Algebric topology

This webpage contains exercises for the algebraic topology course and a brief plan of each lecture.

Lecture 1. Summary of general topology: topological spaces, metric spaces, continuous maps, homeomorphisms, what means compact, what means connected, connected components, $\pi_0$ as the set of connected components, examples of spaces distinguished by $\pi_0$.

Lecture 2. Hex game and non-simply-connectedness of $\mathbb{R}^2\setminus{0}$. What means simply connected? Convex sets are simply connected. Paths in a space, homotopy of paths, associativity and invertibility (paths up to homotopy form a groupoid).

Lecture 3. Categories. Category of topological spaces. Homotopy equivalence of maps. Homotopy category. What means contractible. Deformation retract. Convex sets are contractible. Definition of the fundamental group. Idea of proof of $\pi_1(S^1,*)=\mathbb{Z}$ via covering space $\mathbb{R}$ and path lifting property.

Lecture 4. Various questions (connected vs. path connected, contractible vs. deformation retracts to a point. Details of the proof of $\pi_1(S^1,*)=\mathbb{Z}$. Constructions of topological spaces and their universal properties: product, quotient, disjoint union, pushout.

Exercise set 1 (due 19.03.2019)

Lecture 5. Cone, suspension, balls and spheres. Definition of a CW complex. Invariance of the fundamental group under change of basepoint, functoriality, invariance under deformation retractions and homotopy equivalences.

Lecture 6. Details on invariance of $\pi_1$ under homotopy equivalences. Construction of factorizations by fragmentation of paths and proof of surjectivity in Seifert-van Kampen theorem. Vanishing of $\pi_1$ for spheres of dimension at least $2$. Formulation of Seifert-van Kampen using moves. Fragmentation of homotopies. Invariance of factorizations with respect to refinements and choices of paths connecting to base points.

Exercise set 2 (due 26.03.2019)

Lecture 7. Complete the proof of Seifert-van Kampen. Main ideas: fragmentation of paths and homotopies; factorization depends on the choice of partition of interval, the choice of open subset for each part, the choice of paths from the basepoint; factorization modulo moves does not depend on these choices. Free product (coproduct) of groups, amalgamated product of groups, their universal properties, abstract formulation of Seifert-van Kampen using these operations on groups. Examples of computation of $\pi_1$: wedge of circles and oriented surface of genus $g$.

Lecture 8. Surfaces of genus $g$. Cases $g=1$ (commutative), $g=2$ (not commutative). Fundamental group of a product. Picture hanging exercise. Technical results on CW complexes: $X^n$ is closed, normality, compact sets intersect only finitely many cells.

Exercise set 3 (due 02.04.2019)

Lecture 9. Warmup: connected components of a CW complex. Homotopy extension property for CW complexes. Proof of the fact that $X/A$ is homotopy equivalent to $X$ for a contractible subcomplex $A$. Fundamental group of a graph is free. Description of the fundamental group of an arbitrary CW complex by generators and relations. Setup of the proof.

Lecture 10. Continuation: fundamental group of a CW complex, complete the proof. Every group is a fundamental group. Covering spaces: definition, picture, homotopy lifting property.

Exercise set 4 (due 09.04.2019)

Lecture 11. Recollection on group actions and their classification (orbit decomposition, transitive actions correspond to subgroups). Covering spaces: action by deck transformations arising from a covering, construction of the universal covering space of $X$ as the space of homotopy classes of paths $[\gamma]$ with appropriate topology pulled from $X$. Universal covering space corresponds to the action of the fundamental group on itself by left multiplication.

Lecture 12. Quotients by properly discontinuous group actions. Construction of the covering space corresponding to a given fundamental group action. Functoriality of the bijection between group actions and covering spaces. Fundamental group of $\tilde X$ when $\tilde X\to X$ is a covering. Characterization of the universal covering space as a simply connected covering space. Applications: fundamental group of a quotient by properly discontinuous group action. Subgroup of a free group is free. Overview of the theory of higher homotopy groups: statement of Whitehead’s theorem and construction of an element of $\pi_3(S^2)$.

Exercise set 5 (due 30.04.2019)

Lecture 13. Homology: motivation through low dimensional examples and Stokes theorem. Simplicial complexes: Definition and examples (2-sphere, torus, Klein bottle, $\mathbb{RP}^2$). The boundary maps $\partial_n$. Proof of $\partial^2=0$. Homology. Computation of the homology of $S^2$ using decomposition into two triangles.

Exercise set 6 (due 07.05.2019)

Lecture 14. Computation of the simplicial homology of the torus and $\mathbb{RP}^2$. Simplicial homology with rational coefficients, Betti numbers and Euler characteristic. Definition of singular homology. $H_0$. Singular homology of a point. Functoriality and homotopy invariance (proof next time).

Lecture 15. Triangulations of prisms and construction of maps $h : C_{i}(X)\to C_{i+1}(Y)$ coming from a homotopy between maps $f,g:X\to Y$, proof of the equation $h\partial+\partial h = g_*-f_*$, proof of invariance of homology under homotopy equivalences. Idea of the Mayer-Vietoris sequence.

Exercise set 7 (due 14.05.2019)

Lecture 16. Introduction to homological algebra: complexes, maps between complexes, homology, homotopies, long exact homology sequence. Mayer-Vietoris sequence for a sphere.

Lecture 17. Barycentric subdivision. Pictures, examples. Proof that simplices of the barycentric subdivision become arbitrarily small. Definition of the subdivision operator as a sum over permutations. Construction of an explicit homotopy between the identity and the subdivision operator. Conclusion (Mayer-Vietoris theorem).

Exercise set 8 (due 21.05.2019)

Lecture 18. Idea that for CW complexes the homology can be computed inductively similar to the way we did it for spheres. Reduced homology. Reduced homology of spheres. Long exact homology sequence for a good pair $X, Z$.

Lecture 19. Homology of a wedge of spheres. Proof that singular homology is isomorphic to simplicial homology: induction for finite dimensional, passage from finite to infinite by a compactness argument. Examples of homology computations with SAGE using explicit triangulations of the torus, $RP^2$.

Exercise set 9 (due 28.05.2019)

Lecture 20. Euler characteristic for long exact sequences. Basic principles of computations with long exact sequences (if one is zero, the other two are isomorphic; if two are isomorphisms, the third one is an isomorphism). Relative homology: homotopy invariance, excision, long exact sequence. Homology relative to a point is the reduced homology. For good pairs relative homology is the reduced homology of the quotient. For all pairs it is the reduced homology of the mapping cone. Local homology at a point and how it recognizes dimensions of manifolds. Example with non-trivial local homology.

Lecture 21. Degree of a map $S^n \to S^n$. Construction of the cellular homology complex and proof of the isomorphism between cellular and simplicial homology.

Exercise set 10 (due 04.06.2019)

Lecture 22. Dualization functor $Hom(-,R)$, correspondence between free abelian groups and groups of functions. Definition of cohomology (singular, simplicial, etc.) Main ideas about how we can deduce statements for cohomology from the corresponding statements for homology. Computation of homology and cohomology of real projective spaces.

Lecture 23. An example how $T^3=S^1\times S^1 \times S^1$ can be obtained from surgery along Borromean rings. Universal coefficient theorem for cohomology.

Lecture 24. The problem of defining intersection of homology classes. Construction of cup and cap products. Poincare duality and the main idea of proof by defining cohomology with compact supports and building up manifold $X$ out of balls.

Written final exercise set 11 (due 24.06.2019, 12:00) Please, submit by email or in person on 24.06.2019.