Warsaw district

Warsaw district

In mathematics , the Warsaw Circle (named after the place of activity of its discoverer Karol Borsuk ) is a topological space, which among other things serves as a counterexample for generalizations of various topological theorems of CW complexes to any topological space .

construction

The Warsaw Circle is a closed subset of the plane , which is created from a part of the graph and the segment of the y-axis by adding a curve connecting both parts. ${\displaystyle W\subset \mathbb {R} ^{2}}$ ${\displaystyle y=\sin({\tfrac {1}{x}})}$ ${\displaystyle Y=\left\{(x,y)\colon x=0,-1\leq y\leq 1\right\}}$

properties

• ${\displaystyle W}$is not a CW complex and also not homotopy equivalent to a CW complex.
• ${\displaystyle W}$is not locally route-related .
• ${\displaystyle W}$is simply connected .
• The Čech homology and Čech cohomology of coincide with that of the circle . However, the singular homology and cohomology of are trivial. (In contrast, for spaces of the homotopy type of a CW complex, the Čech cohomology is always isomorphic to the singular cohomology.) ${\displaystyle {\breve {H}}_{*}(W)}$ ${\displaystyle {\breve {H}}^{*}(W)}$ ${\displaystyle W}$ ${\displaystyle W}$
• ${\displaystyle W}$does not have a universal overlay . The generalized universal overlay is a half-open interval. ${\displaystyle {\widetilde {W}}}$
• The generalized universal overlay is a fiber and has the "unique path lifting property" (for each path there is a clear elevation). But (because of ) it is not a homeomorphism and therefore (because of ) it cannot be a superposition either . ${\displaystyle {\widetilde {W}}\to W}$ ${\displaystyle {\breve {H}}^{1}(W)\not =0}$ ${\displaystyle \pi _{1}W=0}$
• The quotient space is homeomorphic to the circle , the quotient mapping cannot be raised to a mapping . This is remarkable, on the one hand, because the induced homomorphism can of course be elevated to a homomorphism . On the other hand, it proves that the mapping is not null homotopic (because the projection is a Serre fiber ), so it does not apply to the relationship known for CW complexes that homotopy classes of maps are classified by the singular cohomology . ${\displaystyle W/Y}$ ${\displaystyle S^{1}}$ ${\displaystyle W\to S^{1}}$ ${\displaystyle W\to \mathbb {R} ^{1}}$ ${\displaystyle \pi _{1}W=0}$ ${\displaystyle \pi _{1}W\to \pi _{1}S^{1}}$ ${\displaystyle \pi _{1}W\to \pi _{1}\mathbb {R} ^{1}}$ ${\displaystyle W\to S^{1}}$ ${\displaystyle \mathbb {R} ^{1}\to S^{1}}$ ${\displaystyle W}$ ${\displaystyle W\to S^{1}}$ ${\displaystyle H^{1}(W;\mathbb {Z} )}$
• There is a fibration with base in which and the homotopy have a CW complex, the base is not. (In contrast, it is known that the homotopy a CW complex has, if this is on , and holds and that has the homotopy a CW complex when this on and the case.) Further, in this fibrillation and contractible, the base is not. ${\displaystyle F\to E\to B}$ ${\displaystyle B=W}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle W}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle B}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle W}$

Individual evidence

1. ↑ Occasionally the term "Polish Circle" is also found, for example in Mardešić, S .: A survey of the shape theory of compacta. General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971), pp. 291-300. Academia, Prague 1972. online (PDF)
2. ↑ Remark 2.7 in: Kryszewski, Wojciech; Szulkin, Andrzej: Infinite-dimensional homology and multibump solutions. J. Fixed Point Theory Appl. 5 (2009), no. 1, pp. 1-35.
3. ↑ Schön, Rolf: Fibrations over a CWh-base. Proc. Amer. Math. Soc. 62 (1976), no. 1, pp. 165-166 (1977). online (PDF)
4. ↑ Section 4.4, Example 8 in: Spanier, Edwin H .: Algebraic topology. Corrected reprint. Springer-Verlag, New York-Berlin 1981. ISBN 0-387-90646-0