Topic outline
General
Algebraic Topology
(Topologia Algebrica)
a.a. 2016/17 INSTRUCTOR:
Prof. Alberto ALBANO
email: alberto.albano@unito.it
tel.: 011 670 2890GENERAL INFOS:
Credits: 6 ECTS
Lectures: 48 hours
Period: FIRST semester
Language: the course is taught in ENGLISH
SCHEDULE:
- MON 12.30-14.30 (Aula 3), THU 14.30-16.30 (Aula 2)
OFFICE HOURS:
- by appointment (call or send an email)
SYLLABUS:
1. Homotopy: Fundamental group - Cellular spaces - Seifert-van Kampen theorem - Covering spaces -
2. Homology: Singular homology - homotopic invariance - relative homology - exact sequences and homological algebra - Mayer-Vietoris sequence - Eilienberg-Steenrod axioms for homology - CW-complexes and cellular homology - singular cohomology
TEXTBOOKS:
There are many introductory books in algebraic topology suitable for this course. We will mainly follow the book:
Yves Félix, Daniel Tanré, Topologie algébrique. Cours et exercices corrigés, Dunod, 2010
Other useful books:
C. KOSNIOWSKI, Introduzione alla Topologia Algebrica, Zanichelli, 1988.
W. FULTON, Algebraic Topology - A First Course, Springer, 1995.
A. HATCHER, Algebraic Topology, Cambridge University Press, 2001 (free download)
M. GREENBERG & J. HARPER, Algebraic Topology - A First Course, Perseus Publishing, 1981.
J. LEE, Introduction to Topological Manifolds, second edition, Springer, 2011.USEFUL LINKS
These are links to web pages of courses similar to this one. They often contain freely downloadable material.
- Mikhail Khovanov, Introduction to Algebraic Topology, Columbia University, New York, 2013
- David Wilkins, Lecture Notes, Trinity College, Dublin. Lecture notes of various courses, including one on Algebraic Topology
- Algebraic Topology, OCW (Open Courseware) di MIT, Cambridge, MA - Course given in 2006
EXAM:
The final exam consists in an ORAL examination.
There will be questions that require the solution of exercises chosen from a list that will be available before the end of the course.
EXAM SCHEDULE:
1^ date: tuesday 24 january 2017, ore 9:30
2^ date: tuesday 7 february 2017, ore 9:30
3^ date: monday 20 february 2017, ore 9:30Seminars: these are the arguments for the seminars, from the book of J.-C. Pont
1. Les pseudo-précurseurs Partie I Chapitre 1 - sections 1, 2
Eulero-Legendre Partie I Chapitre 2 - sections 2.1, 2.2, 2.3
2. Cauchy - Lhuilier - von Staudt Partie I Chapitre 2 - sections 2.5, 2.6, 2.7
3. Gauss Partie I Chapitre 2 - section 3
4. Riemann Partie II Chapitre 2 - section 1
5. Moebius Partie II Chapitre 3 - section 1Lesson 1 - Monday, 26 September 2016
Course introduction.
Review of basic concepts: homotopy relation between functions, homotopy equivalent spaces, the fundamental group of a topological space, induced homomorphisms, functioriality.
Contractible spaces. \(\mathbf{R}^n \) is contractible. In general any star-shaped subspace of \(\mathbf{R}^n \) is contractible.
Retracts and deformation retracts.
The fundamental group of \( S^1 \) is non trivial. Consequence: \( S^1 \) is not a retract of the disc \( D^2 \).
Lesson 2 - Thursday, 29 September 2016
Properties of the exponential map from \(\mathbf{R}\) to the circle \( S^1 \)
Definition of covering map \(p : E \to X\). The exponential map is a covering map.The Lebesgue's Number Lemma (without proof).
Lifting of paths and of homotopies: existence and uniqueness.
The degree of a closed path in \( \mathbf{S}^1 \).
Theorem. The degree map gives an isomorphism \(\deg : \pi_1(S^1, 1) \to \mathbf{Z}\)
Application: Brouwer's fixed point theorem.
