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The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ind …

Segal wrote another paper in which this question came up. Cohomology of topological groups In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377-387. Academic Press, London, (1970). In …

This is an example of twisted cohomology. In general, for a generalized cohomology theory (spectrum) E and a space X you can talk about E-twists over X. This is a certain structure on X. Given a parti …

This is not an answer, but these are my thoughts so far and hopefully they will lead someone to a correct answer (hence the community wiki). My vague recollection is that the algebraic loop space only …

In the case that $Y = A \times X$ is an untwisted sheaf, then there is an easy description (for reasonable spaces X and top. abelian groups A (say Hausdorff, compactly generated, locally contractible) …

For a category enriched in topological spaces, the usual classifying space can be made to take into account the topology on morphisms. More generally this works for categories internal to top and is d …

Here is another interesting class to throw in the mix in between CW-complexes and general spaces. They are defined exactly like CW-complexes, by inductively attaching cells, except that you are allow …

It is independent of the choice of base point. Let $Map((X,x_0), (G_F,id))$ be the based mapping space (based at $x_0$). Let $Map(X, G_F)$ be the free mapping space. Then we have a split short exact …

Let $(X, x_0)$ be a pointed space. Then we can define the homotopy groups $\pi_i(X, x_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi_1(X, …

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point. However for general spaces it is well-known that just because a space is contractible, it does n …

It is well known that if a category has all coequalizers and all (small) coproducts then in fact it has all (small) colimits. More important is the proof which shows that every colimit can be built by …

This question concerns diffeomorphism of manifolds. Let $f: M \to M$ be a self-diffeomorphism. We will say that it is isotopic to the identity if there is a continuous one-parameter family of diffeomo …

Yes, but it is much better to look at the representing spectrum. Cohomology in degree n is represented by the (pointed) space K(Z, n), as you pointed out. Then the product R of all the K(Z, n) where n …

Computing the cohomology of Eilenberg Maclane spaces is a feasible but difficult problem in algebraic topology. The general answer is quite complicated (see the MO answer and the reference therein htt …

The Whitehead tower of a (pointed) space is a tower of spaces which successively kills the bottom homotopy groups. The first two spaces can be constructed functorially (at least for suitably nice spac …