algebraictopology

Table of Contents

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List of Notations

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- Prerequisites:
- Theorem 1.1 (Jordan Curve Theorem):
- Theorem 1.2 (Poincare Bendixon):
- Theorem 1.3
.
- Definition 2.1:
- Theorem 2.1(Heine Borel):
- Theorem 2.2:
- Proof:
- Definition 2.2 (The Lebsesgue number for a cover):
- Theorem 2.3:
- Proof:
- Definition 2.3 (Locally compact spaces):
- Examples:
- One point compactification:
- Theorem 2.4:
- Definition 2.4 (Proper maps):
- Theorem 2.5:
- Proof:
- Theorem 2.6:
- Proof:
- Stereographic projection:
- Theorem 2.7:
- Theorem 2.8:
- Proof:
- Theorem 2.9:
Exercises
.
- Definition 3.1:
- Examples 3.1:
- Theorem 3.1:
- Example 3.2:
- Example 3.3:
- Definition 3.2 (Path connectedness):
- Theorem 3.3:
- Proof:
- Corollary 3.4:
- Theorem 3.5:
- Proof:
- Definition 3.3:
- Theorem 3.6:
- Proof:
- The Tietze's extension theorem:
- Theorem 3.7:
- Remarks:
Exercises:
.
- Quotient Spaces:
- Definition 4.1:
- Theorem 4.1 (Universal property of quotients):
- Proof:
- Definition 4.2:
- Example 4.1:
- Theorem 4.2:
- Proof:
- Corollary 4.3:
- Identification spaces:
- Theorem 4.4:
- Proof:
- The real projective spaces $\mathbb{R}P^n$ :
- Theorem 4.5:
- Proof:
- Theorem 4.6:
- The Möbius band and the Klein's bottle:
- The torus:
- Example:
- The wedge:
- Theorem 4.7:
- Proof:
- Surfaces:
- Hausdorff Quotients:
- Definition 4.3:
- Theorem 4.8:
- Proof:
- Corollary 4.9:
- Proof:
Exercises
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- Definition 5.1:
- Theorem 5.1:
- Proof:
- Theorem 5.2:
- Proof:
- Theorem 5.3:
- Proof:
- Theorem 5.4:
- Proof:
- Theorem 5.5:
- Proof:
- Remark:
Exercises
- Definition 7.1 (homotopy of paths):
- Theorem 7.1:
- Proof:
- Notation:
- Theorem 7.2 (Reparametrization theorem):
- Proof:
- Juxtaposition of paths:
- Lemma 7.3:
- Proof:
- Corollary 7.4:
- Definition 7.2:
- The Inverse Path and the constant path:
- Lemma 7.5:
- Proofs:
- Theorem 7.6:
- Proof:
- Definition 7.3 (The fundamental group $\pi_1(X, x_0)$ ):
- Terminology:
- Theorem 7.7:
- Definition:
- Theorem 7.8:
- Proof:
- Remarks:
- Theorem 7.9:
- Proof:
- Definition:
- Definition 7.4 (Convex and star-shaped domains):
- Theorem 7.10:
- Proof:
Exercises:
- Example 8.1:
- Definition 8.1 (Covariant functor):
- Definition 8.2 (Contravariant functor):
- Example 8.2:
- Example 8.3:
- Example 8.4:
- Example 8.5:
- Category of pairs:
- Definition 8.3:
Exercises:
- Definition 9.1 (The category of pointed topological spaces):
- Lemma 9.1:
- Proof:
- Definition 9.2 (Retraction):
- Example 9.1:
- Theorem 9.2:
- Lemma 9.3:
- Proof:
- Corollary 9.4 (No retraction theorem):
- Proof:
- Corollary 9.5 (Brouwer's fixed point theorem):
- Proof:
- Fundamental group of a Product:
- Theorem 9.6:
- Proof:
- Corollary 9.7:
Exercises
- Definition 10.1:
- Theorem 10.1:
- Proof:
- Examples 10.1:
- Theorem 10.2 (Brouwer's fixed point theorem):
- Proof:
- Remark:
- Theorem 10.3 (Perron-Frobenius):
- Proof:
- Fundamental groups of spheres:
- Theorem 10.4:
- Proof:
- Theorem 10.5:
- Proof:
Exercises
- Definition 11.1 (Homotopies of maps):
- Theorem 11.1:
- Theorem 11.2:
- Theorem 11.3:
- Proof:
- Corollary 11.4:
- Theorem 11.5:
- Definition 11.2 (Homotopy equivalence):
- Theorem 11.6:
- Proof:
- Deformation retract:
- Theorem 11.7:
- Proof:
