Search Results

The Becker--Gottlieb transfer implies that $\pi^*$ is rationally a (split) monomorphism unless the Euler characteristic of the fibre is zero. Thus any proposed example must have this property.

I think one can do something like the following. Let $M = map([0,1], B)$ and $e:M \to B$ be evaluation at 0: this is a Hurewicz fibration and a homotopy equivalence. Now form the pullback fibration $e …

Yes, such a thing exists, but I don't know an explicit example. To see that it exists, it is clearest to me to consider the universal situation. For any $k \in \mathbb{Z}$ there is a space $\mathcal{S …

Let $G=\mathbb{Z}$ act on $S^1$ by an irrational rotation: this defines a flat fibre bundle $S^1 \to E \to S^1$. As $\mathbb{Z}$ is a free group this action can be deformed to the trivial action, and …

This is closely related to (and can be proved by the same methods as) the fact that if we fix another manifold $N$ then the restriction map $$\mathrm{Emb}(M, N) \to \mathrm{Emb}(S, N)$$ is a locally t …

Yes, you can conclude that, because the construction $$(\wedge^2 \xi \setminus \text{zero section})/\mathbb{R}_{>0} \to X/(\mathbb{Z}/2)$$ recovers the original double cover.

No. let $F$ have a $G$-action, take $B=EG$ and $E=EG \times F$ with the diagonal action. Then $\eta$ is the Borel construction $EG \times_G F \to BG$ so need not be trivial.

No it isn't, but I had to dig quite deep to get a counterexample. Let us look at smooth $(D^7, \partial D^7)$-bundles over $S^2$, i.e $D^7 \to E \overset{\pi} \to S^2$ with an identification $\partial …

Let me not Poincare dualise, and work in cohomology. Let me generalise the setting you have described, and consider the universal surface bundle $\pi : E \to M_g^1$ over the moduli space of surfaces w …