Theorem of Dold

In mathematics , Dold's theorem is a generalization of Borsuk-Ulam's theorem , which has numerous applications in topological combinatorics .

Theorem of Dold

If a continuous map

${\displaystyle f\colon S^{m}\to S^{n}}$

equivariant for free effects of a nontrivial finite group on the spheres and is, then is ${\displaystyle S^{m}}$ ${\displaystyle S^{n}}$

${\displaystyle n\geq m}$.

If is, then is not null homotop . ${\displaystyle n=m}$ ${\displaystyle f}$

Special case: Borsuk-Ulam's theorem

If you and your effect by antipodal mapping ${\displaystyle G=\mathbb {Z} /2\mathbb {Z} }$

${\displaystyle x\to -x}$

to and viewed, is then obtained from the set of Dold the following variant of Borsuk-Ulam . ${\displaystyle S^{m}}$ ${\displaystyle S^{n}}$

For there is no continuous map ${\displaystyle m>n}$

${\displaystyle f\colon S^{m}\to S^{n}}$,

the

${\displaystyle f(-x)=-f(x)}$

fulfilled for all . ${\displaystyle x\in S^{m}}$

generalization

A nontrivial finite group works in a space and freely in a space . ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle Y}$

For the dimension applies . ${\displaystyle n:=\mathrm {dim} (Y)}$ ${\displaystyle \pi _{0}X=\pi _{1}X=\ldots =\pi _{n-1}X=\pi _{n}X=0}$

Then there is no continuous - equivariant mapping . ${\displaystyle G}$ ${\displaystyle f\colon X\to Y}$

history

The set was published in 1983 by Albrecht Dold . The generalization (assuming that the effect on is also free) was also made by Dold with the remark "Essentially the same proof gives the following result." formulated. He also noted that for paracompact and an equivariant continuous mapping cannot be null homotopic. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \mathrm {dim} (Y)=1+\mathrm {conn} (X)<\infty }$

A proof of generalization can be found in.

literature

• Albrecht Dold : Simple proofs of some Borsuk-Ulam results. Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), 65-69, Contemp. Math., 19, Amer. Math. Soc., Providence, RI, 1983. online
• Pavle Blagojević, Aleksandra Dimitrijević Blagojević, John McCleary: Spectral sequences in combinatorial geometry: cheeses, inscribed sets, and Borsuk-Ulam type theorems. Topology Appl. 158 (2011), no.15, 1920-1936. on-line

Individual evidence

1. ↑ Dold, op.cit., P. 65
2. ↑ Dold, op.cit., P. 68
3. ↑ The connectivity of a topological space is the largest number for which applies. ${\displaystyle \mathrm {conn} (X)}$ ${\displaystyle m}$ ${\displaystyle \pi _{0}X=\pi _{1}X=\ldots =\pi _{m}X=0}$
4. ↑ Blagojević et al., Op. Cit.