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Here's how I explain Steenrod squares to geometers. First, if $X$ is a manifold of dimension $d$ then one can produce classes in $H^n(X)$ by proper maps $f: V \to X$ where $V$ is a manifold of dimens …

Here's how I explain Steenrod squares to geometers. First, if $X$ is a manifold of dimension $d$ then one can produce classes in $H^n(X)$ by proper maps $f: V \to X$ where $V$ is a manifold of dimens …

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 …

I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question: CW complexes conne …

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 …

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 …

Yes, this iterated bar construction model should be better known, and can be expressed even more geometrically than Ravenel and Wilson do. If $A$ is an abelian group then $K(A,n)$ is modeled by a spa …

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 …

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 homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more acc …

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 …

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 …

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 …

As a topologist, my view is that group cohomology of interest to number theorests seems to generally be with non-trivial module coefficients. Many of the tricks topologists employ to study spaces do …

Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all compact subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the …