A basic course in algebraic topology

A basic course in algebraic topology [Massey]......Page Massey W.S. A basic course in algebraic topology (Springer, 1991)(444s).djvu-page1.djvu
Title......Page 9713_to_0021aaa.tif.djvu
Copyright......Page 9713_to_0021aab.tif.djvu
Preface......Page 9713_to_0021aac.tif.djvu
Contents......Page 9713_to_0021aag.tif.djvu
Notation and Terminology......Page 9713_to_0021aam.tif.djvu
§1. Introduction......Page 9713_to_0021aao.tif.djvu
§2. Definition and examples of n-manifolds......Page 9713_to_0021aap.tif.djvu
§3. Orientable vs. nonorientable manifolds......Page 9713_to_0021aaq.tif.djvu
§4. Examples of compact, connected 2-manifolds......Page 9713_to_0021aas.tif.djvu
§5. Statement of the classification theorem for compact surfaces......Page 9713_to_0041aac.tif.djvu
§6. Triangulations of compact surfaces......Page 9713_to_0041aah.tif.djvu
§7. Proof of theorem 5.1......Page 9713_to_0041aaj.tif.djvu
§8. The Euler characteristic of a surface......Page 9713_to_0041aat.tif.djvu
Notes......Page 9713_to_0061aaf.tif.djvu
References......Page 9713_to_0061aah.tif.djvu
§1. Introduction......Page 9713_to_0061aai.tif.djvu
§2. Basic notation and terminology......Page 9713_to_0061aaj.tif.djvu
§3. Definition of the fundamental group of a space......Page 9713_to_0061aal.tif.djvu
§4. The effect of a continuous mapping on the fundamental group......Page 9713_to_0061aap.tif.djvu
§5. The fundamental group of a circle is infinite cyclic......Page 9713_to_0081aaa.tif.djvu
§6. Application: The Brouwer fixed-point theorem in dimension 2......Page 9713_to_0081aad.tif.djvu
§7. The fundamental group of a product space......Page 9713_to_0081aaf.tif.djvu
§8. Homotopy type and homotopy equivalence of spaces......Page 9713_to_0081aah.tif.djvu
Notes......Page 9713_to_0081aal.tif.djvu
References......Page 9713_to_0081aam.tif.djvu
§2. The weak product of abelian groups......Page 9713_to_0081aan.tif.djvu
§3. Free abelian groups......Page 9713_to_0081aaq.tif.djvu
§4. Free products of groups......Page 9713_to_0101aae.tif.djvu
§5. Free groups......Page 9713_to_0101aai.tif.djvu
§6. The presentation of groups by generators and relations......Page 9713_to_0101aal.tif.djvu
§7. Universal mapping problems......Page 9713_to_0101aao.tif.djvu
Notes......Page 9713_to_0101aaq.tif.djvu
References......Page 9713_to_0101aar.tif.djvu
§1. Introduction......Page 9713_to_0101aat.tif.djvu
§2. Statement and proof of the theorem of Seifert and Van Kampen......Page 9713_to_0121aaa.tif.djvu
§3. First application of theorem 2.1......Page 9713_to_0121aae.tif.djvu
§4. Second application of theorem 2.1......Page 9713_to_0121aai.tif.djvu
§5. Structure of the fundamental group of a compact surface......Page 9713_to_0121aaj.tif.djvu
§6. Application to knot theory......Page 9713_to_0121aaq.tif.djvu
§7. Proof of lemma 2.4......Page 9713_to_0141aab.tif.djvu
Notes......Page 9713_to_0141aag.tif.djvu
References......Page 9713_to_0141aaj.tif.djvu
§2. Definition and some examples of covering spaces......Page 9713_to_0141aak.tif.djvu
§3. Lifting of paths to a covering space......Page 9713_to_0141aaq.tif.djvu
§4. The fundamental group of a covering space......Page 9713_to_0141aat.tif.djvu
§5. Lifting of arbitrary maps to a covering space......Page 9713_to_0161aaa.tif.djvu
§6. Homomorphisms and automorphisms of covering spaces......Page 9713_to_0161aad.tif.djvu