Easy part of van Kampen's theorem: generators. Consequence: the spheres \(S^n\) are simply connected for \( n \ge 2 \)
Lesson 3 - Monday, 3 October 2016
Review of the quotient topology.
Definition of \(X/A\): collapsing (or contracting) \(A\) to a point.
Examples (Proposition 2.2)
Attaching spaces: definition of the attachment of two spaces \(X\) and \(Y\) along \(f : A\to X\) and \(f': A \to Y\).
Example: bouquet of 2 (or \(n\)) spaces.
Attaching a cell to a space.
Example of cellular decomposition: spheres, topological surfaces.
Théorème 2.8. Extension of homotopy for cellular spaces.
Lesson 4 - Thursday, 6 October 2016
Théorème 2.9. If \(Y = X \cup_{f_1} e^{n_1}_1 \cup \dots \cup_{f_k} e^{n_k}_k\) then the quotient map \(q : Y \to Y/X\) is a homotopy equivalence.
Topological groups: definition and action on topological spaces. Orbit spaces.
Proposition 2.13. The quotient map \(q : X \to X/G\) is open. If \(X\) and \(G\) are both compact Hausdorff, then \(X/G\) is Hausdorff (and compact).
Proposition 2.15. Let \(G\) is compact act on \(X\) Hausdorff. If the action is transitive, there are homeomorphisms \(G/G_x \cong X\).
Examples: \(\text{O}(n) / \text{O}(n - 1) \cong S^{n-1} \)
Projective spaces: various description of \(\mathbf{P}^n(\mathbf{R})\) as quotient of
- \(\mathbf{R}^{n+1} - \{0\}\),
- \(S^n\) by the antipodal map
- the closed ball \(E^n\).
Cellular decomposition of \(\mathbf{P}^n(\mathbf{R})\)
- \(\mathbf{R}^{n+1} - \{0\}\),
Lesson 5 - Monday, 10 October 2016
Definition of category, examples: Sets, Grp, Ab, \(R\)-mod, \(k\)-vect, Top, Top*
Some categorial constructions: products and coproducts in various categories.
Definition of concrete category. Free objects in concrete categories. Equivalence of free objects.
Example: all vector spaces are free objects (over one of their bases).
Free objects in Grp: free groups. \(F_n\) = free group on \(n\) generators.
\(F_1 = \mathbf{Z}\), \(F_2\) already quite complicated.
Fact: \(F_n\) is a subgroup of \(F_2\), for any \(n \ge 2\).
Every group is the quotient of a free group.
Definition of presentation of a group.
Lesson 6 - Thusday, 13 October 2016
Free product of groups, amalgamated product and their universal properties.
Generators and relations for the amalgamated product.
Théorème 3.12 (Seifert-van Kampen). Discussion of the theorem (no proof).
Definition of correctly pointed space.
Example: topological manifolds are correctly pointed.
Proposition 3.14. A finite cell complex is correctly pointed.
Proposition 3.15. Fundamental group of bouquet of correctly pointed spaces.
Examples: bouquets of circle, graphs.
Proposition 3.17. Fundamental group and attachment of cells.
Examples:
- every finitely presented group is the fundamental group of a finite cell complex
- the fundamental groups of topological surfaces.
Lesson 7 - Monday, 17 October 2016
Definition of covering space \( p : E \to X\). Various remarks about the conditions on the spaces \(E\) and \(X\).
Examples:- omeomorphisms, trivial covering
- the exponential map \(\mathbf{R} \to S^1\)
- power map \(S^1 \to S^1\) given by \( z \mapsto z^n\)
- quotient map \(S^n \to \mathbf{RP}^n \)
- the orbit space \(G/H\) of a topological group \(G\) modulo a discrete subgroup \(H\)
First properties (Proposition 4.3)
Proposition 4.4. Uniqueness of lifting of maps.
General problem about the existence of liftings.
Théorème 4.5. Path can be lifted.
Proposition 4.6. Homotopies can be lifted.