- Example 11.1:
- Theorem 11.8:
- Example 11.2:
- Example 11.3:
- Corollary 11.9:
Exercises:
- Lemma 12.1:
- Proof:
- Corollary 12.2:
- Proof:
- Theorem 12.3:
- Proof:
- Lemma 12.4 (The lifting lemma):
- Proof:
- Note:
- Definition 12.1:
- Lemma 12.5:
- Proof:
- Lemma 12.6:
- Proof:
- Corollary 12.7 (Generators for $\pi_1(S^1, 1)$ ):
- Proof:
- Definition 12.2 (Degree of a map):
- Theorem 12.8:
- Proof:
- Theorem 12.9 (The Borsuk Ulam Theorem):
- Proof for the case :
- Corollary 12.10:
- Proof:
- Theorem 12.11 (Fundamental theorem of algebra):
- Proof:
Exercises:
- Definition 15.1:
- Remark:
- Examples 15.1:
- Theorem 15.1:
- Proof:
- The lifting problem:
- Example 15.2:
- Theorem 15.2 (uniqueness of lifts):
- Proof:
Exercises:
- Theorem 16.1 (path lifting lemma):
- Proof:
- Lifting of homotopies:
- Theorem 16.2 (Covering homotopy property):
- Proof:
- Theorem 16.3:
- Proof:
- Remark:
- Theorem 16.4:
- Proof:
Exercises
- Definition 17.1:
- Theorem 17.1:
- Proof:
- Theorem 17.2:
- Proofs:
- Corollary 17.3:
- Proof:
- Regular coverings:
- Theorem 17.4:
- Proof:
- Definition 17.2:
- Corollary 17.5:
Exercises
- Theorem 18.1:
- Proof:
- Continuity of the lift ${\tilde f}$ :
- Theorem 18.2 (Uniqueness of simply connected covers):
- Proof:
- Example 18.1 (Some applications to complex analysis):
- Theorem 16.3:
- Theorem 16.4 (The Little Picard Theorem):
- Proof:
Exercises:
- Definition 19.1 (Deck transformations):
- Examples 19.1:
- Theorem 19.1:
- Proof:
- Remark:
- Definition 19.2:
- Action of Deck $(\widetilde X, X)$ on the fibers $p^{-1}(x_0)$ :
- Theorem 19.2:
- Proof:
- Theorem 19.3:
- Corollary 19.4:
- Corollary 19.5:
- Corollary 19.6:
- Existence of a simply connected covering space:
- Definition 19.3:
- Definition 19.4:
Exercises
- Fundamental groups of orbit spaces:
- Definition 20.1:
- Theorem 20.1:
- Proof:
- Theorem 20.2:
- Proof:
- Definition 20.2 (Lens spaces):
- Definition 20.3 (Generalized lens spaces):
- The Mobius band:
- Klein's bottle:
- Theorem 20.3:
Exercises:
- Theorem 22.1
- Definition 22.1:
- Lemma 22.2:
- Lemma 22.3:
- Lemma 22.4:
- Lemma 22.5:
- Proof:
- Lemma 22.6:
- Proof:
- Remark:
- Topological structure of $SO(4, \mathbb{R})$ :
- Lemma 22.7:
- Proof:
- Theorem 22.8:
- Proof:
- Corollary 22.9:
Exercises
- Definition 23.1:
- Theorem 23.1:
- Proof:
- Theorem 23.2:
- Proof:
- Definition 23.2 (Coproduct of abelian groups or the direct sum):
- Theorem 23.3:
- Proof:
- Definition 23.3 (free groups):
- Theorem 23.4:
- Proof:
- Generators and relations:
- Example 23.1 (Presentation of some groups):
- Push-outs:
- Definition 23.4:
- Remark:
- Theorem 23.5:
- Proof:
- Example:
- Existence of push outs:
- Theorem 23.6:
- Proof:
Exercises
- Definition 25.1:
- Example 25.1:
- Example 25.2
- Theorem 25.1:
- Proof:
- Corollary 25.2:
- Proof:
- Definition 25.2:
- Example 25.3 (The torus and the Klein's bottle):
- Example 25.4 (The projective plane):
- Example 25.5 (Real projective spaces):
- Theorem 25.3:
- Definition 25.3 (The cone over a space):
- Theorem 25.5:
- Proof:
Exercises
- Theorem 26.1 (Seifert and Van Kampen - version I):
- Theorem 26.2 (Seifert and Van Kampen - version II):
- Corollary 26.3:
- Fundamental groups of spheres:
- Corollary 26.4:
- Wedge of two circles:
- Corollary 26.5
- Theorem 26.6:
- Proof:
- Corollary 26.7:
- Proof:
- The projective plane:
