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By a theorem of Thom {{cite|???}} the diffeomorphism type of $X(f_1, \dots, f_k)$ depends only u ...x = i^*(c_1(L))$ where $c_1(L)$ is the [[Wikipedia:Chern_class|first Chern class]] of $L$. Let

...by $\xi : X \to BO(r)$. A $B_r$-structure on $\xi$ is a vertical homotopy class of maps $\bar \xi : X \to B_r$ such that $\gamma_r \circ \bar \xi = \xi$. ...e a fibred stable vector bundle. A $B$-structure on $M$ is an equivalence class of $B_r$-structure on $M$ where two such structures are equivalent if they

forms a ring under cartesian products of manifolds. Thom {{cite|Thom1954}} has shown that this ring is a polynomial ring over $\math Using the results by Thom {{cite|Thom1954}} Dold shows that these manifolds give ring generators of $

...n_structure|spin structures]] are isomorphic to the homotopy groups of the Thom spectrum $MSpin$. The spin bordism class of a manifold is detected by $\Zz_2$-cohomology (Stiefel-Whitney) and KO-th

... of closed oriented manifolds are isomorphic to the homotopy groups of the Thom spectrum $MSO$. [[Wikipedia:Pontryagin class#Pontryagin numbers|Pontryagin number]] $p_J$ of a closed, oriented manifold

... were laid in the pioneering works of Pontrjagin \cite{Pontryagin1959} and Thom \cite{Thom1954}, and the theory experienced a spectacular development in th We denote the bordism class of $M$ by $[M]$, and denote by

...motopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained belo ... $BU\left<6 \right>$. If we choose a complex orienatation the lift gives a class $f$ in the cohomology

...opf, <!--E. van Kampen, K. Kuratowski, S. MacLane,--> L. S. Pontryagin, R. Thom, H. Whitney, M. Atiyah, F. Hirzebruch, R. Penrose, J. H. C. Whitehead, C. Z ...y or may not lead to readily calculable classification results for a given class of manifolds. One of the motivations for Kreck's modified surgery theory \

...es|the first Stefel-Whitney class]] $w_1(N)$ (which equals to the first Wu class $v_1(N)$ \cite[Theorem 11.4]{Milnor&Stasheff1974}). Since $N$ is non-orien ...d invariant of $E^{2n}(S^1 \times S^{n-1})$ can be defined by the homotopy class of the map

...ary manifolds $(M, \bar \nu)$ are isomorphic to the homotopy groups of the Thom spectrum $MU$, $\Omega_*^{U} \cong \pi_n(MU)$.--> class of any manifold $M$ which bounds and whose stable tangent

cohomology class $x+y$. first <i>Conner-Floyd Chern class</i> of the complex line bundle

=== Mapping class groups === ...om spectrum of the virtual bundle $-\gamma_6^{\mathrm{Spin}}$. Pontrjagin--Thom theory provides a map

...he fundamental group of a smooth closed 4-manifold. On the other hand, the class of simply connected (topological or smooth) 4-manifolds still appears to be ...e the rank of $H^2(S_d;\mathbb{Z})$. Likewise, by computing the Pontryagin class and using the [[Hirzebruch signature theorem]], stating that for a closed o

generated by the Thom class.

{| class="wikitable sortable" width="100%" {| class="wikitable sortable" width="100%"

* [[Thom space]] * [[Pontrjagin-Thom isomorphism]]

...momorphism, given by $[M,f] \to f_{*}([M])$ where $[M]$ is the fundamental class of $M$. The elements in the image of $\Phi$ are called representable. In certain situations it is convenient to assume that a homology class is representable. In dimensions $0$ and $1$ it is clear that $\Phi$ is surj

{{beginthm|Exercise}} \label{exrcs:S-duality-of-Thom-class-is-fund-class}
 ...Thom class $u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the fundamental class $[X] \in C_{n} (X)$. Show that


...ntinuous''', if for each $x \in M$ there is an open neighborhood $U$ and a class $\alpha \in H_n(M, M-U;\mathbb Z)$ such that the map induced by the inclusi ...riented real vector space''' $V$, i.e. $V$ is equipped with an equivalence class of bases $v_1,...,v_n$, where two bases are called equivalent, if and only

...ity of a closed connected manifold $M$ is the existence of the fundamental class $[M]\in H_n(M)$. It is clear that this definition is very suitable to gener We denote the $i$th [[Wikipedia:Stiefel-Whitney_class|Stiefel-Whitney class]] by $w_i$.