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These operations in the singular setting were fully and explicitly developed and generalized beautifully by McClure and Smith (who also credit Benson and Milgram) in their paper "Multivariable cochain …

Brown Representability combined with the Landweber Exact Functor theorem allows one to construct homotopy types out of purely algebro-geometric data, and is in particular the starting point for theori …

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 …

You're going to get many different answers depending on the tastes of the topologist answering... I like to think about homotopy groups of spheres through framed cobordism. Theories like unoriented …

A Point in $RP^n$ corresponds to a pair of antipodal points on $S^n$ - so just practice visualizing two antipodal points on a sphere every time you say Point. Such an approach is clearly equivalent t …

Let me offer another definition not far from obstruction theory (as Ilya gave), but without referring to obstruction theory and thus more elementary. Suppose for simplicity that $X$ is a simplicial c …

First an important distinction: "Each element in $\pi_3(\vee S^2)$ has description in terms of linking number of point preimages (circles in $S^3$) of map $S^3 \to S^2$" is not a fully correct stateme …

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 …

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 …

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. …

The classifying space of the nth symmetric group $S_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S_n$ have related models. For examp …

The title pretty much says it. In a follow-up to my question about alternating groups, does anyone know of a "geometric" model for $BGL_n(F_q)$? By "geometric" I mean "a space you would have heard a …

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 …

On a closed, oriented manifold, homology and cohomology are represented by similar objects, but their variance is different and there is an important change in degrees. For simplicity, consider homol …

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 …