Thom room

The Thom space or Thom complex , named after René Thom , is a topological space assigned to a vector bundle in algebraic topology and differential topology .

Construction of the Thom room

Let a k-dimensional real vector bundle over a paracompact space be through ${\displaystyle E}$ ${\displaystyle B}$

${\displaystyle p\colon E\to B}$

given. Then for each point of the base the fiber of the vector bundle is a k-dimensional real vector space. A corresponding bundle of spheres can be formed by separate single-point compacting of each fiber. The Thom complex is obtained from the bundle by identifying all newly added points with the point , the base point of . ${\displaystyle b}$ ${\displaystyle B}$ ${\displaystyle F_{b}}$ ${\displaystyle \operatorname {Sph} (E)\to B}$ ${\displaystyle \operatorname {Sph} (E)}$ ${\displaystyle T(E)}$ ${\displaystyle \infty }$ ${\displaystyle T(E)}$

Thom isomorphism

The meaning of the Thom space results from the sentence about the Thom isomorphism from the theory of fiber bundles (here formulated using - cohomology in order to avoid complications from questions of orientation ). ${\displaystyle \mathbb {Z} _{2}}$

With is referred to as a real vector bundles in the previous section. Then there is an isomorphism, the Thom isomorphism ${\displaystyle p\colon E\to B}$

${\displaystyle \Phi \colon {H}^{i}(B;\mathbf {Z} _{2})\to {\tilde {H}}^{i+k}(T(E);\mathbf {Z} _{2})}$,

for all , where the right hand side is the reduced cohomology . ${\displaystyle i\geq 0}$

The isomorphism can be interpreted geometrically as integration over the fibers . In the special case of a trivial bundle, the -fold suspension is the base and the Thom isomorphism follows from the suspension isomorphism . The Thom isomorphism also applies to generalized cohomology theories . ${\displaystyle T(E)=S^{k}B_{+}}$ ${\displaystyle k}$ ${\displaystyle {\tilde {H}}^{i}(B)=H^{i+1}(SB)}$

The theorem was proven by René Thom in his dissertation in 1952.

Thom class

Thom also gave an explicit construction of the Thom isomorphism. This maps the neutral element from to a class in the th cohomology group of the Thom space, the Thom class . With this one can calculate the isomorphism for a cohomology class in the cohomology of the base space via the retraction of the bundle projection and the cohomological cup product : ${\displaystyle H^{*}(B)}$ ${\displaystyle U}$ ${\displaystyle k}$ ${\displaystyle b}$

${\displaystyle \Phi (b)=p^{*}(b)\smile U.}$

Thom further showed in his 1954 paper that the Thom class, the Stiefel-Whitney classes, and the Steenrod operations are linked. He also showed that the cobordism groups can be calculated as homotopy groups of certain rooms , which can themselves be constructed as Thom rooms. In terms of homotopy theory, they form a spectrum , called the Thom spectrum . That was an important step towards modern stable homotopy theory. ${\displaystyle MSO(n)}$ ${\displaystyle MSO}$

If Steenrod operations can be defined, one can use them and the Thom isomorphism to construct Stiefel-Whitney classes. By definition, the Steenrod operations (mod 2) are natural transformations

${\displaystyle Sq^{i}\colon H^{m}(-;\mathbf {Z} _{2})\to H^{m+i}(-;\mathbf {Z} _{2})}$,

defined for all natural numbers . If so, Sq ⁱ matches the square of the cup. The -th Stiefel-Whitney classes of the vector bundle are then given by: ${\displaystyle m}$ ${\displaystyle i=m}$ ${\displaystyle i}$ ${\displaystyle w_{i}(p)}$ ${\displaystyle p\colon E\to B}$

${\displaystyle w_{i}(p)=\Phi ^{-1}(Sq^{i}(\Phi (1)))=\Phi ^{-1}(Sq^{i}(U)).\,}$

literature

• JP May : A Concise Course in Algebraic Topology. University of Chicago Press, Chicago IL et al. 1999, ISBN 0-226-51182-0 , pp. 183-198 ( Chicago Lectures in Mathematics Series ).
• Dennis Sullivan : René Thom's Work on Geometric Homology and Bordism. In: Bulletin of the American Mathematical Society. 41, 2004, pp. 341-350, online .
• René Thom : Espaces fibers en spheres et carrés de Steenrod. In: Annales scientifiques de l'École Normale Supérieure. Ser. 3, 69, 1952, pp. 109-182, online .
• René Thom, Quelques propriétés globales des variétés differentiables. In: Commentarii Mathematici Helvetici. 28, 1954, pp. 17-86, online .