All Questions

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...

I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...

Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results, $$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$ and $$\Omega^...

If $X$ is a space, we can form $QX=\varinjlim \Omega^n\Sigma^nX$ which is an infinite loop space with homotopy groups $\pi_i(QX)=\pi^{s}_i(X)$ the stable homotopy groups of $X.$ But these are the ...

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum $$ \Sigma^t X\cong \bigvee _{k=1}^\infty Y_k. $$ (1). Does this imply $$ X\to \Sigma^tX\to \bigvee _{k=1}^\...

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...