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Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--Whitn …

This is similar to Dan Petersen's answer, but more elementary. A fact I always mention when talking to students about matrix groups is that the diagonalisable matrices (over $\mathbb{C}$) are dense in …

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 …

There is a ``Mayer--Vietoris" sequence $$\cdots \to \pi_2(Z, z) \to \pi_1(E, e) \to \pi_1(X, x) \times \pi_1(Y, y) \to \pi_1(Z,z) \to \pi_0(E) \to \cdots$$ that can be developed by fitting together th …

For any (connective) spectrum $E$ one may rationalise it to get a rational spectrum $E_\mathbb{Q}$, and a map $E \to E_\mathbb{Q}$. Now rational spectra split as wedges of Eilenberg-Mac Lane spectra, …

All current proofs of Mumford's conjecture in fact prove a far stronger result, the "Strong Mumford conjecture", first formulated by Ib Madsen. This says the following (where by "moduli space" in the …

No. The space of homeomorphisms of a compact manifold is locally contractible: A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79 (121):307–356 …

This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi_k(MTO(n) \wedge X_+)$ is in bijection with tuples of a $(n+k)$-manifold $M$, an $n$-dimensional vector bund …

The statement that $M \to \Omega BM$ is a weak equivalence when $M$ is a group-like topological monoid is indeed easier: the map $EM = B(M \wr M) \to BM$ is then a quasi-fibration, has geometric fibre …

Suppose $\overline{M}_i$, $i=0,1$, are compact smooth manifolds with boundary whose interiors are diffeomorphic: let $\psi$ be such a diffeomorphism, and $M$ for either interior (identified via $\psi$ …

Yes. By Smale-Hirsch theory it is enough to find a bundle injection $T\Sigma \to \epsilon^8$, so it is enough to find a trivialisation of $T\Sigma \oplus \epsilon^1$. It is a theorem of Kervaire and M …

Your condition determines the map $a_1 \vee a_2 : S^n \vee S^n \to S^n$ on the $n$-skeleton, so the question is when does this extend over the $2n$-cell of $S^n \times S^n$. The $2n$-cell is by defini …

According to Characteristic Classes and Homogeneous Spaces, I A. Borel and F. Hirzebruch, Section 17, they are not even homotopy equivalent: $G_2 / SO(4)$ has mod $2$ homology in degree 2, whereas …

Yes, $B \beta_\infty$ is homology equivalent to $\Omega^2_0 S^2$, the zero component of the double loop space of $S^2$. The map $B_\infty \to S_\infty$ induces the obvious stablisation map $\Omega^2_0 …

I think I have been able to reproduce the "argument by Wagoner" (perhaps it was removed from the published version?). It certainly holds in more generality that what I have written below, using the no …