Algebraic topology : videos 1 35 + Review
The Algebraic Topology series was recorded from a UNSW course in the subject meant for 3rd year students. It adopts a simple minded beginner's approach to the subject, with (of course!) a minimum of "infinite this" and "infinite that". Isn't it surprising how much mathematics can be developed without all that infinite stuff?
This is the full introductory lecture of a beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three wellknown but perhaps challenging problems for you to try.
The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties. The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia, well known for his discovery of Rational Trigonometry, explained in the series WildTrig, the development of Universal Hyperbolic Geometry, explained in the series UnivHypGeom, and for his other YouTube series WildLinAlg and MathFoundations. He also has done a fair amount of research in harmonic analysis and representation theory of Lie groups. 

This is the full first lecture of this beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. Here we begin to introduce basic one dimensional objects, namely the line and the circle. However each can appear in rather a remarkable variety of different ways.


This is the full second lecture in this beginner's course on Algebraic Topology. We give the basic definition of homeomorphism between two topological spaces, and explain why the line and circle are not homeomorphic.
Then we introduce the group structure on a circle, or in fact a general conic, in a novel way. This gives a gentle intro to the definition of a group. It also uses Pascal's theorem in an interesting way, so we give some background on projective geometry. This course is given by N J Wildberger at UNSW, also the discoverer of Rational Trigonometry, and a leading advocate of logically correct thinking in mathematics. 

After the plane, the twodimensional sphere is the most important surface, and in this lecture we give a number of ways in which it appears. As a Euclidean sphere, we relate it to stereographic projection and the inversive plane.
This is the full third lecture in this beginner's course on Algebraic Topology. The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. 

This is the fourth lecture of this beginner's course in Algebraic Topology given by N J Wildberger of UNSW. This lecture continues our discussion of the sphere, relating inversive geometry on the plane to the more fundamental inversive geometry of the sphere, introducing the Riemann sphere model of the complex plane with a point at infinity.
Then we discuss the sphere as the projective line over the (rational!) complex numbers. 

This is the 5th lecture of this beginners course in Algebraic Topology. We introduce some other surfaces: the cylinder, the torus or doughnut, and the nholed torus. We define the genus of a surface in terms of maximal number of disjoint curves that do not disconnect it. We discuss how the plane covers the cylinder and the torus, and the associated group of translations.
This arose in the work of Riemann in complex function theory. This course is given by N J Wildberger of UNSW, who is also the discoverer of Rational Trigonometry (see his WildTrig series of videos). 

A surface is nonorientable if there is no consistent notion of right handed versus left handed on it. The simplest example is the Mobius band, a twisted strip with one side, and one edge. An important deformation gives what we call a crosscap.
This is the sixth lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. 

The Klein bottle and the projective plane are the basic nonorientable surfaces. The Klein bottle, obtained by gluing together two Mobius bands, is similar in some ways to the torus, and is something of a curiosity. The projective plane, obtained by gluing a disk to a Mobius band, is one of the most fundamental of all mathematical objects. Of all the surfaces, it most closely resembles the sphere.
This is the seventh lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. 

We investigate the five Platonic solids: tetrahedron, cube, octohedron, icosahedron and dodecahedron. Euler's formula relates the number of vertices, edges and faces. We give a proof using a triangulation argument and flow down a sphere.
This is the eighth lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. 

We use Euler's formula to show that there are at most 5 Platonic, or regular, solids. We discuss other types of polyhedra, including deltahedra (made of equilateral triangles) and Schafli's generalizations to higher dimensions. In particular in 4 dimensions there is the 120cell, the 600cell and the 24cell. Finally we state a version of Euler's formula valid for planar graphs.
This is the ninth lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW. 

We discuss applications of Euler's formula to various planar situations, in particular to planar graphs, including complete and complete bipartite graphs, the Five neighbours theorem, the Six colouring theorem, and to Pick's formula, which lets us compute the area of an integral polygonal figure by counting lattice points inside and on the boundary.
This is the tenth lecture of this beginner's course in Algebraic Topology by N J Wildberger of UNSW. 

