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
Another nice book you can find online, with plenty of problems in:
And a couple of other recommended books (not online):
W.A.Sutherland, Introduction to metric and topological
spaces, Clarendon Press, Oxford. (For basic material on
Munkres, Topology, Pearson Education.
Notes on continuity
short document summarizes facts about continuous maps that are
useful for giving proofs of continuity.
Summary of singular
This short document
summarizes without proofs the basic definitions and theorems on
Plans for lectures
I'll put here what I intend to cover in individual lectures. I won't
always judge the timing perfectly, and I'll add annotations
in bold if plans change.
Until further notice, page number references are to Hatcher's notes on
topological spaces. I won't always cover precisely what's in the
notes, and will try to give examples and alternative points of view
not in the notes.
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
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
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.
Mon 4 Nov. Description of free products in terms of reduced
words. First part of van Kampen's Theorem (surjectivity)
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
Wed 13 Nov. Isomorphisms of covering spaces. Uniqueness of
path-connected covering space associated to a given subgroup of the
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
Mon 25 Nov. Simplices. Definition of singular homology
(p.108-109). Calculation of (unreduced and reduced) homology of a point
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.
There is no exam for this unit, but there will be five assignments
that each count for 16 percent of the final mark (plus one optional
assignment early on, which won't count towards the final mark). You'll
have at least two weeks for each assignment.
The timetable for
the compulsory homework assignments is as follows.
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.
I'm allowing more time for the last three assignments because you'll
also be preparing presentations (see below).
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 working days.