Lecture: Degree of a Map and Homology

Introduction

• Topic: The degree of a continuous map from ( S^n ) to ( S^n ).
• Induced Map on Homology: Focus on the nth homology group, the only interesting one (( H_n(S^n) = \mathbb{Z} )).
• Homomorphism from ( \mathbb{Z} ) to ( \mathbb{Z} ): The map is multiplication by some integer ( d ), referred to as the degree of the map (degree of f = ( d )).

Properties of Degree

• Degree of the Identity Map: ( ext{deg}( ext{Id}) = 1 ).
• Homotopic Maps: If ( f \sim g ), then ( ext{deg}(f) = ext{deg}(g) ).
  • Converse is also true: Homotopic classification of maps between spheres.
• Composition of Maps: ( ext{deg}(f imes g) = ext{deg}(f) imes ext{deg}(g) ).

Examples

• Reflection: Degree of a reflection map is ±1. Specifically, ( ext{deg}( ext{reflection}) = -1 ).
• Antipodal Map: Maps every point ( x ) to ( -x ); degree is ( (-1)^{n+1} ).

Application: Hairy Ball Theorem

• Statement: ( S^n ) has a non-zero continuous tangent vector field iff ( n ) is odd.
• Proof Sketch:
  1. Define function ( F_t(x) = ext{cos}(t)x + ext{sin}(t) rac{v(x)}{|v(x)|} ).
  2. Maps identity to antipodal map showing ( ext{deg(Id)} = ext{deg}( ext{antipodal}) ).
  3. Conclude ( n ) must be odd for non-zero continuous tangent vector field on ( S^n ).

Cellular Homology

• Review of CW Complexes: Building spaces dimension by dimension.
  • 0-skeleton: Points.
  • 1-skeleton: Attach 1-cells (open 1-dimensional disks).
  • Higher skeleta: Attach higher dimensional cells.
• Cellular Chain Complex: Long exact sequence involving relative homology groups.
• Cellular Homology Definition: ( H_n^{cell}(X) = ext{ker}(d_n) / ext{im}(d_{n+1}) ).
  • Reduced to homology of wedges of spheres.

Calculation Examples

• Torus: Calculation aligns with singular and simplicial homology.
  • Sequences: [ 0 \to ext{Z (2-cells)} \to ext{Z}^2 ( ext{1-cells}) \to ext{Z (0-cells)} \to 0 ]
  • Homology: ( H_0 = Z, H_1 = Z^2, H_2 = Z ).
• Klein Bottle: Similar steps but different attachments.
  • Sequences: [ 0 \to ext{Z (2-cells)} \to ext{Z}^2 ( ext{1-cells}) \to ext{Z (0-cells)} \to 0 ]
  • Result: ( H_0 = Z, H_1 = Z imes Z/2Z, H_2 = 0 ).

Key Takeaways

• Degree and Homology: Degree helps to understand homotopy classes of maps, crucial in cellular homology calculation.
• Efficient Calculations: Cellular homology provides a fast way to calculate the homology groups of CW complexes.

Next Steps

• Explore further examples, reinforcing calculation techniques for various CW complexes.