Search Results

This question was inspired by the question posed by John Baez here: https://mathoverflow.net/questions/67209?sort=votes#sort-top and Neil Strickland's answer to that question. Let $X$ be a CW complex …

Here are some thoughts on the question of the posting: Suppose a principal $G$-bundle $E\to B$ for $G$ a topological group has a section $s:B\to E$. Then $(g,b)\mapsto g s(b)$ is a trivialization (r …

Computing the integral cohomology of $K(\pi,n)$'s is feasible but a bit tricky. In fact the only reference I know is exposé 11 of H. Cartan's seminar, year 7. I'd be interested if there are other sour …

A map $f:X\to Y$ of CW-complexes is called a phantom if $f$ restricted to the $n$-skeleton of $X$ is contractible for all $n$. The first non-trivial example of such a map, with $X=\Sigma\mathbb{P}^\in …

Set $X=D,Y=D\setminus D',n=\dim D$. I presume you are interested in real algebraic curves (in the complex case everything simplifies). The problem is that $X\setminus Y$ is not smooth. But there is an …

If $X$ is a simply connected compact manifold, then one sufficient condition for the existence of maps $X\to X$ of any sufficiently divisible degree is formality: there is a commutative differential g …

Ok, here is a simple example when this fails: let $X$ be $\mathbf{R}^2$ minus the origin and consider the automorphism of $X$ given by $(x,y)\mapsto (2x,y/2)$. This automorphism generates a group $G$. …

Here is a partial answer to question 1: the necessary and sufficient condition for a (sufficiently reasonable, say a CW-complex) space to be homotopy equivalent to a topological group is that it shoul …

The sphere minus 4 points is doubly covered by the torus minus 4 points. This double cover gives a representation of $\pi_1(S^2\setminus\mbox{4 points})$ in the symmetric group on two letters. Namely, …

Here is a sketch of a solution. As Petya tells us, $\mathbf{R}^4\setminus M$ is invariant under the action of $\mathbf{Z}^4$ by integral translations, so it suffices to show that the image of $\mathbf …

It is hard to answer the question without actually knowing what the group is. Here are some remarks: For classical congruence subgroups of $PSL_2(\mathbf{Z})$ there is a formula for the number of cu …

A pseudomanifold is a finite-dimensional topological space $X$ (say Hausdorff and locally compact) that admits a closed subspace $Y$ of dimension $\dim X-2$ such that $X-Y$ is a manifold (see e.g. Gor …

Here is a down-to-earth cohomological interpretation of the usual linking number. Let $M$ be an oriented manifold (possibly non-compact) of real dimension $p$ and let $X\subset M$ be a closed subset, …

It all depends on which bundles we are considering. If we take topological vector bundles with structure group a Lie group $G$, then there aren't any non-trivial bundles over the 3-sphere: we have $\p …

The integral cohomology rings of both $BO(n)$ and $BSO(n)$ were computed by E. H. Brown, Proceedings AMS, 85, 2, 1982, p. 283-288. These rings are generated by the Pontrjagin classes, Bocksteins of mo …