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...ia:Spin_manifold|spinable manifolds]]. The calculation of $\mathcal{M}_5^\Spin$ was first obtained by Smale {{cite|Smale1962}} and was one of the first ap === The general spin case ===

* $H_n$ is a [[Wikipedia:Spin-manifold|spinable]] if and only if $n$ is even. ...P^1)$ is the Euler class of $L_{-n}$ {{cite|Milnor&Stasheff1974|Problem 11-C}}.

=== Bordism classes === As every homotopy sphere is stably parallelisable, homotopy spheres admit [[B-Bordism|$B$-structures]] for any $B$. If $B$ is such that $[S^n, F] \mapsto 0 \in

...lar and we discuss it briefly [[B-Bordism#Piecewise linear and topological bordism|below]]. ...{cite|Kreck&Lück2005|18.10}}. See also the [[Wikipedia:Bordism|Wikipedia bordism page]].

The theory of bordism is one of the deepest and most influential parts of ...elopment in the 1960s. In particular, Atiyah \cite{Atiyah1961} showed that bordism is a [[Wikipedia:homology theory|generalised homology theory]] and related

...dism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below. ...e $String(n)$ appears as the realization of a smooth 2-group extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2

For an oriented manifold a [[Wikipedia:Spin_structure|spin structure]] is a reduction of the structure group of its ...m the connected topological group $SO(n)$ to the double (universal) cover $Spin(n)$.

==[C] Talks 6-8== *[[Normal bordism - definitions (Ex)]]

...smooth manifold $M$ is $KO$-orientable if and only if it admits a $\textup{Spin}$-structure. This holds, in turn, iff $w_1(M)=0=w_2(M$. This condition is p Note that complex manifold are $E$-oriented for all $E$ from (a,b,c) (but not (d, e) below).