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16 votes
Accepted

Are Eilenberg-MacLane spaces limits of manifolds?

Any homotopy type that is represented by a countable CW complex is also represented by an increasing union of closed manifolds: First consider a finite CW complex $X$. Let $M\sim X$ be a compact ...
Tom Goodwillie's user avatar
12 votes

Where is the Steenrod Realization problem at?

A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, ...
Mark Grant's user avatar
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11 votes
Accepted

Steenrod powers of the Thom class

I don't know a reference, but you can proceed as follows. By the splitting principle, it suffices to give the formula for vector bundles which are sums of complex line bundles, and we may as well then ...
Oscar Randal-Williams's user avatar
9 votes

Bott & Tu differential forms Example 10.1

In my version of Bott and Tu, the example reads (emphasis mine): "Moreover, if $V \subset U$ is an inclusion of CONTRACTIBLE open sets, then $\rho^U_V: H^q(\pi^{-1} U) \to H^q( \pi^{-1} V)$ is an ...
Greg Friedman's user avatar
6 votes
Accepted

Steenrod operations on classifying spaces

With apologies for reviving this old question: the following paper by Borel and Serre seems to be the canonical reference for Steenrod powers in the cohomology of $BU(n)$, $BSO(n)$ and $BSp(n)$: Borel,...
Mark Grant's user avatar
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6 votes
Accepted

Loop-space functor on cohomology

If you are willing to read a little French, look at page 434 of Serre's "Homologie singuliere des espaces fibres" (1951). For an exercise in English, with a hint, try Exercise 2 on page 155 ...
John Rognes's user avatar
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5 votes
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Surjection onto $H_{2}(\mathrm{PGL}(2,\mathbb{C}),\mathbb{Z})$

This answer has been edited so that it not only reads better but also gives a better answer. I'll refer to Dupont and Sah, "Scissors congruences, II" as [DS] and the paper of Sah and Wagoner,...
Dave Benson's user avatar
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5 votes

Relation between 16 $\mathbf{CP}^2$ and $\overline{K3}$

[Note: This answer was updated according to @MarcoGolla's comment to avoid a potential circularity.] (1) The existence of such a bordism is already answered by your question: "$\mathbb{CP}^2$ ...
Leo's user avatar
  • 643
4 votes
Accepted

Gluing $n$ $2(n-1)$-simplices

I wonder whether you're looking for the following. Interpret simplicial complexes as determined by subsets of vertices that correspond to simplices, in the usual fashion. Take a set $S$ of $2n$ ...
Dave Benson's user avatar
  • 12.8k
3 votes

Loop-space functor on cohomology

For Q2, when searching it might be useful to know two things: The map $\omega$ is often called the cohomology suspension (not to be confused with the suspension isomorphism in cohomology!), and ...
Mark Grant's user avatar
  • 35.4k
3 votes

Co-index of a Space

Given a finite group $G$ acting freely on a paracompact space $X$, a principal ideal domain $L$, and a positive integer $n$, Conner and Floyd construct a cohomology class $c^n(X; L)$ on $X/G$ (see ...
Michael Albanese's user avatar
1 vote

Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$

This should be a comment but turns out too long... According to Bott's periodicity, the next layer of your Whitehead tower is $\cdots\to \mathrm{MFivebrane}\to B^9\mathbb{Z}/2$. Hence your guess is ...
Leo's user avatar
  • 643

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