Number of cuts

In differential topology and in algebraic topology , the intersection number denotes an integer that indicates the intersection multiplicity that can be assigned to the intersection points of oriented submanifolds or homology classes of oriented manifolds .

Differential topology

In differential topology, one first considers the intersection numbers of images with submanifolds. Intersection numbers of submanifolds of complementary dimensions are calculated as the intersection number of the inclusion map of one submanifold with the other submanifold.

definition

Let be differentiable manifolds , compact and a submanifold, and be a differentiable map that is too transversal . In addition, applies . Then is called ${\ displaystyle X, Y}$ ${\ displaystyle X}$ ${\ displaystyle Z \ subseteq Y}$ ${\ displaystyle f \ colon X \ rightarrow Y}$ ${\ displaystyle Z}$ ${\ displaystyle \ dim X + \ dim Z = \ dim Y}$

{\ displaystyle I (f, Z): = \ sum _ {x \ in f ^ {- 1} (Z)} \ mathrm {sign} (x)}

the number of cuts in the figure with . ${\ displaystyle f}$ ${\ displaystyle Z}$

Transversality and compactness guarantee that the sum is finite. The signum is defined as follows: ${\ displaystyle \ mathrm {sign} (x)}$

${\ displaystyle \ mathrm {sign} (x) = + 1}$ if the orientation is given as a direct sum of oriented vector spaces, ${\ displaystyle d_ {x} f (T_ {x} X) \ oplus T_ {f (x)} Z = T_ {f (x)} Y}$
${\ displaystyle \ mathrm {sign} (x) = - 1}$ if the orientation reverses as a direct sum of oriented vector spaces. ${\ displaystyle d_ {x} f (T_ {x} X) \ oplus T_ {f (x)} Z = T_ {f (x)} Y}$

With the help of the homotopy transversality theorem, the definition can also be extended to mappings that are not transversal: Let be differentiable manifolds , be compact and a submanifold and be a differentiable map. In addition, applies . According to the homotopy transversality theorem , there is a differentiable mapping , which is transversal to and homotopic to . Man sets: . ${\ displaystyle X, Y}$ ${\ displaystyle X}$ ${\ displaystyle Z \ subseteq Y}$ ${\ displaystyle f \ colon X \ rightarrow Y}$ ${\ displaystyle \ dim X + \ dim Z = \ dim Y}$ ${\ displaystyle g \ colon X \ rightarrow Y}$ ${\ displaystyle Z}$ ${\ displaystyle f}$ ${\ displaystyle I (f, Z): = I (g, X)}$

properties

Let be a compact differentiable manifold with boundary and be a differentiable map. Then for for every submanifold of that holds . ${\ displaystyle W}$ ${\ displaystyle \ partial W = X}$ ${\ displaystyle F \ colon W \ rightarrow Y}$ ${\ displaystyle f = \ partial F \ colon X \ rightarrow Y}$ ${\ displaystyle Z}$ ${\ displaystyle Y}$ ${\ displaystyle I (f, Z) = 0}$

The cut numbers of the homotopic images match.

Auto cut number

In the event that compact, oriented submanifolds are an oriented differentiable manifold, with , the number of intersections can be defined, with the canonical inclusion mapping denoting. ${\ displaystyle X, Z}$ ${\ displaystyle \ dim X + \ dim Z = \ dim Y}$ ${\ displaystyle I (X, Z): = I (i, Z)}$ ${\ displaystyle i \ colon X \ hookrightarrow Y}$

One can show that is true. In the case of , the self-cutting number is defined and for odd it follows . ${\ displaystyle I (Z, X): = (- 1) ^ {\ dim X \ cdot \ dim Z} \ cdot I (X, Z)}$ ${\ displaystyle \ dim X = {\ frac {1} {2}} \ dim Y}$ ${\ displaystyle I (X, X)}$ ${\ displaystyle \ dim X}$ ${\ displaystyle I (X, X) = 0}$

Let now be a compact oriented manifold, denote the diagonal. According to the above consideration it is well defined and one can show with the help of the Lefschetz fixed point theory that the manifold agrees with the Euler characteristic . ${\ displaystyle Y}$ ${\ displaystyle \ Delta: = \ left \ {(y, y) \ ,: y \ in Y \ right \} \ subset Y \ times Y}$ ${\ displaystyle I (\ Delta, \ Delta)}$ ${\ displaystyle I (\ Delta, \ Delta)}$

Number of cuts mod 2

The number of cuts ${\ displaystyle \ mathrm {mod} \, 2}$ is independent of the orientation of the manifolds, which is the sign that occurs in the definition of the number of cuts, and the calculation of the number of cuts is reduced to counting the points of intersection . Of course, this does not allow as precise statements as with the number of intersections of oriented manifolds, but it also enables the calculation for non-orientable manifolds. ${\ displaystyle \ mathrm {mod} \, 2 = 1}$ ${\ displaystyle \ mathrm {mod} \, 2}$ ${\ displaystyle \ mathrm {mod} \, 2}$

Application example

As an application it is shown that the Möbius strip cannot be oriented. denote the center line of the Möbius strip, which is diffeomorphic to the circular line . The self-intersection number of is 1. If the Möbius strip were orientable, then it would have to apply. , so the Möbius strip cannot be orientable. ${\ displaystyle X}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle \ mathrm {mod} \, 2}$ ${\ displaystyle X}$ ${\ displaystyle I (X, X) = 0}$ ${\ displaystyle I (X, X) = 0 \, \ mathrm {mod} \, 2 \ neq 1}$

Algebraic topology

The algebraic topology enables the extension of the concept of the intersection number to oriented topological manifolds, where the intersection numbers are defined with the help of singular homology .

literature

John W. Milnor: Topology from the differentiable viewpoint. Revised edition, 1st printing. Princeton University Press, Princeton NJ 1997, ISBN 0-691-04833-9 .
Victor Guillemin , Alan Pollack: Differential topology. Prentice-Hall, Englewood Cliffs NJ 1974, ISBN 0-13-212605-2 .
Ralph Stöcker, Heiner Zieschang : Algebraic Topology. An introduction. 2nd revised and expanded edition. BG Teubner, Stuttgart 1994, ISBN 3-519-12226-X .