Consequences:
Corollaire 4.7. The homomorphism \(p_* : \pi_1(E, \tilde x) \to \pi_1(X, x_0) \) is injective.
Corollaire 4.8. The fiber of a covering map is discrete, and all fibers have the same cardinality.
Lesson 8 - Thursday, 20 October 2016
Existence of liftings for arbitrary maps.
Théorème 4.10. Let \(p : E \to X\) be a covering map, and \(f : Y \to X\) a continuos map such that \(f(y_0) = x_0\). Assume \(Y\) path connected and locally path connected. Given any point \(\tilde x_0 \in E\) such that \( p(\tilde x_0) = x_0\), there exists a (unique) lifting \(\tilde f\) of \(f\) such that \(\tilde f(y_0) = \tilde x_0\) if and only if\( f_* \pi_1(Y, y_0) \subseteq p_* \pi_1(E, \tilde x_0) \)Group actions and coverings.
Definition of totally discountinuos, proper and free action.
Proposition 4.12. If the action of a discrete group \(G\) on \(X\) is totally discontinous, then the quotient map \(q : X \to X/G\) is a covering map.
Proposition 4.15. If the action of a discrete group \(G\) on \(X\) is proper and free, and \(X\) is Hausdorff and locally compact, then the quotient map \(q : X \to X/G\) is a covering map.
The group \(A(p)\) of automorphisms of a covering map \(p : E \to X\).
Proposition 4.16. \(A(p)\) acts on \(E\) in a free and totally discountinuos way.
Few remarks about quotients \(E/G\), where \(G\) is a subgroup of \(A(p)\).
Covering maps and fundamental groups.
Théorème 4.17. Let \(p : E \to X\) be a covering map, \(x_0 \in X\). The set of all subgroups of the form \( p_* \pi_1(E, \tilde x_0) \) as \(\tilde x_0\) varies in the fiber \(p^{-1}(x_0)\) is a conjugacy class of subgroups of \( \pi_1(X, x_0) \).
Monodromy action of \( \pi_1(X, x_0) \) on the fiber \( p^{-1}(x_0) \).
Proposition 4.20. The monodromy action is transitive, and the stabilizers are the images \( p_* \pi_1(E, \tilde x_0) \). As a consequence, there is an equivariant bijection\( p^{-1}(x_0) \cong \pi_1(X, x_0) / p_* \pi_1(E, \tilde x_0) \)Lesson 9 - Monday, 24 October 2016
Théorème 4.21 Relations beween the action of \(A(p)\) on the total space \(E\) and the action of \(\pi_1(X, x_0)\) on the fiber \(p^{-1}(x_0)\).
Corollaire 4.22 Consequence: a simply connected space has only trivial coverings.
Théorème 4.27 Definition of Galois covering. Some equivalent conditions.
Théorème 4.29 A Galois cover is given by a group action: if \(p : E \to X\) is Galois, then \(X \cong E/A(p)\).
Proposition 4.30 If \(p : E \to X\) is Galois then the action of the fundamental group of \(X\) on the fiber induces a group isomorphism \(\pi_1 (X,x) / p_* \pi_1 (E, \tilde x) \cong Aut(E, p, X) \)
Lesson 10 - Thursday, 27 October 2016
Homology. Algebraic preliminaries:
- exact sequences of \(R\)-modules, examples of short exact sequences
- definition of chain complex \((C_*, d)\)
- definition of cycles, boundaries, homology groups of a complex
- definition of morphism of complexes, induced maps in homology
Definition of chain homotopy between morphisms of complexes.
Proposition 5.6. If there is a chain homotopy between \(f\) and \(g\), then they induce the same homomorphisms in homology.
Short exact sequences of complexes (def. 5.7) and long exact sequence in homology.
Definition of the connecting homomorphism. Proof that is well defined.
- exact sequences of \(R\)-modules, examples of short exact sequences
Lesson 11 - Thursday, 3 November 2016
Théorème 5.8. A short exact sequence of complexes induces a long exact sequence in homology.
