If $A$ is ...
27 votes
2 answers
2k views

Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory?)

The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...
18 votes
1 answer
539 views

Is there a cotangent bundle of a stable $\infty$-category?

Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following? When $C$ is the ...
2 votes
0 answers
160 views

Infinite loop space as an endofunctor of compactly generated weak hausdorff topological spaces?

I am trying to see whether it is possible to define smash product of infinite loop spaces using the space $S^{\infty}$. Let C be the category of compactly generated weak Hausdorff topological spaces. ...
3 votes
1 answer
135 views

Can a phantom map have finite cofiber?

Let $f : X \to Y$ be a nonzero phantom map between spectra. Can the cofiber of $f$ be a finite spectrum? Recall that a map $f$ is said to be phantom if $f \circ i = 0$ whenever $i : F \to X$ is a map ...
1 vote
0 answers
204 views

Properties of colim Ωⁿ Σⁿ X

I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of the endofunctor $Q: ...
3 votes
0 answers
46 views

Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups

I am a physicist knowing a bit of algebraic topology, and trying to answer the following question. This is perhaps not appropriate as a question on MO, in which case I apologize. I posted this ...
1 vote
1 answer
216 views

Symmetric-monoidal-associative smash product up to homotopy

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above. Recall that a sequential ...
32 votes
1 answer
2k views

What happened to the last work Gaunce Lewis was doing when he died?

In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
15 votes
1 answer
735 views

If homotopy groups of spaces are identical, then stable ones are also identical?

Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$? In particular, is this ...
3 votes
1 answer
152 views

Which positive flat stable model structures on (flavors of) spectra have the property that cofibrant operad-algebras forget to cofibrant spectra?

Let $M$ be a monoidal model category and $O$ an operad valued in $M$, and the category of $O$-algebras inherits a model structure from $M$ where a map $f$ is a weak equivalence (resp. fibration) if ...
2 votes
0 answers
63 views

Equivariant $K$-theory and proper actions of discrete groups

The work of Lück and Oliver describes the generalization of equivariant $K$-theory to infinite discrete groups. When $X$ is a finite proper $G$-CW complex, there exist Bott isomorphisms $K^n_G(X)\cong ...
7 votes
0 answers
262 views

Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
7 votes
0 answers
138 views

Explicit framed null bordism realizing $\eta\nu =0$ in stable homotopy group of spheres

There are many standard results in the stable homotopy group of spheres (or equivalently framed bordism groups), about which I would like to acquire better geometric understanding. For example I ...
6 votes
1 answer
220 views

Does a complex-oriented $E_1$ ring spectrum (not assumed to have graded-commutative homotopy groups) receive a map from $MU$?

It's well-known that complex cobordism $MU^\ast$ is universal among complex-oriented associative, graded-commutative cohomology theories $E$. This means that if $E$ is a multiplicative cohomology ...
23 votes
3 answers
2k views

What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero. Question: What are some "toy models" ...
2 votes
0 answers
64 views

What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category). Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
4 votes
0 answers
91 views

What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
4 votes
1 answer
156 views

The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...
8 votes
1 answer
334 views

Telescopic localisation of Eilenberg-MacLane spaces

Fix a prime $p$ and an integer $n>0$. Let $K$ be the corresponding Morava $K$-theory spectrum, and let $T$ be the telescope of a $v_n$-self map of a finite spectrum of type $n$, and let $X$ be the ...
13 votes
1 answer
335 views

Is $KU\otimes S^1_+$ isomorphic to $F(S^1_+,KU)$ as $E_\infty$ rings?

There are various ways to construct $KU$ as an $E_\infty$ ring spectrum; I will take that as given. Using this, we can make $KU\otimes G_+$ into an $E_\infty$ ring for any commutative topological ...
4 votes
0 answers
426 views

An $E_{\infty}$-algebra is a $C_{\infty}$-algebra?

Past this question in MO have raised the following questions for me. Question In characteristic $0$, it is well-known that a Kadeishvili‘s $C_{\infty}$-algebra is an $E_{\infty}$-algebra. However, do ...
13 votes
3 answers
1k views

What is so 'coloured' on Chromatic Homotopy Theory

As the title suggest, I would like know the motivation/ historical background why chromatic homotopy theory is called 'chromatic'. Literally, what analogy to colors it might have. Accordings to ...
13 votes
2 answers
483 views

How many automorphisms are there of the category of filtered spectra?

Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...
2 votes
1 answer
248 views

Filtered homotopy colimits of spectra

Let $\mathcal{I}: \mathbb{N} \to \operatorname{Sp}$ be a diagram in the infinity category of spectra. Let $\pi_0(\mathcal{I})$ denote the corresponding $1$-categorical diagram (i.e. compose $\mathcal{...
3 votes
1 answer
390 views

Homotopy colimit commutes with homotopy groups

I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem Let $(E, \mu, \iota)$ be a ring spectrum. For $\beta \in \pi_n(E)$ an element of the $n$th stable ...
2 votes
2 answers
241 views

The complex $K$-theory of the Thom spectrum $MU$

The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
5 votes
0 answers
175 views

Preservation of (co)limits under taking derived categories

Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects). ...
2 votes
0 answers
180 views

The derived category of $p$-complete abelian groups is comonadic over the derived category of $\mathbb F_p$-vector spaces?

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ext{Ext}$Let $p$ be a prime. The adjunction $$\mathbb F_p \otimes_\mathbb{Z} (-) : \Mod(\mathbb Z) \rightleftarrows \Mod(\mathbb F_p) : U $$ descends ...
2 votes
1 answer
197 views

What is the homotopy type of the smash power of Moore spectra $(S/2)^{\otimes n}$?

Let $S/2$ be the mod $2$ Moore spectrum, and let $n \in \mathbb N$. Question: What is the homotopy type of the $n$th smash power $(S/2)^{\otimes n}$? Notes: When $p$ is odd, we have $S/p \otimes S/p =...
2 votes
0 answers
126 views

What is the colimit of the punctured $k$-cube $\{X^{\epsilon_1 + \dotsb + \epsilon_k} \mid \epsilon_i \in \{0,1\} \text{ not all } 1\}$?

Let $\mathcal C$ be a stably monoidal $\infty$-category, and let $I \xrightarrow f X \to Y$ be a fiber sequence where $I$ is the unit. Then for each $k \in \mathbb N$, we can form a canonical cubical ...
7 votes
1 answer
594 views

Image of J in the classical Adams Spectral Sequence

Hey all, I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of ...
3 votes
1 answer
347 views

Spectral sequence in Adam's book, Theorem 8.2

I am having trouble in understanding Theorem 8.2 of Adams's book and the application afterwards of constructing the spectral sequence. I think I should prove somehow that the spectral sequence in this ...
2 votes
2 answers
292 views

Does the homotopy category of finite spectra act on stable homotopy categories?

Assume that C is a stable infinity category; $SH_{fin}$ is the homotopy category of finite spectra. Is there a canonical bi-functor (action? module structure?) $SH_{fin}\times hC \to hC$? Is there any ...
7 votes
1 answer
307 views

Stable splitting of $\Omega SU(n)$

The space $\Omega SU(n)$ is homotopy-equivalent to $SL_n(\mathbb{C}[z,z^{-1}])/SL_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F_k$ which can ...
10 votes
0 answers
365 views

What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?

The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
22 votes
3 answers
1k views

Why the stable module category?

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
6 votes
1 answer
162 views

When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups?

Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\...
4 votes
1 answer
292 views

Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces

Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
10 votes
1 answer
299 views

Which spectra have a universal connective quotient?

Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...
3 votes
0 answers
126 views

Which spectra have a homotopy-universal connective quotient?

Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...
6 votes
2 answers
377 views

How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism

The Pontryagin-Thom construction gives an isomorphism from the stable homotopy groups of spheres and framed cobordism groups. It seems to be well-established that for dimension 1 (see this question), ...
9 votes
1 answer
277 views

Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$

Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \...
5 votes
1 answer
189 views

Homotopy coherent localisation of a ring spectrum $E$ at a subset of $\pi_0E$

Homotopy coherent Invertibility. Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we ...
4 votes
1 answer
363 views

The “field of fractions” of the sphere spectrum (localization at $\pi_0(\mathbb{S})\setminus\{0\}$, the non-zero integers)

Perhaps the most common construction of the rational numbers is the one given by taking the field of fractions $\mathrm{Frac}(\mathbb{Z})\cong\mathbb{Q}$ of the ring $\mathbb{Z}$ of integers. I'm ...
4 votes
1 answer
208 views

Equivariant complex $K$-theory of a real representation sphere

Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
8 votes
1 answer
339 views

When can I extend a map of spectra?

Suppose I have a commutative ring $R$. Given an element $(x_1,x_2)\in R^2$ there exists a homomorphism $\mathbb{Z} \to R\otimes R$ taking $1$ to $x_1\otimes x_2$, so there exists a map $f:S^0 \to HR \...
3 votes
0 answers
133 views

Equivariant classifying space and manifold models

The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
14 votes
1 answer
936 views

Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?

The question is inspired by an answer to The concept of Duality It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...
6 votes
0 answers
217 views

Compatible algebraic Spanier-Whitehead dual

Let me first ask an intuitive version of the question: Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...

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