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am taking the liberty of adding pointer to
Well, from a cursory look, there aren’t too many other references to choose. Cruikshank’s thesis has a relevant Chap. 7, perhaps the source for the entry included in cohomotopy
I think that’s just the stable version.
Thanks. I have added the pointer.
added statement of one form of what we like to call the “twisted Pontrjagin-Thom theorem”, currently it reads as follows:
Let
$X^n$ be a closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the scanning map constitutes a weak homotopy equivalence
$\underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{scanning map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }$between
the J-twisted (n+k)-Cohomotopy space of $X^n$, hence the space of sections of the $(n + k)$-spherical fibration over $X$ which is associated via the tangent bundle by the O(n)-action on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$
the configuration space of points on $X^n$ with labels in $S^k$.
added pointer to
Soren Galatius, Theorem 6.3 in: Stable homology of automorphism groups of free groups (arXiv:0610216)
Oscar Randal-Williams, Embedded Cobordism Categories and Spaces of Manifolds, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)
Also took the liberty of adding pointer to
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