The course will most closely follow parts of the following notes and book by Hatcher:

- A good, leisurely set of notes on the basics of topological spaces by Hatcher.
- An excellent book, "Algebraic Topology" by Hatcher. This is available as a physical book, published by Cambridge University Press, but is also available (legally!) for download for noncommercial personal use from the link given. Chapters 0,1,2 will be especially useful for this unit.

- Elementary Topology. Textbook in Problems by Viro, Ivanov, Kharlamov, Netsvetaev. Material on topological spaces and algebraic topology with lots of nice exercises.

- W.A.Sutherland,
*Introduction to metric and topological spaces*, Clarendon Press, Oxford. (For basic material on topological spaces.) - Munkres,
*Topology*, Pearson Education.

- Mon 30 Sep. Introduction. Review of topological spaces (p.1-5). Basis for a topology. (p.7-8).
- Tue 1 Oct. Neighbourhoods (p.9). Closures, interiors, boundaries (p.5-7). Continuity (p12-13). Homeomorphisms (p.12-13) including example to show a continuous bijection is not necessarily a homeomorphism.
- Wed 2 Oct. Compactness (p.30-31). Hausdorff spaces (p.34). Subspaces (p.10-12)
- Mon 7 Oct. More on compact and Hausdorff spaces (34-36). Definition of connectedness and path-connectedness (p.18-19).
- Tue 8 Oct. More on (path-)connectedness (p.20-25). Disjoint unions. Product topology (p.13-16).
- Wed 9 Oct. Quotient spaces (p.44-52).
From now on, page references are to Hatcher's Algebraic Topology book.

- Mon 14 Oct. More examples of quotient spaces. Odds and ends on continuity. Basic definitions of homotopy (p.3-4).
- Tue 15 Oct. Examples of homotopy. Homotopy equivalence and
contractibility (p.1-4). Deformation retracts (p.1-2). Cell complexes
(only finite dimensional ones) (p.5-8).
**[Didn't do cell complexes; that will be on Wednesday.]** - Wed 16 Oct.
**Cell complexes. (p.5-8)**Homotopy extension property (p.14-17): Examples, reformulation for closed subspaces, sketch of "CW pair has HEP" (p.14-16). - Mon 21 Oct. Application of HEP to homotopy equivalence. (Prop. 0.17). Homotopy of paths (p.25-26)
- Tue 22 Oct. Definition and basic properties of fundamental group (p.26-28).
- Wed 23 Oct. More on fundamental groups: homomomorphisms induced by continuous maps; homotopy equivalent path connected spaces have isomorphic fundamental groups (p.34-37).
- Mon 28 Oct. Fundamental group of circle (p. 29-30), and derivation from Homotopy Lifting Property.Applications of fundamental group of circle (Brouwer Fixed Point Theorem in 2 dimensions, Fundamental Theorem of Algebra)(p. 31-32).
- Tue 29 Oct. Proof of Homotopy Lifting Property for circle (p.30). Introduction to van Kampen Theorem.
- Wed 30 Oct. Define free products. Universal property ("Key Theorem"). Free groups. (p. 41-42).
- Mon 4 Nov. Description of free products in terms of reduced words. First part of van Kampen's Theorem (surjectivity) (p. 43-44).
- Tue 5 Nov. Statement of second part of van Kampen (identifying the kernel) (p.45-46) Calculations of some fundamental groups.
- Wed 6 Nov. More calculations of fundamental groups. Outline of proof of second part of van Kampen.
- Mon 11 Nov. Definition and examples of covering spaces (p. 56-60). Covering map induces injective map on fundamental groups; Homotopy Lifting Property (p.60-61), and corollaries for lifting paths and homotopies between them. Number of sheets of a covering of a path connected space is well-defined.
- Tue 12 Nov. Number of sheets is index of subgroup of fundamental group associated to the covering. Lifting Criterion and Unique Lifting Property. (p.61-63).
- Wed 13 Nov. Isomorphisms of covering spaces. Uniqueness of path-connected covering space associated to a given subgroup of the fundamental group.
- Mon 18 Nov. Universal covers. Deck transformations and action of fundamental group on universal cover (p.70-72, but I only discussed the universal cover).
- Tue 19 Nov. Orbit spaces and construction of covering spaces (p.72-73). Examples, including the torus and Klein bottle as orbit spaces of the plane.
- Wed 20 Nov. Algebra of chain complexes (definition, chain maps, homology of a chain complex, chain homotopy, chain homotopic maps induce the same map on homology) (p.106, 110-111). Free abelian groups and bases.
- Mon 25 Nov. Simplices. Definition of singular homology (p.108-109). Calculation of (unreduced and reduced) homology of a point (p.110).
- Tue 26 Nov. Map on homology induced by a continuous map (homology is a functor) (p.100-111). Statement of homotopy invariance, and idea of proof. The consequence that homotopy equivalent spaces have the same homology (p.111). Definition of relative homology.
- Wed 28 Nov. Definition of exact sequences (p.113). Long exact sequence of homology associated to a short exact sequence of chain complexes, and application to relative homology (p.115-117). Reduced homology is homology relative to a point.

- Homework 1: Set Wednesday 16 Oct, due Wednesday 30 Oct.
- Homework 2: Set Wednesday 30 Oct, due Wednesday 13 Nov.
- Homework 3: Set Wednesday 13 Nov, due Wednesday 4 Dec.
- Homework 4: Set Wednesday 27 Nov, due Wednesday 18 Dec.
- Homework 5: Set Wednesday 11 Dec, due Wednesday 15 Jan.

Please hand in work by 12 noon on the due date, to the secure blue box in G90 in the Fry building.

I'll attempt to return marked work within a week, but certainly within eight

- Homework 0 (optional)
- Homework 1
- Homework 2
- Homework 3
- Homework 4
- Homework 5