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Ben Walter and I make the functors $\Gamma$ and $A$ more explicit, by using an explicit model for the cofree Lie Coalgebra functor, in this paper. We do not discuss the application to $\infty$-algebr …

Some of the best results known along these lines are in the paper Pascal Lambrechts, Don Stanley, Algebraic models of Poincaré embeddings, Algebr. Geom. Topol. 5 (2005) 135-182, doi:10.2140/agt.2005. …

Bott and Tu do this completely, in the de Rham theoretic setting of course. Here's an alternate proof I have used when I teach this material, which I find slightly more clean and direct than using Tho …

Ryan's answer generalizes. I prefer $\iota_1, \iota_2$ for the inclusions of wedge summands, and then $\omega_1, \omega_2$ for forms which generate cohomology supported on each of the wedge summands. …

In a related and highly relevant comment thread, Mike Miller pointed me to this preprint of Lipyanskiy. I'm sure there are arguments which work, such as what Joshua and Dmitri and I discuss in the co …

Let $M$ be a smoothly triangulated compact $d$-dimensional manifold. Consider the subcomplex $C_*^{\pitchfork T}(M)$ of smooth singular chains which are transverse to the triangulation. An inductive …

Behrens's monograph "The Goodwillie tower and the EHP sequence" reproduces some of the Toda calculations (out to the k~20 range as you cite) using a modern toolset, as named in the title. Depending o …

Homotopy groups of spheres correspond to framed submanifolds of Euclidean space through the Pontrjagin-Thom construction. For example, the Hopf map corresponds to a circle in $\mathbb{R}^3$ framed “w …

For Q2, my collaborators and I show that all mod-two cohomology of symmetric groups is generated by Stiefel-Whitney classes of standard representations, if you allow both cup product and transfer (ind …

By Pontryagin's Theorem, you are asking for the preimages of any chosen points in each wedge factor of $S^p \vee S^q \vee S^r$ for the iterated Whitehead product map $p : S^{p+q+r-2} \to S^p \vee S^q …

Parallel to the OP's two examples, if a cohomology class is defined through intersection with a submanifold (or subvariety with fundamental class in locally finite homology) then the pushforward is de …

The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its suspens …

The set of maps $S^n \to S^m$ is better known as $\pi_n(S^m)$. The one other (than n=m) relatively easy case is when we tensor by the rational numbers (and thus answer the question "when is f homotop …

Let me elaborate on part of Tyler's answer, and "second" the importance of constructing examples yourself in a setting where that is feasible, namely that of bicomplexes. Even though as you say they …

Recall Thom's result that cobordism groups are homotopy groups of Thom spaces/ spectra. The characteristic numbers of $[M]$ are encoding the image of the corresponding element of $\pi_* MO$ under t …