# Tagged Questions

**9**

votes

**1**answer

180 views

### Algebraic $K$-theory of algebras in symmetric spectra: reference

I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines ...

**5**

votes

**0**answers

122 views

### Reference for analyticity of $K$-theory

This is a follow-up to my last question, Homotopy excision for structured ring spectra -- reference?. The immediate reason why I care about Blakers-Massey theorems for ring spectra is to prove that ...

**5**

votes

**1**answer

319 views

### Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...

**7**

votes

**1**answer

271 views

### Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...

**23**

votes

**4**answers

1k views

### What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...

**13**

votes

**1**answer

529 views

### Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$,
a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...

**4**

votes

**1**answer

126 views

### On closed model categories: standard arguments and fibrantly cogenerated categories

Some not very clever questions on closed model categories.
For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when ...

**4**

votes

**1**answer

131 views

### Yoneda embeddings of stable model categories; composition with Bousfield localizations

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question ...

**16**

votes

**2**answers

513 views

### Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras:
$$\begin{array}{ccc}
n & Cl_n & M_n/i^{*}M_{n+1}\\
1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\
2 & \mathbb{H} & ...

**10**

votes

**3**answers

706 views

### Plus construction considerations.

In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = ...

**21**

votes

**1**answer

738 views

### Is every ''group-completion'' map an acyclic map?

I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...

**12**

votes

**2**answers

889 views

### Finiteness of stable homotopy groups of spheres

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or ...

**9**

votes

**0**answers

382 views

### Two constructions for BU×Z

Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$:
1) Take the groupoid of finite dimensional complex inner product spaces with isometries ...

**11**

votes

**2**answers

729 views

### Twists of K-theory and tmf

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How ...

**10**

votes

**3**answers

932 views

### Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi
(this is my very first question here, so please don't hurt me...)
for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...

**15**

votes

**2**answers

910 views

### Proof of Bott Periodicity in twisted K-theory

I have a question about the Proof of Bott Periodicity in twisted K-theory
by Atiyah and Segal in their paper Twisted K-theory.
Following their notation, to prove Bott periodicity in this context it ...

**25**

votes

**0**answers

711 views

### Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting)
I start by recalling the analytic definition of KO-theory:
The following ...

**24**

votes

**1**answer

1k views

### A question about the topological proofs of Bott periodicity

There are purely topological proof of Botts periodicity theorem, the first one given by Dyer and Lashof. I am heading to discuss the proof
in my lecture course on homotopy theory (as a final chord ...

**6**

votes

**0**answers

263 views

### homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...

**7**

votes

**0**answers

262 views

### The residue class functor from a Frobenius category to its stable category induces a functor on cube-shaped diagrams - is it dense?

Let $\mathcal{E}$ be a Frobenius category, i.e. an exact category with sufficiently many bijective objects. (Such as e.g. the category of complexes over an additive category.)
Let ...

**3**

votes

**3**answers

333 views

### Sequential colim vs sequential hocolim

Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but ...

**6**

votes

**1**answer

627 views

### Do homotopy colimits always commute with homotopy colimits?

Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A ...

**45**

votes

**6**answers

3k views

### What are surprising examples of Model Categories?

Background
Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three ...

**16**

votes

**3**answers

1k views

### How do you relate the number of independent vector fields on spheres and Bott Periodicity for real K-Theory?

The theory of Clifford algebras gives us an explicit lower bound for the number of linearly independent vector fields on the $n$-sphere, and Adams proved that this is actually always the best ...

**10**

votes

**1**answer

374 views

### BU with tensor product H-space structure

Hi,
I came across the space $BU_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, BU_\otimes]$ ...

**102**

votes

**7**answers

7k views

### Proofs of Bott periodicity

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...

**8**

votes

**1**answer

637 views

### Constructing Twisted K-theory

There is a simple, intuitive "construction" of twisted K-theory if we are allowed to ignore that many things only hold up to homotopy. We know that maps to $K(Z,2)$ give line bundles on a space and ...

**12**

votes

**2**answers

483 views

### Is there a constructive description of type in the p-local stable homotopy category?

The title pretty much sums it up - but let me give a little bit of background first.
In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) ...