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I will explain that as long as $n>2$ the real vector bundles $\Omega^{even}$ and $\Omega^{odd}$ over $M$ are isomorphic. If $n>2$ then $dim(\Omega^{even}) = dim(\Omega^{odd}) = 2^{n-1} > n$ and so $\O …

Some further data points: (2') For a finite field $\mathbb{F}_{p^r}$ with $p^r \neq 2$ one has $\mathbf{N}(\mathbb{F}_{p^r}, j) \leq \max(\lceil\tfrac{j}{2}\rceil, j-r(p-1)+3)$. This is by combining T …

Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Le …

This is regarding your second question. In dimensions $\geq 5$, where the $s$-cobordism theorem applies, $h$-cobordisms are invertible in the following sense: if $W : M \leadsto M'$ is an $h$-cobordis …

I will assume that by $\wedge$ you meant $\times$, and did not mean to write reduced cohomology (because I don't think the $H$-space structure gives you a map out of the smash product, and $b_k \otime …

This comes from the choice of the $K$-theory Thom class for complex vector bundles. Firstly, recall that $K$-theory $K^0(X)$ can be described as the group of bounded chain complexes of vector bundles …

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 …

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, …