The emphasis of the course is on the interconnections of modern topology with other branches of mathematics and on concrete topological spaces (manifolds, vector bundles) rather than the most general abstract categories.
Prerequisites
Topology I: topological spaces, continuity, connectedness, compactness,homeomorphisms; Linear Algebra; Abstract Algebra I: abelian groups, subgroups, and quotient groups.
Curriculum:
- Homology Theory. Chain complexes, cycles, boundaries, and homology groups. Polyhedra, triangulations, simplicial homology. Betti numbers and Euler characteristic. Homology of classical surfaces: sphere, torus, Klein bottle, projective plane. Cell complexes (CW-complexes). Cell chains, incidence coefficients. Cell homology coincides with simplicial (statement without proof). Examples: spheres, tori, projective spaces. Exact sequences: long exact sequence of a pair, relative homology. Mayer–Vietoris exact sequence. Computations. Topological constructions: product, quotient, cone, wedge, suspension, join, loop space – their homology.
- Homotopy and Degree. Classification of maps \(S^{1}\to S^{1}\) by degree. Index of isolated singular point of a plane vector field. Poincaré index theorem: sum of indices equals Euler characteristic. Applications: hairy ball theorem (brushing a sphere), Brouwer fixed point theorem.
- Singular Homology. Definition, homotopy invariance. Exact sequences of pairs and triples. Homology of spheres. Coincidence of singular, cellular, and simplicial homology.
- Intersections and Lefschetz. Intersection of submanifolds and cycles. Homological interpretation of the index of a vector field. Lefschetz fixed point theorem and applications.
- Cohomology. Cohomology groups, Poincaré duality for closed oriented manifolds. De Rham cohomology. Cup product.
- Seifert Surfaces and Knot Theory. Compact oriented surfaces in \(S^{3}\) with boundary a knot \(K\). Seifert’s algorithm (crossings → discs + twisted bands). Genus \(g(K)\) as a knot invariant; formula \(g = \frac{c – s + 1}{2}\). Homology \(H_{1}(F)\cong \mathbb{Z}^{2g}\), Euler characteristic \(\chi(F) = 1 – 2g\). Seifert form and Alexander polynomial. Relation to Milnor fibration: for an isolated singularity \(x^{p} + y^{q} = 0\), the Milnor fibre is a Seifert surface with \(\chi = 1 – \mu\), where \(\mu = (p-1)(q-1)\) is the Milnor number.
- Zariski Topology. Closed sets = zero sets of polynomials (over \(\mathbb{A}^{n}, \mathbb{P}^{n}\)). Properties: non-Hausdorff, quasicompact, irreducible components. Comparison with Euclidean topology: strictly coarser. \(H^{1}_{\mathrm{Zar}}(X, \mathbb{Z}) = 0\) for a curve of positive genus.
- From Zariski to Grothendieck Topologies (Motivation). Problem: Zariski cohomology of constant sheaf misses genus. Idea: étale covers – abstract “coverings” not necessarily subsets. Leads to \(\ell\)-adic cohomology, which satisfies Poincaré duality and recovers Betti numbers. Deligne’s proof of Weil conjectures.
- Geometric Applications (Connecting with Previous Material). Index of vector field via divisors (Lefschetz connection). Lefschetz fixed point formula for algebraic curves. For smooth complex varieties, topological Euler characteristic (Euclidean) equals \(\chi\) of complexification. Example: \(\mathbb{P}^{1}\) has \(\chi = 2\).
Grading & Assessment
Your final grade is based on a 10‑point scale, with homework contributing up to 4 points, the midterm exam up to 3 points, and the final exam up to 3–4 points. As for the approximate weekly workload, you can expect around 5–6 homework exercises per week, or fewer depending on difficulty.
Example Seminar Problem Set
Problem 1. Trefoil Knot
For the trefoil knot \(T(2,3)\):
- Draw its standard diagram.
- Construct a Seifert surface using Seifert’s algorithm.
- Compute its genus and Euler characteristic.
- Show that the same knot appears as \(x^{2} + y^{3} = 0\) in \(\C^{2} \cap S_{\epsilon}^{3}\).
Problem 2. Torus Knots and Milnor Number
Let \(f(x,y) = x^{p} + y^{q}\) with \(p,q\) coprime.
- Find the link \(f^{-1}(0) \cap S_{\epsilon}^{3}\).
- Compute the Milnor number \(\mu = (p-1)(q-1)\).
- Compute the Euler characteristic of the Milnor fibre using \(\chi = 1 – \mu\).
- Relate this to the genus of the torus knot \(T(p,q)\).
Problem 3. Zariski Topology and Homology
Consider the curve \(y^{2} = x^{3} – x\) in \(\C^{2}\).
- Show it is irreducible in Zariski topology.
- Compare its Zariski-open subsets with Euclidean-open subsets.
- Why does singular homology in Zariski topology not see the genus?
Textbooks
- S.Matveev, Lectures on Algebraic Topology, AMS, 1999.