Snake Lemma, Corollaire 5.9, see link below for a proof
Five Lemma, Lemme des cinq, Corollaire 5.10
To gain a better understanding of the concepts of complex and homology, do some of the exercices at the end of chapter 5: at least 5.1, 5.2, 5.4, 5.5. To go deeper do 5.6, 5.7, 5.8. (solutions to all the exercises are in the book. Try to solve the exercises before reading them!)
Definition of convex hull, simplex in \(\mathbf{R}^n\), standard simplex, finite simplicial complex in \(\mathbf{R}^n\) (from 5.12 to 5.20)
Simplicial chain complex: oriented simplices, boundary map, simplicial homology.
Example: simplicial homology of a circle and a sphere.
Lesson 12 - Monday, 7 November 2016
Singular homology.
Definition of singular simplex and face of a simplex.
The sinular chain complex: definition of the boundary map \(d\).
Proposition 5.26. \(d^2 = 0\).
Some computations: homology of a point, \(H_0(X)\) for \(X\) path connected (Proposition 5.28), direct sum decomposition with respect to path components. (Proposition 5.29).
Lemme 5.31. A continuos function \(f: X \to Y\) induces a homomorphism \(S(f): S_*(X) \to S_*(Y)\) between the singular chain complexes and hence maps in homology
\(H_n(f): H_n(X) \to H_n(Y), \qquad \forall n \ge 0\)
Théorème 5.32. If \(f, g :X \to Y\) are homotopic, then they induce the same maps in homology, i.e., \(H_n(f) = H_n(g)\) for all \(n\ge 0\).
Important consequence. Singular homology is a homotopy invariant, i.e., if \(X\) and \(Y\) are homotopically equivalent, then \(H_n(X) \cong H_n(Y)\) for all \(n\ge 0\). Even more, if \(f: X \to Y\) is a homotopy equivalence, then all the induced homomorphisms \(H_n(f)\) are isomorphisms.
Lesson 13 - Thursday, 10 November 2016
Théorème 5.32: proof.
Reduced homology (Definition 5.30). The reduced complex \(\tilde S_*(X)\) and its homology \(\tilde H_*(X)\).Relative homology of a pair \( (X, A)\). Relative cycles (= chains whose boundary is supported on \(A\)) and relative boundaries (= boundaries up to chains supported on \(A\)).
The exact sequence of complexes:
\(0 \to S_*(A) \to S_*(X) \to S_*(X, A) \to 0\)
and the long exact homology sequence. The connecting homomorphism \(\delta : H_q(X, A) \to H_{q-1}(A)\) is induced by the boundary map of chains in \(X\).
Comparisons between reduced and relative homology. Relative homology of the pair \(X, \{x_0\} )\). The homomorphism (at the chain level)\( q_X : \tilde S_*(X) \to S_*(X) \to S_*(X, \{x_0\})\)
Proposition 6.5. \(q_X\) induces an isomorphism in homology. Hence
\(\tilde H_*(X) \cong H_*(X, \{x_0\})\)
Théorème 6.6. (\(\mathcal{U}\)-small chains theorem). The inclusion of complexes \(i : S^{\mathcal{U}}_* \to S_*(X)\) induces an isomorphism in homology.
Discussion of the statement and (very vague) idea of the proof. Full details are in Hatcher's book (Proposition 2.21) and they are NOT required for the exam.
Lesson 14 - Monday, 14 November 2016
Corollaire 6.7. Relative version of Théorème 6.6
Consequences of théorème 6.6:
Corollaire 6.8. (Mayer-Vietoris exact sequence)
Example: homology of a bouquet of two spaces
Corollaire 6.9 (Excision theorem, version 1)
Corollaire 6.10 (Excision theorem, version 2)
Example: excision for cellular spaces. In this case, there is a close relation between the relative homology of the pair \((X, A)\) and the homology of the quotioent space \(X/A\):
Proposition 6.11. Let \(X = A \cup_f e^n \dots\) and \(p: X \to X/A\) the quotient map.
