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Let $ T = \mathbb{G}_m$ be the torus, and let $ \tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $ T$ and $ \tilde{T}$ have a well-defined étale homotopy type. Explicitly, the homotopy types are $ ét(T) = B\hat{\mathbb{Z}}$ and $ ét(\tilde{T}) = *,$ for $ *$ theRead more

I would like to know where about I can find the most updated results on the unstable groups $ \pi_{2k+1}P_{k+1}$ and $ \pi_{2k}P_k$ . I think there would be definitely computations Mahowald’s AMS memoir, but I presume there has been some work after this masterpiece! I also wonder how is known about the stable homotopyRead more

If we fix a finite group $ G$ , there are two different useful homotopy theories on the set of $ G$ -equivariant topological spaces (which are CW complexes, say). One, the “weak” homotopy theory, is given by inverting all maps of spaces $ X\to X’$ which are homotopy equivalences, and it produces a homotopyRead more

In particular I’m interested in this situation. Let $ sSet^{2}$ denote the category of bisimplicial sets with diagonal model structure (weak equivalences are diagonal weak equivalences and cofibrations are monomorphisms) and $ sSet$ be the category of simplicial sets with usual model structure. Then we know $ d$ (the diagonal functor) is a right QuillenRead more

One can show that spectrifications of maps between $ \Sigma$ -cofibrant prespectra that are spacewise homotopy equivalences are homotopy equivalences of spectra. I’m interested in the following: Does this result hold if only the source prespectrum is assumed to be $ \Sigma$ -cofibrant? The reason for my interest is the following. The adjoint from prespectraRead more

Suppose $ \mathcal{C}$ is a stable $ \infty$ -category whose homotopy category is a well-generated triangulated category in the sense of Neeman’s book. Must $ \mathcal{C}$ be a presentable $ \infty$ -category? Rosicky proved that if $ \mathcal{M}$ is a combinatorial stable model category then $ Ho(\mathcal{M})$ is well-generated, so I’m asking for a kindRead more

It is well-known that any Lie group $ G$ has $ \pi_2(G)=0$ : see this question. Is the same true for any compact (Hausdorff) topological group? Or even for locally compact ones? Maybe there is a way of showing it by expressing the group as an inverse limit of Lie groups? It seems that itRead more

Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it’s not known if they exist between finite complexes (Freyd’s Generating Hypothesis). Unstably, it’s easy to find maps which are the same on homotopy but not homotopic (even between finite complexes), however I don’t know an example of a map which is theRead more

Suppose that $ G$ is a finite group, $ M$ is a right $ G$ -set and $ N$ is a left $ G$ -set. Then we have a simplicial set $ B(M,G,N)$ whose $ n$ -simplicies are $ M \times G^n \times N$ . Now suppose that $ H \subseteq G$ is a subgroup.Read more

Assume given a pullback square of simplicial categories $ $ \begin{array}[c]{ccc} A&{\rightarrow}&B\ \downarrow&&\downarrow\ C&{\rightarrow}&D. \end{array}$ $ Suppose further that one of the induced arrows $ Ho (B) \to Ho(D)$ or $ Ho(C) \to Ho(D)$ is an isofibration, and for each couple of objects $ x,y \in A$ , the induced pullback square of simplicial mappingRead more