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thinking about invariants of 2-knots ... is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$? Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live? Have …

As Andrew Stacey pointed out, certainly there is no representability theorem for arbitrary cohomology theories (even ordinary). For instance, singular cohomology is not representable on compact metric …

Let me try again. The original question Is there a version of Brown representability for arbitrary pointed topological spaces? is interesting, and I don't see how it has been fully answered. Tha …

not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction Both questions are answered in the book b …

I will construct a closed simply-connected $8$-manifold $M$ and an $a\in H^3(M;\Bbb Z)$ such that the Poincare dual $b$ of $a$ is not realizable by a map $S^5\to M$, and a Hom-dual element in $H^5(M;\ …

How about $X=\Bbb RP^2\times L^2_3$, where $L^2_3$ is the cone of the $3$-fold cover $S^1\to S^1$. Here $H^4(X;\Bbb Z)\simeq\Bbb Z/2\otimes\Bbb Z/3=0$, but $H^4(X;\mathcal L)\simeq\Bbb Z\otimes\Bbb Z/ …

There's a lot of literature on classifying spaces of the $p$-adic integers. For instance, there are two 45-year old "computations" of the complex K-theory of the $p$-adic integers, resulting in two di …

To get some intuition behind the double suspension theorem, you can try three pages with a lot of pictures in Ferry's notes (starting with p.166, in Chapter 26). He gives a rough sketch of proof in th …

It is not hard to reconstruct a finite regular CW complex from its face poset. This is essentially the order complex construction. One just has to understand the order complex $\Delta(P)$ of the poset …

This addresses the modified question in Jeremy's comments, on keeping the preferred CW-structure. If the CW complex happens to be regular and PL (i.e. the attaching maps are injective and piecewise-l …

Immersion theory has been "explained" by the Compression Theorem, with new proofs arguably being much more elementary and intuitive: Rourke and Sanderson, A master's thesis with another exposition

Not only a covering, but every piecewise linear map $f$ between finite graphs can be realized in $\Bbb R^3$ in your sense. To see this, pick a triangulation $T$ of the mapping cylinder of $f$ that pro …

In defining his finiteness obstruction (for a finite dimensional finitely dominated CW-complex to be homotopy finite), Wall amends the given finite-dimensional complex by a homotopy equivalence so as …

1) The book "2D homotopy and combinatorial group theory" (1993) is decidedly oriented towards applications and problem solving, and does discuss crossed modules in Chapters 2 and 4, and further mentio …

Edit: The following simplifies the original answer (which unnecessarily used singular cohomology). If $f:\Bbb R^n\to S^n$ is a fibration, then as Mark noted, a fiber $F$ of $f$ is weak homotopy equiv …