## Online lectures

Lectures are slide-shows consisting of videos (screencasts) mostly alternating with quizzes. You can make the videos full-screen. It is recommended that you watch with "quality" at least 720p to avoid fuzzy writing (click on the setting at the bottom of the YouTube video to specify this).

- Lecture 1:
**Introduction.**Here we give an overview of what we study in topology, and formulate what we mean by topological problems. We then clarify what it means to

*translate*a topological problem to an algebraic one, and sketch the approach taken in algebraic topology. - Lecture 2:
**Paths and Homotopy.**At the core of Algebraic topology is a relation between paths, and more generally continuous functions, called

*homotopy*. We introduce paths and*multiplication*and*inverse*operations on paths. A key construction is an equivalence relation on paths,*homotopy fixing endpoints*. Paths up to this equivalence relation have nice algebraic structures, and the collection of paths up to equivalence has a more manageable size. This is based on nice properties of the algebraic operations up to equivalence, which we prove. - Lecture 3:
**Groupoids.**This lecture is a digression introducing and giving examples of an algebraic structure called a

*groupoid*. Groupoids generalizes groups, and are more natural in various contexts. Indeed a further generalization, $\infty$-groupoids (which we do not introduce) are algebraic structures that in some sense are as rich as topological spaces. - Lecture 4:
**Fundamental Groups; Covering spaces; Path lifting.**Building on the algebraic operations and their nice properties up to homotopy fixing base points, we define the fundamental group, $\pi_1(X, x_0)$ of a (based) topological space. The fundamental group is the central notion of the core of this course. One of the two main ways of computing fundamental groups is

*covering spaces*, which we introduce. After sketching the key geometric ideas, we prove the first result relating fundamental groups to covering spaces, so called*path lifting*. Our proof is chosen to generalize easily and illustrate relevant hypothesis. - Lecture 5:
**Homotopy lifting; Fundamental group of the circle.**We prove a key property relating fundamental groups to covering spaces, that homotopies (fixing end points) lift to homotopies in covers. Using this, we show that $\pi_1(S^1, 1) = \Z$. The proof of the latter also involves using symmetries of the covering map, which we will later formalize and understand as deck transformations.

- Lecture 6:
**Functoriality; Based spaces.**We now establish functorial properties of the fundamental group. The group $\pi_1(X, x_0)$ is not a functor on the category of topological spaces, but on

*based spaces*. We define based spaces and see this. On the other hand, the morphisms induced by a based map $f: (X, x_0) \to (Y, y_0)$ depend only on the homotopy class of $f$ as a map between based spaces, i.e., we have a functor on the category where morphisms are homotopy classes of maps. Finally, as we really want to study topological spaces, not based topological spaces, we see how the fundamental group depends on the choice of basepoint. - Lecture 7:
**Applications.**We see two applications of the fundamental group, specifically of $\pi_1(S^1, 1) =\Z$ together with functoriality. The first is the $2$-dimensional

*Brouwer fixed point theorem*, which says that any map $f$ from the $2$-disc to itself has a fixed point. The second is that $\R^2$ is not homeomorphic to $\R^3$. This involves showing that the fundamental group of $S^2$ is trivial, which involves ideas relevant to understanding the fundamental group of unions. - Lecture 8:
**More covering maps.**We look at more covering maps for the circle - disconnected covers and $n$-fold covers. We will eventually see that the $n$-fold covers, together with the standard cover $p: \R \to S^1$ give all connected covers of the circle, and there is a nice descriptions of all covers of all reasonable spaces. We also prove the first result in this general theory - that covering maps induce injections on the fundamental group.

- Lecture 9:
**Graphs.**We introduce graphs (which may contain loops and multiple edges) as combinatorial objects, and associate to them topological spaces called

*geometric realizations*. Further, we define morphisms of graphs, so that geometric realization becomes a functor. Finally, there is a simple combinatorial characterization of when graph morphisms are covering maps. Graphs are very useful for constructing examples of covering spaces, and also for applications of topology to group theory. - Lecture 10:
**Free Groups.**We introduce free groups. These will give us interesting examples. Further as all groups are quotients of free groups, we can define so called

*presentations*of groups. These allow us to describe fundamental groups of most nice enough spaces. - Lecture 11:
**Fundamental groups, Trees and Free groups.**We construct spaces with fundamental groups free groups on a given set of generators. The proof that the fundamental group is free involves studying a specific class of graphs,

*Trees*, and proving that these are contractible. The proof we give for contractibility of trees is a special case of methods of Whitehead, which are applicable in much greater generality, and our goal is to illustrate these methods. Given contractibility of the tree, the rest of the proof involves ideas similar to those used for the fundamental group of the circle, as well as an interplay between group theory and topology. As a consequence we obtain a topological proof of an algebraic result concerning the free group. - Lecture 12:
**Map lifting.**The

*map lifting*theorem says that the lifting problem for a covering map has a solution if and only if the corresponding lifting problem on groups obtained by passing to fundamental groups has a solution, provided the spaces involved are reasonable. We prove this and give an application. Map lifting will also be used in the classification of covering spaces. - Lecture 13:
**Isomorphisms of based covers.**We begin the classification of covering maps for reasonable spaces, concretely connected, locally path-connected and

