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Mentor: Artur B. Saturnino
Degree and Intersection Theory
Directed Reading Program Fall 2020
30 April 2020
AIRIKA YEE
Text: John Milnor, "Topology from the Differentiable Viewpoint" / Victor Guillemin and Alan Pollack, "Differential Topology.”
University of Pennsylvania
The aim of this presentation is to prove the Jordan Brouwer Separation Theorem.
OVERVIEW
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
DEGREE OF A MAP
DEFINITIONS
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
DEGREE OF A MAP
DEFINITIONS
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
DIRECTIONAL MAP
Given a compact, connected manifold X and a smooth map , the directional map of any not in the image of f(x) is defined as:
WINDING NUMBER
DEFINITIONS
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
JORDAN BROUWER SEPARATION THEOREM
JORDAN BROUWER SEPARATION THEOREM
The complement of the compact, connected manifold X consists of two connected open sets: the outside D
0 and an inside D
1 .
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
Any fixed point in Rn - X can be joined to a point in a neighborhood of some without intersecting X.
STEP ONE
JORDAN BROUWER SEPARATION THEOREM
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
STEP TWO
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
Rn - X has, at most, 2 connected components.
JORDAN BROUWER SEPARATION THEOREM
Rn - X has, at most, 2 connected components.Points in the same connected component have the same winding number.
STEP TWO
Degree is invariant under homotopy.
Homotopy between directional maps.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
JORDAN BROUWER SEPARATION THEOREM
Consider a ray,that intersects X.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
STEP THREE
JORDAN BROUWER SEPARATION THEOREM
Consider a ray,that intersects X.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
STEP THREE
JORDAN BROUWER SEPARATION THEOREM
STEP FOUR
Rn - X has precisely two components.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
JORDAN BROUWER SEPARATION THEOREM
STEP FOUR
Rn - X has precisely two components.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
JORDAN BROUWER SEPARATION THEOREM
STEP FIVE
If z is very large, then (i.e., D
0 is the “outside” of X).
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
JORDAN BROUWER SEPARATION THEOREM
THEOREM
JORDAN BROUWER SEPARATION THEOREM
We’ve shown that a simple, closed curve in Rn can be separated into an “inside” and “outside,” which can be identified by the mod 2 winding number.
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
● The idea of a direction map is seen in the proof of other theorems.
Ex: Poincare-Hopf Theorem
● Really interesting results from counting points!
● Thank you Artur for the past two semesters in the Directed Reading Program!
FINAL THOUGHTS
Airika Yee, Degree and Intersection Theory Directed Reading Program, April 2020
O U T L I N E D E F I N I T I O N S J O R D A N B R O U W E R C O N C L U S I O N
THANK YOU
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