§7. The action of the group π(X.x) on the set p[-1](x) ......Page 9713_to_0161aag.tif.djvu
§8. Regular covering spaces and quotient spaces......Page 9713_to_0161aai.tif.djvu
§9. Application: The Borsuk-Ulam theorem for the 2-sphere......Page 9713_to_0161aal.tif.djvu
§10. The existence theorem for covering spaces......Page 9713_to_0161aan.tif.djvu
Notes......Page 9713_to_0161aar.tif.djvu
References......Page 9713_to_0161aat.tif.djvu
§2. Summary of some of the basic properties of homology theory......Page 9713_to_0181aaa.tif.djvu
§3. Some examples of problems which motivated the development of homology theory in the nineteenth century......Page 9713_to_0181aac.tif.djvu
Notes......Page 9713_to_0181aaj.tif.djvu
References......Page 9713_to_0181aak.tif.djvu
§2. Definition of cubical singular homology groups......Page 9713_to_0181aal.tif.djvu
§3. The homomorphism induced by a continuous map......Page 9713_to_0181aaq.tif.djvu
§4. The homotopy property of the induced homomorphisms......Page 9713_to_0181aat.tif.djvu
§5. The exact homology sequence of a pair......Page 9713_to_0201aac.tif.djvu
§6. The main properties of a relative homology group......Page 9713_to_0201aag.tif.djvu
§7. The subdivision of singular cubes and the proof of theorem 6.4......Page 9713_to_0201aal.tif.djvu
Notes......Page 9713_to_0201aar.tif.djvu
§2. Homology groups of cells and spheres ─ Applications......Page 9713_to_0201aat.tif.djvu
§3. Homology of finite graphs......Page 9713_to_0221aaf.tif.djvu
§4. Homology of compact surfaces......Page 9713_to_0221aao.tif.djvu
§5. The Mayer-Vietoris exact sequence......Page 9713_to_0241aaa.tif.djvu
§6. The Jordan-Brouwer separation theorem and invariance of domain......Page 9713_to_0241aae.tif.djvu
§7. The relation between the fundamental group and the first homology group*......Page 9713_to_0241aak.tif.djvu
Notes & References......Page 9713_to_0241aar.tif.djvu
§2. Adjoining cells to a space......Page 9713_to_0241aas.tif.djvu
§3. CW-complexes......Page 9713_to_0261aab.tif.djvu
§4. The homology groups of a CW-complex......Page 9713_to_0261aaf.tif.djvu
§5. Incidence numbers and orientation of cells......Page 9713_to_0261aal.tif.djvu
§6. Regular CW-complexes......Page 9713_to_0261aaq.tif.djvu
§7. Determination of incidence numbers for a regular cell complex......Page 9713_to_0261aar.tif.djvu
§8. Homology groups of a pseudomanifold......Page 9713_to_0281aac.tif.djvu
References......Page 9713_to_0281aag.tif.djvu
§2. Chain complexes......Page 9713_to_0281aah.tif.djvu
§3. Definition and basic properties of homology with arbitrary coefficients......Page 9713_to_0281aap.tif.djvu
§4. Intuitive geometric picture of a cycle with coefficients in G......Page 9713_to_0281aat.tif.djvu
§5. Coefficient homomorphisms and coefficient exact sequences......Page 9713_to_0301aaa.tif.djvu
§6. The universal coefficient theorem......Page 9713_to_0301aac.tif.djvu
§7. Further properties of homology with arbitrary coefficients......Page 9713_to_0301aah.tif.djvu
References......Page 9713_to_0301aal.tif.djvu
§1. Introduction......Page 9713_to_0301aam.tif.djvu
§2. The product of CW-complexes and the tensor product of chain complexes......Page 9713_to_0301aan.tif.djvu
§3. The singular chain complex of a product space......Page 9713_to_0301aap.tif.djvu
§4. The homology of the tensor product of chain complexes (The Künneth theorem)......Page 9713_to_0301aar.tif.djvu
