**31**

votes

**2**answers

2k views

### What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its $p$-...

**51**

votes

**6**answers

5k views

### third stable homotopy group of spheres via geometry?

It is ''well-known'' that the third stable homotopy group of spheres is cyclic of order $24$. It is also ''well-known'' that the quaternionic Hopf map $\nu:S^7 \to S^4$, an $S^3$-bundle, suspends to a ...

**38**

votes

**2**answers

3k views

### Are spectra really the same as cohomology theories?

Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. ...

**23**

votes

**4**answers

3k views

### What is a simplicial commutative ring from the point of view of homotopy theory?

Let $k$ be a field. There are two natural categories to consider:
The category of simplicial commutative $k$-algebras.
The category of connective $E_\infty$ $k$-algebras (i.e., chain complexes of $...

**27**

votes

**1**answer

556 views

### 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 ...

**85**

votes

**4**answers

4k views

### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
$\mathbb{Z}\...

**23**

votes

**3**answers

2k views

### Modern Source for Spectra (including Ring Spectra)

I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been ...

**49**

votes

**1**answer

948 views

### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...

**31**

votes

**4**answers

2k views

### Integral cohomology (stable) operations

There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...

**26**

votes

**1**answer

854 views

### Early stabilization in the homotopy groups of spheres

Thanks to Freudenthal we know that $\pi_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the table on Wikipedia of some of the homotopy groups of spheres and noticed ...

**20**

votes

**0**answers

682 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 $n \geq 3$ (...

**7**

votes

**2**answers

583 views

### Absence of Maps Between p-local and q-local spectra

Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that $[X,Y]_\ast=...

**23**

votes

**1**answer

814 views

### From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...

**17**

votes

**1**answer

1k views

### The cell structure of Thom spectra

I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...

**12**

votes

**0**answers

409 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:
third-stable-...

**15**

votes

**2**answers

1k views

### Are $G$-spectra the same as modules over a “group ring spectrum”?

Let $G$ be a finite group (maybe this will also work when $G$ is compact, or something, but to be safe we'll let it be finite). I imagine it's quite natural to ask: is the category of $G$-spectra ...

**13**

votes

**1**answer

597 views

### Geometric interpretation of families in the stable homotopy groups of spheres

There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image ...

**11**

votes

**2**answers

440 views

### The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory $...

**8**

votes

**1**answer

890 views

### A heart for stable equivariant homotopy theory

Let $G$ be a finite group. I wonder whether the following statement is true, known and written down:
There is a t-structure on the stable $G$-equivariant homotopy category such that the associated ...

**14**

votes

**2**answers

908 views

### Exotic spheres and stable homotopy in all large dimensions?

Given that the kervaire invariant one problem has been solved in (almost) all dimensions....my question is whether there exists an exotic sphere in all sufficently lagre dimensions? Given the Kervaire-...

**13**

votes

**3**answers

860 views

### Isomorphism between two universal p-typical formal group laws

EDIT: I've tried to alter the question so that its basic nature is clearer, as it's been unclear to a number of people now.
At any prime p, there is a graded polynomial ring $V \cong {\mathbb Z}_{(p)}...

**5**

votes

**2**answers

940 views

### Homotopy limit-colimit diagrams in stable model categories

It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...

**9**

votes

**0**answers

113 views

### t-structures on the tensor product of stable $\infty$-categories

It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...

**14**

votes

**2**answers

657 views

### Must a weak homotopy equivalence induce an isomorphism between stable homotopy groups?

I'm confused by the following question:
$f:X\to Y$ is a weak homotopy equivalence, that is $f_*:\pi_*(X)\to \pi_*(Y)$ is an isomorphism for any dimensional homotopy groups. However, for the stable ...

**11**

votes

**1**answer

309 views

### Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.
Not every ghost map ...

**5**

votes

**1**answer

295 views

### endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra:
Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let $\...