Hopf-Whitney theorem

In the mathematical field of algebraic topology , the Hopf-Whitney theorem is a classification theorem for mapping a -dimensional simplicial complex into the -dimensional sphere. ${\displaystyle n}$ ${\displaystyle n}$

Theorem: Let be a -dimensional simplicial complex. Then there is a bijection between the set of homotopy classes of mappings and the -th cohomology of : ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle f\colon X\to S^{n}}$ ${\displaystyle n}$ ${\displaystyle X}$

${\displaystyle \left[X,S^{n}\right]\cong H^{n}(X;\mathbb {Z} )}$.

The bijection is mediated by assigning the withdrawn to the fundamental class of an image . ${\displaystyle f\colon X\to S^{n}}$ ${\displaystyle f^{*}\left[S^{n}\right]\in H^{n}(X;\mathbb {Z} )}$ ${\displaystyle \left[S^{n}\right]\in H^{n}(S^{n};\mathbb {Z} )}$

If there is an orientable , closed manifold , and you get the bijection given by the degree of mapping . ${\displaystyle X=M^{n}}$ ${\displaystyle H^{n}(M^{n};\mathbb {Z} )\cong \mathbb {Z} }$ ${\displaystyle \left[M^{n},S^{n}\right]\cong \mathbb {Z} }$

Hopf-Whitney's theorem applies more generally to mappings of CW-complexes into -contiguous spaces . Be a -dimensional CW-complex and a -contiguous space. Be . Then there is an element which corresponds to the identity morphisus under the correspondence , and the assignment defines a bijection ${\displaystyle (n-1)}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle Y}$ ${\displaystyle (n-1)}$ ${\displaystyle \pi =\pi _{n}Y}$ ${\displaystyle \iota \in H^{n}(Y;\pi )}$ ${\displaystyle H^{n}(Y;\pi )\cong \left[Y;K(\pi ,n)\right]\cong \mathrm {Hom} (\pi _{n}Y,\pi )}$ ${\displaystyle f\to f^{*}\iota }$

${\displaystyle \left[X,Y\right]\cong H^{n}(X;\pi )}$.

literature

• H. Hopf: The classes of the mapping of the n-dimensional polyhedra on the n-dimensional sphere , Comm. Math. Helv. 5, 39-54, 1933
• H. Whitney: The maps of an n-complex into an n-sphere , Duke Math. J. 3, 51-55, 1937