Then \(p\) induces an isomorphism:\( H_*(p) : H_*(X, A) \to H*(X/A, \{A\}) \cong \tilde H_*(X/A) \)
Computation in simplicial homology: the simplicial homology of \(\Delta^n\) and of \(\partial \Delta^n\) (example on pages 132-133)Lesson 15 - Thursday, 17 November 2016
Comparison between simplicial and singular homology.
Théorème 6.13. For a (finite) simplicial complex \(K\), there is a map \(\phi_K : C_*(K) \to S_*(|K|) \) (where \(|K|\) is the topological space underlying the simplicial complex \(K\)) which induces an isomorphism in homology.
Application: the Euler-Poincaré characteristic (from Definition 6.30 to Théorème 6.33).
Some computation in singular homology:
Théorème 6.15. (Reduced) homology of spheres.
Consequences:- \(S^n\) homeomorphic to \(S^m \implies n = m\)
- \(S^n\) is not a retract of the closed ball \(D^{n+1}\)
- (Brouwer fixed point theorem) Any continuos map \(f : D^n \to D^n\) has (at least) a fixed point. (Théorème 6.43)
- local homology a a topological manifold (Proposition 6.26)
- (Invariance of dimension) Let \(U \subseteq \mathbf{R}^n\) and \(V \subseteq \mathbf{R}^m\) be open sets. If \(U\) is homeomorphic to \(V\) then \(n = m\). (Théorème 6.27)
Homology of cellular complexes:
Proposition 6.28. Let \(Y = X \cup_f e^n \), where \(f : S^{n-1} \to X\) is the attachment map. Then
- \( \tilde H_q(X) \cong \tilde H_q(Y)\), per \(q \ne n, n-1\)
- there is an exact sequence\( 0 \to \tilde H_n(X) \to \tilde H_n(Y) \to \tilde H_{n-1}(S^{n-1}) \to \tilde H_{n-1}(X) \to \tilde H_{n-1}(Y) \to 0\)
The file below is the original article by Eilenberg and Steenrod with the axioms for a homology theory and the proof of its uniqueness for simplicial complexes. We have proved most of the axioms for singular homology (i.e., singular homology is a homology theory).
Exercise: read the article.
Lesson 16 - Monday, 21 November 2016
Various properties of cellular complexes: let \(X = \bigcup_{n=0}^N X_n\), where \(X_n\) is the \(n\)-skeleton, i.e., the subspace
made of all cells of dimension less than or equal to \(n\). Then Proposition 6.28 and the long exact sequence of the pair \((X_n, X_{n-1})\) give:- \(H_q(X_n) = 0\), for \(q > n\)
- \(H_q(X) \cong H_q(X_{n+1})\), for \(q \le n\)
- an exact sequence
\(0 \to H_n(X_n) \to H_n(X_n, X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0 \)
The cellular chain complex: Cell\(_n(X) = H_n(X_n, X_{n-1})\), with boundary maps \(d_n\) obtained form the previuos excat sequence.
to "see" the proof, look at the diagram in Hatcher, p. 139.
Théorème 6.29. The homology of the cellular chain complex is isomorphic to the singular homology of \(X\).Examples:
- homology of spheres
- homology of topological surfaces, in the oriented and non oriented case (more details later)
- homology of complex projectives spaces
- homology of real projective spaces (more details later)
Explicit generators for \(\tilde H_0(S^0)\) and in general for \(\tilde H_n(S^n)\).Definition of degree of a map between sphers of the same dimension
Various geometrical consequences:
Proposition 6.20. An orthogonal reflection with respect to a hyperplane in \(\mathbf{R}^{n+1}\) induces a map of degree \(-1\) on the sphere \(S^n\).
Proposition 6.21. The antipodal map on a sphere of dimension \(n\) has degree \((-1)^{n+1}\).Proposition 6.22. A continuos map \(f : S^n \to S^n\) with no fixed points is homotopic the the antipodal map.
Proposition 6.23. If \(f : S^{2n} \to S^{2n}\) has no fixed points, then there exists \(x_0\in S^{2n}\) such that \(f(x_0) = - x_0\).