- The torus and the Klein's bottle:
- The double torus:
- Fundamental groups of some adjunction spaces:
- Theorem 26.8:
Exercises
- Some motivation for singular homology:
- Definition 29.1 (The standard simplex):
- Lemma 29.1:
- Definition 29.2 (Singular chains):
- Definition 29.3 (Boundary of a singular simplex):
- Theorem 29.2:
- Proof:
- Definition 29.4:
- Theorem 29.3:
- Proof:
- The category of chain complexes:
- Definition 29.5:
- Definition 29.6:
- Theorem 29.4:
- Proof:
- Theorem 29.5:
- Proof:
- The long exact homology sequence:
- Definition 29.7:
- Theorem 29.6:
- Proof:
- Exactness of (29.17):
Exercises
- The homology groups :
- Definition 31.1:
- Theorem 31.1:
- Corollary 31.2:
- Theorem 31.3:
- The augmentation map $\epsilon:S_0(X)\longrightarrow \mathbb{Z}\;$ :
- Definition 31.2:
- Theorem 31.4:
- Proof:
- Theorem 31.5:
- Proof:
- Convex sets and barycentric coordinates:
- Theorem 31.6:
- Theorem 31.7:
- Proof:
Exercises
- Theorem 32.1:
- Lemma 32.2:
- Proof:
- Lemma 32.3:
- Proof:
- Lemma 32.4:
- Proof:
- Lemma 32.5:
- Proof:
- Lemma 32.6:
- Proof:
- Definition 32.1 (Natural transformation):
Exercises
- The cross product:
- Theorem 33.1:
- Proof:
- Homotopy and chain homotopy:
- Theorem 33.2:
- Proof:
- Definition 33.1:
- Corollary 33.3:
Exercises
- Affine simplicies and barycentric subdivision:
- Theorem 34.1:
- Definition 34.1 (Barycenter of an affine simplex):
- Definition 34.2:
- Theorem 34.2:
- Proof:
- Theorem 34.3:
- Proof:
- Theorem 34.4:
- Proof:
- Definition 34.3:
- Lemma 34.5:
- Proof:
- Theorem 34.6:
- Proof:
- Theorem 34.7 (Mayer Vietoris sequence):
- Proof:
- Theorem 34.8 (Naturality of the Mayer Vietoris sequence):
- Proof:
Exercises
- Theorem 35.1:
- Interpretation of the connecting homomorphism:
- Corollary 35.2:
- Proof:
- Corollary 35.3:
- Proof:
- Homology groups of adjunction spaces:
- Theorem 35.4
- Proof:
- Theorem 35.5:
- Corollary 35.6 (Homology groups of $\mathbb{R}P^2$ ):
- Proof:
Exercises
- Theorem 36.1 (No retraction theorem):
- Proof:
- Corollary 36.2 (Brouwer's fixed point theorem):
- Proof:
- Degree of a map:
- Definition 36.1:
- Theorem 36.3:
- Proof:
- Corollary 36.4:
- Proof:
- Theorem 36.5:
- Proof:
- The anti-podal map and its properties:
- Theorem 36.6:
- Proof:
- Corollary 36.7:
- Proof:
- Theorem 36.8:
- Proof:
- Corollary 36.9:
- Proof:
- Corollary 36.10 (Hairy ball theorem):
- Proof:
- Suspension:
- Theorem 36.11:
- Proof:
Exercises
- Definition 37.1:
- Theorem 37.1:
- Theorem 37.2:
- Lemma 37.3:
- Theorem 37.4 (Naturality):
- Proof:
- Retraction:
- Definition 37.2:
- Lemma 37.5:
- Proof:
- Theorem 37.6:
- Proof:
- Example 37.1
Exercises
- Theorem 38.1 (The five lemma):
- Proof:
- Theorem 38.2 (Excision):
- Proof:
- Example 38.1:
- Example 38.2:
- Lemma 38.3 (Barrett and Whitehead):
- Corollary 38.4 (Mayer Vietoris):
- Proof:
- Definition 38.2 (Local homology groups):
- Theorem 38.5:
- Proof:
Exercises
- Definition 40.1 (Directed systems):
- Example 40.1:
- Lemma 40.1:
- Definition 40.2 (Inductive limit):
- Notation:
- Theorem 40.2:
- Proof:
- Remarks:
Exercises:
- Theorem 41.1:
- Proof:
- Theorem 41.2:
- Proof:
- Corollary 41.3:
- Proof:
- Corollary 41.4:
- Proof:
- Corollary 41.5 (Invariance of domain):
- Proof:
- Corollary 41.6 (Jordan Curve theorem):
Exercises
Bibliography
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nisha 2012-03-20