*semi-locally simply connected*spaces. The first step is to prove that a based covering map is determined by the subgroup of the fundamental group, which we do in this lecture. We also describe the fibre of the based cover in terms of the fundamental group of the base and its subgroup corresponding to the cover. - Lecture 14:
**Isomorphisms of covers and Deck transformations.**From the classification of based covers we obtain a classification of covers by understanding the dependence on base points. Further, we get a description of symmetries of covers, which are called

*deck transformations*. - Lecture 15:
**Spaces without Universal covers.**Here we see examples of spaces that do not have covers corresponding to certain subgroups of the fundamental groups. This lets us identify te condition of

*semi-local simple connectivity*needed fot the existence of covers. - Lecture 16:
**Galois theory for covering spaces.**We complete the classification of covering maps and their symmetries, so called

*Galois theory for coverings*. Uniqueness was shown earlier. The goal of this lecture is to construct covers and show that they have the required properties. - Lecture 17:
**Free products and the Seifert-Van Kampen theorem.**The Seifert-Van Kampen theorem gives the fundamental group of a union of spaces in terms of their fundamental groups, provided various intersections are path-connected. The fundamental group of the union $X$ is the quotient of the so called

*free product*. We introduce and construct free products, and construct and prove the surjectivity of a homomorphism from the free product onto $\pi_1(X)$. Finally, we discuss the categorical product and co-product constructions, as free products are co-products. - Lecture 18:
**The Seifert-Van Kampen theorem.**We now state and prove the Seifert-Van Kampen theorem, giving the fundamental groups of unions of spaces.

- Lecture 19:
**Graphs and the Fenchel-Nielsen Theorem.**As an application of the Seifert-Van Kampen theorem, we show that the fundamental group of any graph is free. As a consequence, we obtain a group theoretic result called the Fenchel-Nielsen theorem, which states that any subgroup of a free group is isomorphic to a free group.

- Lecture 20:
**Two-complexes and Surfaces.**Most familiar spaces can be obtained inductively by a process called

*attaching cells*, which we formalize. We show the effect of attaching cells on the fundamental group. As a consequence, we show that every group is the fundamental group of a topological space. We also describe the fundamental group of surfaces. - Lecture 21:
**Topological proof of Grushko's theorem (due to Stallings).**Using topological methods, Stallings gave a beautiful proof of the following algebraic result: Given a free group $F$, groups $G_1$ and $G_2$ and a surjection $\varphi: F \to G_1 * G_2$, there is decomposition $F = F_1 * F_2$ such that $\varphi(F_i) = G_i$. We sketch a proof of this result. The videos in this lecture were made several years ago.

- Lecture 22:
**CW complexes.**A

*CW complex*is a space that is inductively obtained by attaching cells. Most familiar spaces can be described as CW complexes. As discovered by Whitehead (who defined CW complexes), these are very well suited for studying homotopy theory, and specifically for showing that in some context algebraic topology determines the homotopy type. - Lecture 23:
**Higher homotopy groups.**Higher homotopy groups $\pi_n(X, x_0)$ are higher-dimensional analogues of the fundamental groups, given by maps from cubes relative to their boundaries (or equivalently by maps from spheres with a basepoint). We define these and show that they form a group.

We prove two properties of higher-homotopy groups that are easy to show but important (and perhaps surprising). These are that higher homotopy groups are abelian, and that covers induce isomorphisms on higher homotopy groups.

- Lecture 24:
**Higher homotopy groups, Extension problems and Contractibility.**We prove the first results that show how homotopy groups control homotopy type. Specifically, the vanishing of homotopy groups is equivalent to all extension problems of maps from spheres to maps from discs they bound have solutions. As a consequence a CW-complex is contractible if and only if its homotopy groups coincide with those of a point.

- Lecture 25:
**Eilenberg-MacLane spaces and Whitehead's theorem.**Eilenberg-MacLane spaces are CW complexes whose higher homotopy groups vanish. Theorems of Whitehead say that the homotopy type of such a space is determined by its fundamental groups, and maps to such spaces are determined up to homotopy by the induced maps on $\pi_1$. We sketch proofs of these results.

- Lecture 26:
**On Homotopy groups and Homology groups.**This lecture largely consists of general remarks, first explaining (without proofs) why higher homotopy groups are complicated, and how homology is related to these, and then sketching how simplicial homology is defined.

- Lecture 27:
**Simplicial complexes and simplicial homology.**We define

*simplicial complexes*, which are spaces obtained by gluing vertices, edges, triangles, tetrahedra and their higher dimensional analogues in a rigid fashion. These have a purely combinatorial description, and have associated to them topological spaces.We then define

*simplicial homology*for simplicial complexes. - Lecture 28:
**Examples and computations for Simplicial homology.**We conclude the course with some examples of computing simplicial homology, including by using so called $\Delta$-complexes which are more flexible than simplicial complexes but on which simplicial homology can be defined.