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Introduction to Topology Class Notes
Algebraic Topology
Topology, 2nd Edition, James R. Munkres.

Copies of the classnotes are on the internet in PDF format as given below. The “Proofs of Theorems” files were prepared in Beamer. The “Printout of Proofs” are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors). The “Proofs of Theorems” files were prepared in Beamer by Jack Hartsell, spring 2018.

Part II. Algebraic Topology.

Chapter 9. The Fundamental Group.

• Section 51. Homotopy of Paths. PDF.
• Supplement. Proofs of Theorems in Section 51. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 51. PDF.
• Section 52. The Fundamental Group. PDF.
• Supplement. Proofs of Theorems in Section 52. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 52. PDF.
• Section 53. Covering Spaces. PDF.
• Supplement. Proofs of Theorems in Section 53. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 53. PDF.
• Section 54. The Fundamental Group of the Circle. PDF.
• Supplement. Proofs of Theorems in Section 54. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 54. PDF.
• Section 55. Retraction and Fixed Points. PDF. (This section includes a proof of Brouwer’s Fixed Point Theorem for the Disk.)
• Supplement. Proofs of Theorems in Section 55. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 55. PDF.
• Section 56. The Fundamental Theorem of Algebra. PDF. (This section includes a proof of the Fundamental Theorem of Algebra.)
• Section 57. The Borsuk-Ulam Theorem. PDF.
• Supplement. Proofs of Theorems in Section 57. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 57. PDF.
• Section 58. Deformation Retracts and Homotopy Type. PDF.
• Supplement. Proofs of Theorems in Section 58. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 58. PDF.
• Section 59. The Fundamental Group of Sn. PDF.
• Supplement. Proofs of Theorems in Section 59. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 59. PDF.
• Section 60. Fundamental Groups of Some Surfaces. PDF.
• Supplement. Proofs of Theorems in Section 60. PDF (prepared in Beamer).
• Supplement. Printout of the Proofs of Theorems in Section 60. PDF.
• Study Guide 9.

Chapter 10. Separation Theorem in the Plane.

• Section 61. The Jordan Separation Theorem.
• Section 62. Invariance of Domain.
• Section 63. The Jordan Curve Theorem.
• Section 64. Imbedding Graphs in the Plane.
• Section 65. The Winding Number of a Simple Closed Curve.
• Section 66. The Cauchy Integral Formula.
• Study Guide 10.

Chapter 11. The Seifert-van Kampen Theorem.

• Section 67. Direct Sums of Abelian Groups.
• Section 68. Free Products of Groups.
• Section 69. Free Groups.
• Section 70. The Seifert-van Kampen Theorem.
• Section 71. The Fundamental Group of a Wedge of Circles.
• Section 72. Adjoining a Two-Cell.
• Section 73. The Fundamental Groups of the Torus and the Dunce Cap.
• Study Guide 11.

Chapter 12. Classification of Surfaces.

• Section 74. Fundamental Groups of Surfaces.
• Section 75. Homology of Surfaces.
• Section 76. Cutting and Pasting.
• Section 77. The Classification Theorem.
• Section 78. Constructing Compact Surfaces.
• Study Guide 12.

Chapter 13. Classification of Covering Spaces.

• Section 79. Equivalence of Covering Spaces.
• Section 80. The Universal Covering Space.
• Section 81. Covering Transformations.
• Section 82. Existence of Covering Spaces.
• Study Guide 13.

Chapter 14. Applications to Group Theory.

• Section 83. Covering Spaces of a Graph.
• Section 84. The Fundamental Group of a Graph.
• Section 85. Subgroups of Free Groups.
• Study Guide 14.

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