All Questions

We can define the signature of a manifold in $4k$ dimensions. 1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero: $$\sigma(S^...

In Norman Steenrod's book "The Topology of Fibre Bundles", on page 37, one can find the following conjecture: if $n$ is a power of two then the fibre bundle with the projection $SO(n)\to SO(n)/SO(n-1)=...

Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover,...