Winding around : the winding number in topology, geometry, and analysis

Winding Around The Winding Number in Topology, Geometry, and Analysis....  1
	Contents.............................................................  6
	Foreword: MASS and REU at Penn State University...................... 10
	Preface.............................................................. 12
	Chapter 1 Prelude: Love, Hate, and Exponentials...................... 16
		1.1. Two sets of travelers....................................... 16
		1.2. Winding around.............................................. 20
		1.3. The most important function in mathematics.................. 22
		1.4. Exercises................................................... 28
	Chapter 2 Paths and Homotopies....................................... 30
		2.1. Path connectedness.......................................... 30
		2.2. Homotopy.................................................... 34
		2.3. Honiotopies and simple-connectivity......................... 37
		2.4. Exercises................................................... 41
	Chapter 3 The Winding Number......................................... 42
		3.1. Maps to the punctured plane................................. 42
		3.2. The winding number.......................................... 44
		3.3. Computing winding numbers................................... 48
		3.4. Smooth paths and loops...................................... 53
		3.5. Counting roots via winding numbers.......................... 57
		3.6. Exercises................................................... 62
	Chapter 4 Topology of the Plane...................................... 64
		4.1. Some classic theorems....................................... 64
		4.2. The Jordan curve theorem I.................................. 69
		4.3. The Jordan curve theorem II................................. 74
		4.4. Inside the Jordan curve..................................... 79
		4.5. Exercises................................................... 82
	Chapter 5 Integrals and the Winding Number........................... 88
		5.1. Differential forms and integration.......................... 88
		5.2. Closed and exact forms...................................... 95
		5.3. The winding number via integration.......................... 99
		5.4. Homology....................................................102
		5.5. Cauchy’s theorem............................................109
		5.6. A glimpse at higher dimensions..............................110
		5.7. Exercises...................................................112
	Chapter 6 Vector Fields and the Rotation Number......................116
		6.1. The rotation number.........................................116
		6.2. Curvature and the rotation number...........................120
		6.3. Vector fields and singularities.............................122
		6.4. Vector fields and surfaces..................................128
		6.5. Exercises...................................................132
	Chapter 7 The Winding Number in Functional Analysis..................136
		7.1. The Fredholm index..........................................136
		7.2. Atkinson’s theorem..........................................140
		7.3. Toeplitz operators..........................................144
		7.4. The Toeplitz index theorem..................................148
		7.5. Exercises...................................................151
	Chapter 8 Coverings and the Fundamental Group........................154
		8.1. The fundamental group.......................................154
		8.2. Covering and lifting........................................159
		8.3. Group actions...............................................166
		8.4. Examples....................................................168
		8.5. The Nielsen-Schreier theorem................................172
		8.6. An application to nonassociative algebra....................176
	Chapter 9 Coda: The Bott Periodicity Theorem.........................184
		9.1. Hornotopy groups............................................184
		9.2. The topology of the general linear group....................190
	Appendix A Linear Algebra............................................196
		A.l. Vector spaces...............................................196
		A.2. Basis and dimension.........................................199
		A.3. Linear transformations......................................203
		A.4. Duality.....................................................207
		A.5. Norms and inner products....................................209
		A.6. Matrices and determinants...................................212
	Appendix B Metric Spaces.............................................218
		B.l. Metric spaces...............................................218
		B.2. Continuous functions........................................221
		B.3. Compact spaces..............................................223
		B.4. Function spaces.............................................229
	Appendix C Extension and Approximation Theorems......................232
		C.l. The Stone-Weierstrass theorem...............................232
		C.2. The Tietze extension theorem................................236
	Appendix D Measure Zero..............................................238
		D.l. Measure zero subsets of R and of Sl.........................238
	Appendix E Calculus on Normed Spaces.................................244
		E.l. Normed vector spaces........................................244
		E.2. The derivative..............................................246
		E.3. Properties of the derivative................................249
		E.4. The inverse function theorem................................252
	Appendix F Hilbert Space.............................................254
		F.l. Definition and examples.....................................255
		F.2. Orthogonality...............................................258
		F.3. Operators...................................................261
	Appendix G Groups and Graphs.........................................264
		G.l. Equivalence relations.......................................265
		G.2. Groups......................................................266
		G.3. Homomorphisms...............................................269
		G.4. Graphs......................................................273
	Bibliography.........................................................276
	Index................................................................280