Corollaire 6.24. A sphere of dimension \(2n\) has no never vanishing tangent vector field.
Théorème 6.25. A sphere has a never vanishing tangent vector field if and only if has odd dimension.
Lesson 17 - Thursday, 24 November 2016
Computation of the boundary maps in the cellular chain complex.
Cellular boundary formula (Hatcher, p. 140) The maps \(d_n\) are given in terms of the degree of certain maps between spheres.
Computation of the degree of a map \(f: S^n \to S^n\): definition of the local degree \(\deg f|_{x_i}\).
Degree formula (Hatcher, Proposition 2.30, p. 136) If there is \(y \in S^n\) such that \(f^{-1}(y) = \{x_1, \dots, x_m\}\) then\(\deg f = \sum_{i=1}^m \deg f|_{x_i} \)
Example: computation of the homology of real projective spaces and of topological surfaces with the determination of all maps in the cellular chain complex.Lesson 18 - Monday, 28 November 2016
Homology and covering maps. The transfer map (Proposition 6.34)
Galois coverings. Homology and invariants for the action (in characteristics zero)
Proposition 6.36. If \(p : E \to B\) is a Galois cover, i.e. \(B = E/G\), for a finite group \(G\), the transfer map induces an isomorphism from \(H_n(B, \mathbf{Q})\) to \(H_n(E, \mathbf{Q})^{G}\), the invariant subspace for the action of \(G\) on the homology of \(E\).
Transfer in positive characteristics. The exact sequence for double covers.
Proposition 6.38. Computation of homology of real vector spaces with \(\mathbf{Z}_2\) coefficients.
Application: Borsuk-Ulam theorem
Théorème 6.39 (Borsuk-Ulam) If \(m > n \ge 1\) there are no continuous odd maps \(f: S^m \to S^n\), i.e., such that \(f(-x) = - f(x)\) per ogni \(x \in S^m\).
Some corollaries:
Corollaire 6.40. If \(n \ge 1\) every odd continuous function \(f: S^n \to \mathbf{R}^n\) has at least one zero.
Corollaire 6.41. If \(n \ge 1\), for every odd continuous function \(f: S^n \to \mathbf{R}^n\) there exists a point \(x_0\in S^n\) such that \(f(x_0) = f(-x_0)\).Théorème 6.42 (Lusternik-Schnirelmann) If \(S^n\) is covered by \(n + 1\) closed subset \(A_1, \dots A_{n+1}\), then at least one of these contains a pair of antipodal points.
Lesson 19 - Thursday, 1 December 2016
Tensor product of modules over a commutative ring: definition via universal property e construction.
Tensor product of maps. The tensor product is a covariant functor from the category of \(R\)-modules to itself.
First properties:- abelian groups can be tensorized as \(\mathbf{Z}\)-modules
- \(A \otimes_\mathbf{Z} R\) gets an induced structure of \(R\)module
- the tensor product commutes with direct sums
Exactness properties: the tensor product is right exact, i.e., if
\( A \overset{f}{\to} B \overset{g}{\to} C \to 0 \)
is exact, then
\( A\otimes M \overset{f\otimes \text{id}}{\to} B\otimes M \overset{g\otimes \text{id}}{\to} C\otimes M \to 0 \)
is exact.
Split exact sequences and subgroups af free abelian groups (Proposition 5.35 (1), (2), (3))
Free resolutions of \(R\)-modules.
Definition of \(\text{Tor}_p(A, B)\) via a free resolution of \(A\).
First properties:
- \(\text{Tor}_0(A, B) = A \otimes B\)
- for an abelian group \(A\), \(\text{Tor}_p(A, B) = 0\) for all \(p\ge 2\)
Lesson 20 - Monday, 5 December 2016
Lifting of maps \(f : A \to A'\) to maps of free resolutions \( F : L_* \to L'_*\).
Proposition 5.37.- Any map can be lifted
- Any two liftings of the same map are chain-homotopic.
Corollaire 5.38. The definition of \(\text{Tor}(A, B)\) does not depend on the free resolution of \(A\) used.
