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MSc.

2011

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Marston Morse (1892-1977)

MORSE

, , Morse, .

1933 Bo-her Memorial . Morse

( .)

Harold Calvin Marston Morse

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-

( + = )

, ( ),

,

() . :

:

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Leonard Euler 1736 : Seven bridges of Konig-sberg, Euler - Konigsberg ( Kaliningrad)

paper (graph theory).

:http://en.wikipedia.org/wiki/File:Konigsberg_bridges.png

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:) ( )T

) ) ,

) , .

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. -

:

1. (1 1 )2.

3.

( , )TX X ( , )TY Y

: ( , ) ( , )T Tf X X Y Y( , )

TX X ( , )TY Y

ff

1

: ( , ) ( , )T Tf Y Y X X

.

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MORSE (MORSE THEORY)

Morse

.

Morse:

f orse , :

, :

f

det( ( )) 0Hessian p

2 2

2

2 2

2

( ) ( )

( )( ) ( )

f p f px x y

Hessian pf p f p

y x y

, :

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p :

( ) 0f p

f

( ) ( ) ... 0f p f px y

(index)

o Hessian:

0 minimum

1 saddle point2 maximum

:

,

f

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V,

(p).

:f M R:

:

a

M x M

( )f x a

:

) ) 2-cell

)

) -

1 ,

) .

0( ( ))a f p a

M( ) ( )f p a f q aM( ) ( )f q a f r aM( ) ( )f r a f s

aM

( )f s a

a

M

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.

( p q)

.

..

.

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.

.

.

John C. Hart

School of EECS

Washington State University

Computational Topologyfor by

Computer Graphics

5

:

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n-cell

n - -

(n=1,2,3),

.

{ : 1}n ne x R x , : { : 1}n ne x R x

n=0, 0-cell .

n=1, 1-cell . n=2, 2-cell .

n=3, 3-cell ..

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-

: 2 2g

Eulerg

( 0)g 2

( 1)g 0 , , .

, ,

g=0 g=1 g=2 g=3

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(HOMOTOPY)

A ~ B

(DEFORMATION RETRACTION)

retraction(, , ,

)

.

.

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0-cell.

1-cell.

2-cell.

1-cell.

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:

(Milnor 1963)

: Ma = {p M |f(p) a}, (

f

f 1[a, b]

f. Ma Mb.

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p .

( p), .

-cell.

:f M R( )f p c

1[ , ]f c c

0cM cM

(Milnor 1963)

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EULER (EULER CHARACTERISTIC)

V E FV =

=

F =

: 2V E F

g: ( )g V E F ( ) 2 2g g

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BETTI (BETTI NUMBERS)

Betti (Betti numbers),

Enrico Betti (1823-1892).

Betti

,

.

, n - Betti (rank) n - (homology group)

BETTI

1 Klein 2

Mbius 1

1

0

2

MORSE (MORSE INEQUALITIES)

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MORSE (MORSE INEQUALITIES).

Morse

M

(

Morse).

( MORSE)

C = R C

( 1) ( ) ( 1)R M C

( MORSE)

1 2 0 1 2 0( ) ( ) ( ) ... ( ) ...R M R M R M R M C C C C

1 1 0C C: 1 1 0R R R C

= O - BettiR

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1. Elementary concepts of topology, Paul Alexandroff, Dover Publications,

Inc, New York 1960.2. Morse theory by J. Milnor, based on lectures by M. Spivac and R. Wells,

Princeton University Press, 1973

3. An invitation to Morse theory, Liviu Nicolaescu, Springer 2007

4. Differential Geometry and Lie Groups for Physicists, Marian Fecko,

Cambridge University Press 2006

5. , ( Differential Geometry , Martin M.

Lipschutz, McGraw Hill,1974), ,1981

6. Theory and Problems of General Topology, Seymour Lipschutz, Schaums

Outline series, McGraw Hill,1965)

7. Algebraic Topology, Allen Hatcher, Cambridge University Press, 2002.

( ) : http://www.math.cornell.edu/~hatcher/AT/AT.pdf

8. Wikipedia Morse,http://en.wikipedia.org/wiki/Morse_theory

http://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://en.wikipedia.org/wiki/Morse_theoryhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://en.wikipedia.org/wiki/Morse_theoryhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdfhttp://www.math.cornell.edu/~hatcher/AT/AT.pdf