§5. Proof of the Eilenberg-Zilber theorem......Page 9713_to_0301aat.tif.djvu
§6. Formulas for the homology groups of product spaces......Page 9713_to_0321aan.tif.djvu
References......Page 9713_to_0321aaq.tif.djvu
§1. Introduction......Page 9713_to_0321aas.tif.djvu
§2. Definition of cohomology groups ─ Proofs of the basic properties......Page 9713_to_0321aat.tif.djvu
§3. Coefficient homomorphisms and the Bockstein operator in cohomology......Page 9713_to_0341aac.tif.djvu
§4. The universal coefficient theorem for cohomology groups......Page 9713_to_0341aad.tif.djvu
§5. Geometric interpretation of cochains, cocycles, etc.......Page 9713_to_0341aaj.tif.djvu
§6. Proof of the excision property: the Mayer-Vietoris sequence......Page 9713_to_0341aam.tif.djvu
Notes & References......Page 9713_to_0341aap.tif.djvu
§1. Introduction......Page 9713_to_0341aaq.tif.djvu
§3. An overall view of the various products......Page 9713_to_0341aar.tif.djvu
§4. Extension of the definition of the various products to relative homology and cohomology groups......Page 9713_to_0361aac.tif.djvu
§5. Associativity, commutativity, and existence of a unit for the various products......Page 9713_to_0361aag.tif.djvu
§6. Digression: The exact sequence of a triple or a triad......Page 9713_to_0361aaj.tif.djvu
§7. Behavior of products with respect to the boundary and coboundary operator of a pair......Page 9713_to_0361aal.tif.djvu
§8. Relations involving the inner product......Page 9713_to_0361aao.tif.djvu
§9. Cup and cap products in a product space......Page 9713_to_0361aap.tif.djvu
§10. Remarks on the coefficients for the various products ─ The cohomology ring......Page 9713_to_0361aaq.tif.djvu
§11. The cohomology of product spaces (The Künneth theorem for cohomology)......Page 9713_to_0361aar.tif.djvu
Notes & References......Page 9713_to_0381aac.tif.djvu
§1. Introduction......Page 9713_to_0381aad.tif.djvu
§2. Orientability and the existence of orientations for manifolds......Page 9713_to_0381aae.tif.djvu
§3. Cohomology with compact supports......Page 9713_to_0381aal.tif.djvu
§4. Statement and proof of the Poincaré duality theorem......Page 9713_to_0381aan.tif.djvu
§5. Applications of the Poincaré duality theorem to compact manifolds......Page 9713_to_0381aas.tif.djvu
§6. The Alexander duality theorem......Page 9713_to_0401aad.tif.djvu
§7. Duality theorems for manifolds with boundary......Page 9713_to_0401aai.tif.djvu
§8. Appendix: Proof of two lemmas about cap products......Page 9713_to_0401aan.tif.djvu
Notes......Page 9713_to_0421aae.tif.djvu
References......Page 9713_to_0421aaf.tif.djvu
§2. The projective spaces......Page 9713_to_0421aah.tif.djvu
§3. The mapping cylinder and mapping cone......Page 9713_to_0421aam.tif.djvu
§4. The Hopf invariant......Page 9713_to_0421aap.tif.djvu
Notes......Page 9713_to_0421aas.tif.djvu
References......Page 9713_to_0421aat.tif.djvu
§1. Introduction......Page 9713_to_0441aaa.tif.djvu
§2. Differentiable singular chains......Page 9713_to_0441aab.tif.djvu
§3. Statement and proof of De Rham\'s theorem......Page 9713_to_0441aae.tif.djvu
Note & References......Page 9713_to_0441aak.tif.djvu
§1. Basic definitions......Page 9713_to_0441aam.tif.djvu
§2. Homogeneous G-spaces......Page 9713_to_0441aao.tif.djvu
Index......Page 9713_to_0441aar.tif.djvu
Purpose of the Book......Page 9713_to_0444aac.tif.djvu