Obstruction theory

In topology , a branch of mathematics , obstruction theory or obstacle theory describes the obstacles to the existence of cuts in fiber bundles .

Obstructive coccycles

Be a fiber over a simplicial complex with fiber . We assume that an incision has already been made over the skeleton of and ask whether this incision can be continued onto the skeleton. ${\displaystyle p\colon E\to B}$ ${\displaystyle B}$ ${\displaystyle F}$ ${\displaystyle s_{n}\colon B_{n}\rightarrow E}$ ${\displaystyle n}$ ${\displaystyle B}$ ${\displaystyle (n+1)}$

For each simplex , homotopy is equivalent to and the map ${\displaystyle (n+1)}$ ${\displaystyle \sigma \in B_{n+1}}$ ${\displaystyle p^{-1}(\sigma )}$ ${\displaystyle F}$

${\displaystyle s_{n}\mid _{\partial \sigma }\colon S^{n}\simeq \partial \sigma \to p^{-1}(\sigma )\simeq F}$

defines an element of the -th homotopy group of the fiber ${\displaystyle n}$

${\displaystyle o_{n+1}(\sigma )\in \pi _{n}(F)}$.

Obviously the given cut can only be continued if ${\displaystyle s_{n}\mid _{\partial \sigma }\colon \partial \sigma \to E}$ ${\displaystyle \sigma }$

${\displaystyle o_{n+1}(\sigma )=0\in \pi _{n}(F)}$.

One can show that a cocycle is with local coefficients, it is called an obstruction cocycle . Its cohomology class (in cohomology with local coefficients ) ${\displaystyle o_{n+1}\colon C_{n+1}(B)\to \pi _{n}(F)}$

${\displaystyle o_{n+1}\in H^{n+1}(B,\pi _{n}(F))}$

is called -th obstruction class. Although it depends on the cut chosen , it can be shown that it really only depends on its restriction to the skeleton. ${\displaystyle (n+1)}$ ${\displaystyle s_{n}}$ ${\displaystyle (n-1)}$

Sections in vector bundles

The most important application of the obstruction theory is to the question of the existence of linearly independent cuts in a vector bundle of rank , for , or equivalently, the existence of a cut in the - frame bundle ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle 1\leq k\leq n}$ ${\displaystyle k}$

${\displaystyle V_{k}(\mathbb {R} ^{n})\to V_{k}(E)\to B}$,

whose fiber is the boot manifold . ${\displaystyle V_{k}(\mathbb {R} ^{n})}$

Because of , one can construct such a cut on the -Skeleton , the obstacle to the continuation on the -Skeleton is then the obstruction class defined above ${\displaystyle \pi _{i}(V_{k}(\mathbb {R} ^{n}))=0}$ ${\displaystyle 0\leq i\leq n-k-1}$ ${\displaystyle (n-k)}$ ${\displaystyle B_{n-k}}$ ${\displaystyle (n-k+1)}$

${\displaystyle o_{n-k+1}(E)\in H^{n-k+1}(B,\pi _{n-k}(V_{k}(\mathbb {R} ^{n}))).}$

Boots Whitney Classes

The Stiefel-Whitney classes were originally defined as obstruction classes by Stiefel and Whitney. The homotopy group is either isomorphic to (if and is even) or otherwise infinitely cyclic, so it can be mapped to surjectively in each case . The image of the obstruction class below this figure is the Stiefel-Whitney class ${\displaystyle \pi _{n-k}(V_{k}(\mathbb {R} ^{n}))}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle k>1}$ ${\displaystyle n-k+1}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

${\displaystyle w_{n-k+1}(E)\in H^{n-k+1}(B,\mathbb {Z} /2\mathbb {Z} )}$.

Euler class

For is , for orientable vector bundles, the cohomology with local coefficients is isomorphic to and the obstruction class defined in this way is the Euler class ${\displaystyle k=1}$ ${\displaystyle \pi _{n-k}(V_{k}(\mathbb {R} ^{n}))=\pi _{n-1}(V_{1}(\mathbb {R} ^{n}))=\pi _{n-1}(S^{n})=\mathbb {Z} }$ ${\displaystyle H^{n}(B,\pi _{n-1}(V_{1}(\mathbb {R} ^{n})))}$ ${\displaystyle H^{n}(B,\mathbb {Z} )}$

${\displaystyle e(E)\in H^{n}(B,\mathbb {Z} )}$.

The Euler class can be defined analogously for any bundle of spheres , i.e. for fiber bundles with fibers : because of there is a cut on the base of the skeleton and the obstruction for the continuation to the skeleton is the Euler class ${\displaystyle S^{n-1}}$ ${\displaystyle \pi _{i}(S^{n-1})=0}$ ${\displaystyle 0<i<n-1}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$

${\displaystyle e(E)\in H^{n}(B,\mathbb {Z} )}$.

(In the case of the unit sphere bundle of an oriented vector bundle, the Euler class of the spherical bundle coincides with the Euler class of the vector bundle.)

literature

• Norman Steenrod: The Topology of Fiber Bundles. (= Princeton Mathematical Series. Vol. 14). Princeton University Press, Princeton, NJ 1951 (Chapters 25, 35, 38)
• John W. Milnor, James D. Stasheff: Characteristic classes. In: Annals of Mathematics Studies. No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo 1974. (Chapter 12)
• George W. Whitehead: Elements of Homotopy Theory. (= Graduate Texts in Mathematics. 61). Springer Verlag, 1978, ISBN 1-4612-6320-4 .