Théorème 5.39 (Universal coefficients theorem) For any complex \(C_*\) of free abelian group and any commutative ring \(R\) there is a split exact sequence\( 0 \to H_n(C_*) \otimes R \to H_n(C_*\otimes R) \to \text{Tor}(H_{n-1}(C_*), R) \to 0 \)
This shows that (singular) homology with integer coefficients determines the homology with coefficients in an arbitrary ring.Lesson 21 - Monday, 12 December 2016
Properties of \(\text{Tor}(A, B)\) for abelian groups.
Proposition 5.41. Exact sequence.
Proposition 5.42 (2). Commutativity: \(\text{Tor}(A, B) \cong \text{Tor}(B, A) \)
Proposition 5.35 (4). A finitely generated torsion-free abelian group is free.
Proposition 5.42 (3). If \(B\) is a torsion-free abelian group then \(\text{Tor}(A, B) = 0\) for every abelian group \(A\)
Basic computation: \( \mathbf{Z}_p \otimes \mathbf{Z}_q \cong \text{Tor}(\mathbf{Z}_p, \mathbf{Z}_q) \cong \mathbf{Z}_m\), where \(m = \text{GCD}(p, q)\)
These properties and the Universal Coefficients Theorem allow to compute \(H_*(X, R)\) from \(H_*(X, \mathbf{Z})\) for most interesting commutative rings, e.g., for \(R = \mathbf{Q}, \mathbf{R}\) or any finite commutative ring.
Fundamental groups and homology.
Théorème 7.1 (Hurewicz). If \(X\) is path-connected, then \(H_1(X, \mathbf{Z})\) is the abelianizaton of the fundamental group \(\pi_1(X, x_0)\).Lesson 22 - Thursday, 15 December 2016
Definition of singular cohomology of a topological space, dualizing the complex \(S_*(X, \mathbf{Z})\) of singular chains.
Cochain complex, coboundary map, cocycles and coboundaries. Induced map in cohomology from a continuos map \(f : X \to Y\).
Example: \(H^0(X, R) \cong \prod_{i \in I} R\), where \(I\) = set of path-components of \(X\).
Example: if \(X\) is a differentiable manifold, differential forms give singular cochains and the coboundary map coincides with exterior differentiation (Stokes theorem).
Universal coefficients theorem for cohomology. For any complex \(C_*\) of free abelian group and any commutative ring \(R\) there is a split exact sequence\( 0 \to \text{Ext}(H_{n-1}(C_*, R) \to H^n(C_*, R) \to \text{Hom}(H_n(C_*), R) \to 0 \)
This shows that (singular) homology with integer coefficients determines the (additive structure of) cohomology with coefficients in an arbitrary ring.
Exactness properties of the (contravariant) functor \(\text{Hom}(-, R)\): if\(0 \to A \overset{\alpha}{\to} B \overset{\beta}{\to} C \to 0 \)
is exact, then
\( \text{Hom}(A, R) \overset{\alpha^*}{\leftarrow} \text{Hom}(B, R) \overset{\beta^*}{\leftarrow} \text{Hom}(C, R) \leftarrow 0 \)
is exact.
Definition of \(\text{Ext}(A, R)\) for an abelian group \(A\) via free resolutions: for an abelian group \(A\) and any free resolution \(0 \to L_1 \overset{d_1}{\to} L_0 \to A \to 0\):
- \(\text{Ext}^0(A, R) = \text{Hom}(A, R)\)
- \(\text{Ext}^1(A, R) = \text{Hom}(L_1, R) / d_1^*(\text{Hom}(L_0, R))\)
- \(\text{Ext}^p(A, R) = 0\) for \(p \ge 2\)
Basic properties of \(\text{Ext}\):
- \(\text{Ext}(A_1\oplus A_2, R) \cong \text{Ext}(A_1, R) \oplus \text{Ext}(A_2, R)\) (\(\text{Ext}\) commutes with direct sums)
- \(\text{Ext}(A, R) = 0\) for any free abelian group \(A\) and any ring \(R\)
- \(\text{Ext}(\mathbf{Z}_n, R) = R/nR\)
Consequence: suppose that the singular homology of \(X\) is finitely generated and write \(H_n(X, \mathbf{Z}) = F_n \oplus T_n\), where \(T_n\) is the torsion subgroup and \(F_n \cong H_n/T_n\) is a finitely generated free group. Then
\( H^n(X, \mathbf{Z}) \cong (H_n/T_n) \oplus T_{n-1} \cong F_n \oplus T_{n-1} \)
Lesson 23 - Monday, 19 December 2016
The results discussed in this lecture have not been proved in class. A complete treatment can be found in Hatcher's book. The proofs are NOT required for the exam.
