Decomposition theorem by Alexandroff-Borsuk

The decomposition theorem of Alexandroff-Borsuk (sometimes just decomposition Borsuk called) is a mathematical theorem about the topology of the finite-dimensional Euclidean space . It goes back to Paul Alexandroff and Karol Borsuk and gives a characterization of the compacta decomposing Euclidean space using the homotopy theory . The theorem is closely related to the decomposition theorem by Jordan-Brouwer .

Formulation of the decomposition theorem

The decomposition theorem of Alexandroff-Borsuk can be formulated as follows:

Let be a compact subset of   ( ). ${\displaystyle K}$ ${\displaystyle {\mathbb {R} }^{n+1}}$ ${\displaystyle n\in {\mathbb {N} }}$

Then ensure that the ${\displaystyle K}$ ${\displaystyle {\mathbb {R} }^{n+1}}$ decomposed , sufficient and necessary that a substantial continuous mapping of the dimensional sphere exists. ${\displaystyle f\colon K\to S^{n}}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle S^{n}\subseteq {\mathbb {R} }^{n+1}}$

Explanations

It is said that a subset of this decomposes if the complement set in the subspace topology is disconnected , i.e. it consists of at least two connected components. ${\displaystyle K}$ ${\displaystyle {\mathbb {R} }^{n}}$ ${\displaystyle \mathbb {R} ^{n}\setminus K}$

Furthermore, one calls a continuous mapping between two topological spaces and essential if it is not homotopic to a constant mapping . Otherwise it is called insignificant or null homotop . ${\displaystyle f\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle C\colon X\to \{y\}\subseteq Y}$ ${\displaystyle f}$

Application: Jordan-Brouwer's qualitative decomposition theorem

The qualitative decomposition theorem of Jordan-Brouwer says the following:

Is a compact subset by an injective continuous map within mapped and divided the , so the decomposed image set to . ${\displaystyle K\subseteq {\mathbb {R} }^{n+1}}$ ${\displaystyle h\colon K\to {\mathbb {R} }^{n+1}}$ ${\displaystyle {\mathbb {R} }^{n+1}}$ ${\displaystyle K}$ ${\displaystyle {\mathbb {R} }^{n+1}}$ ${\displaystyle h(K)}$ ${\displaystyle {\mathbb {R} }^{n+1}}$

The qualitative decomposition theorem of Jordan-Brouwer is obtained from the decomposition theorem of Alexandroff-Borsuk taking into account the fact that the essential continuous maps of and dimensional n-into the sphere via and correspond to each other clearly reversibly . ${\displaystyle K}$ ${\displaystyle h(K)}$ ${\displaystyle h}$ ${\displaystyle h^{-1}}$

literature

• Paul Alexandroff: Dimension Theory . In: Math. Ann . tape 106 , 1932, pp. 161-238 ( MR1512756 ).
• Karol Borsuk: About sections of the n-dimensional Euclidean sphere . In: Math. Ann . tape 106 , 1932, pp. 239-248 .
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Willi Rinow : Textbook of Topology . German Science Publishers, Berlin 1975.

Notes and individual references

1. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 201 ff . ( MR0533264 ).
2. ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 394 ff .
3. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 149 ( MR0533264 ).
4. ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 151 .
5. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 79-80 ( MR0533264 ).
6. ^ Willi Rinow : Textbook of Topology . Deutscher Verlag der Wissenschaften, Berlin 1975, p. 380 .
7. ↑ Egbert Harzheim : Introduction to Combinatorial Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 203 ( MR0533264 ).