Timeline for Structures between PL and smooth
Current License: CC BY-SA 4.0
6 events
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| Jan 16, 2021 at 1:41 | comment | added | user171227 | ... This is "concrete" in that a $C$-structure is an assignment of $c(\sigma)$ to every 7-simplex $\sigma$ in the triagulation. Integrality of Pontryagin classes of vector bundles gives preferred $C$-structures on smooth manifolds. | |
| Jan 16, 2021 at 1:39 | comment | added | user171227 | Another idea, if a triangulation of $M$ is specified. Cheeger defined explicit real cocycles $L_i \in C^{4i}(M;\mathbb{R})$ in the simplicial cochain complex, representing the Hirzebruch classes. Now $p_2 = 9(L_1^2 + 5L_2)/7 \in C^8(M;\mathbb{R})$ represents the Pontryagin class. If we define a $C$-structure on $M$ to be a simplicial cochain $c \in C^7(M;\mathbb{R}/\mathbb{Z})$ with $\delta(c) = p_2$ mod $\mathbb{Z}$, then $PL/C \simeq K(\mathbb{R}/\mathbb{Z},7)$ ... | |
| Jan 13, 2021 at 16:59 | history | edited | Philip Engel | CC BY-SA 4.0 |
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| Jan 12, 2021 at 21:37 | comment | added | user171227 | The "formal solution" that you're not looking for might define $BC \to BPL$ as a Moore-Postnikov factorization of $BO \to BPL$ and then define a $C$-structure to be a lift of a map classifying the stable tangent microbundle. More geometrically, perhaps a $C$-structure can be thought of as a smooth structure outside a subset of codimension 9. (I'm not sure how to make that precise though.) | |
| Jan 12, 2021 at 20:04 | history | edited | YCor |
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| Jan 12, 2021 at 20:02 | history | asked | Philip Engel | CC BY-SA 4.0 |