Definition of cup product for singular cochains.
The coboundary map is a derivation with respect to the cup product: \(\delta (\phi \smallsmile \psi) = \delta\phi \smallsmile \psi + (-1)^k \phi \smallsmile \delta\psi\), where \(\phi \in S^k(X, R)\).
cocycle \(\smallsmile\) cocycle = cocycle
cocycle \(\smallsmile\) coboundary = coboundary
and hence the cup product descends to cohomology.
The cohomology ring \(H^*(X, R) = \bigoplus_{k \ge 0} H^k(X, R)\).
Theorem. The cup product is (anti)-commutative at the cohomology level, i.e., for \(\phi \in H^(k(X, R)\) and \(\psi \in H^(l(X, R)\)\(\phi \smallsmile \psi = (-1)^{kl} \psi \smallsmile \phi\)Theorem (Künneth formula). Under good hypotheses, the cup product induces an isomorphism\( H^*(X, R) \otimes H^*(Y, R) \cong H^*(X \times Y, R)\)and in particular\( H^n(X \times Y, R) \cong \bigoplus_{k=0}^n \left[H^k(X, R) \otimes H^{n-k}(Y, R)\right]\)
For example, the Künneth formula holds for \(R\) a field and \(X, Y\) compact manifolds. See Hatcher for more details.
Easy consequence: for \(X, Y\) compact manifolds, the Euler characteristics is multiplicative, i.e., \(\chi(X \times Y) = \chi(X) \cdot \chi(Y)\).
The cap product: \(H_n(X, R) \times H^k(X, R) \to H_{n-k}(X, R)\)
Definition: a compact manifold without boundary of dimension \(n\) is orientable if \(H_n(X, \mathbf{Z}) \cong \mathbf{Z}\).
Poincaré duality: let \(M\) be a compact, orientable manifold without boundary of dimension \(n\) and let \([M]\) be the fundamental class, i.e., a generator of \(H_n(X, \mathbf{Z})\).
Then the map \(H^k(X, \mathbf{Q}) \to H_{n-k}(X, \mathbf{Q})\) given by \(\alpha \mapsto [M] \smallfrown \alpha\) is an isomorphism.
Consequences:
Using the universal coefficients theorem and the fact that \(\mathbf{Q}\) is torsion free, we have \(H^k(X, \mathbf{Q}) \cong \left( H_k(X, \mathbf{Q})\right)^*\) and in particular they are isomorphic. Then Poincaré duality gives an isomorphism\(H^k(X, \mathbf{Q}) \cong H^{n-k}(X, \mathbf{Q})\)and a similar statement for homology with \(\mathbf{Q}\) coefficients.
In particular: the Euler characteristics of a compact orientable manifold without boundary of odd dimension is zero.Lesson 24 - Thursday, 12 January 2017
Seminars
1. Les pseudo-précurseurs Partie I Chapitre 1 - sections 1, 2
Eulero-Legendre Partie I Chapitre 2 - sections 2.1, 2.2, 2.3
2. Cauchy - Lhuilier - von Staudt Partie I Chapitre 2 - sections 2.5, 2.6, 2.7
3. Gauss Partie I Chapitre 2 - section 3
4. Riemann Partie II Chapitre 2 - section 1
5. Moebius Partie II Chapitre 3 - section 1