This video introduces an important rescaling of curvature, using the natural geometric unit rather than radians or degrees. We call this the turnangle, or tangle, and use it to describe polygons, convex and otherwise. We also introduce winding numbers and the turning number of a planar curve.
This is the 11th lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J WIldberger at UNSW 

We define the dual of a polygon in the plane with respect to a fixed origin and unit circle. This duality is related to the notion of the dual of a cone. Then we give a purely rational formulation of the Fundamental Theorem of Algebra, and a proof which keeps track of the winding number of the image of concentric circles about the origin. This is an argument every undergraduate math student ought to know!
This is the 12th lecture in this beginner's course in Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW. 

We define the degree of a function from the circle to the circle, and use that to show that there is no retraction from the disk to the circle, the Brouwer fixed point theorem, and a Lemma of Borsuk.
This is the 13th lecture of this beginner's course in Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW. 

We discuss the BorsukUlam theorem concerning a continuous map from the sphere to the plane, and the Ham Sandwich theorem. One application is to show that the two dimensional and three dimensional affine spaces are not homeomorphic.
This is the 14th lecture of this beginner's course in Algebraic Topology, given by Assoc Prof N J WIldberger at UNSW. 

We introduce a new, rational definition of the curvature of a polytope. This removes the usual pi's that occur in such formulas, giving a more direct connection to the Euler number: total curvature equals Euler number.
We use our new normalization of angle called turnangle, or "tangle" to define the curvature of a polygon P at a vertex A. This number is obtained by studying the opposite cone at the vertex A, whose faces are perpendicular to the edges of P meeting at A. A classical theorem of Harriot on spherical triangles is important. This the 15th lecture in this beginner's course on Algebraic Topology given by Assoc Prof N J Wildberger at UNSW. 

We show that the total curvature of a polyhedron is equal to its Euler number. This only works with the rational formulation of curvature, using an analog of the turn angle suitable for the 2 dimensional sphere. We treat Harriot's theorem on the area of a spherical polygon.
This is the first video in the 16th lecture of this beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. 

The central theorem in algebraic topology is the classification of connected compact combinatorial surfaces. In this lecture we introduce this result and indicate the strategy behind the traditional proof.
This is the 17th lecture in this beginner's series on Algebraic Topology given by N Wildberger at UNSW. 

In this lecture we present the traditional proof of the classification of (twodimensional) surfaces using a reduction to a normal or standard form. The main idea is to carefully cut and paste the polygons forming the surface in a particular way, creating crosscaps and handles.
This is the 18th lecture of this beginner's course in Algebraic Topology, given by N J Wildberger of the School of Mathematics and Statistics at UNSW. 

In this lecture we sketch an algebraic version of Conway's ZIP proof of the classification of twodimensional surfaces. One key idea is to replace the basic polygons in the standard proof with spheres with holes in them. We introduce an algebraic notation that allows us to manipulate (ZIP) edges together between holes on spheres.
This is the 19th lecture of this beginner's course in Algebraic Topology, given by N J Wildberger of the School of Mathematics and Statistics at UNSW. 

This lecture relates the two dimensional surfaces we have just classified with the three classical geometries Euclidean, spherical and hyperbolic. Our approach to these geometries is nonstandard (the usual formulations are in fact deeply flawed) and we concentrate on isometries, avoiding distance and angle formulations. In particular we introduce hyperbolic geometry via inversions in circlesthe Beltrami Poincare disk model.
This is the 20th lecture in this beginner's course on Algebraic Topology, given by N J Wildberger at UNSW. 

We describe how the twoholed torus and the 3crosscaps surface can be given hyperbolic geometric structure. For the twoholed torus we cut it into 4 hexagons and then describe a tesselation of the hyperbolic plane (using the Beltrami Poincare model described in the previous lecture) composed of regular hexagons meeting four at a vertex. We will look at an octagon model involving the standard form. Then we briefly look at the 3crosscaps surface in the same way.
This is the 21st lecture in this beginner's course on Algebraic Topology, given by N J Wildberger of UNSW. 

This lecture is an introduction to knot theory. We discuss the origins of the subject, show a few simple knots, talk about the Reidemeister moves, and then some basic invariants, namely minimal crossing number, linking number (for links) and then the AlexanderConway polynomial.
This is part of a beginner's course on Algebraic Topology, given by N J Wildberger at UNSW. 

In the 1930's H. Siefert showed that any knot can be viewed as the boundary of an orientable surface with boundary, and gave a relatively simple procedure for explicitly constructing such `Seifert surfaces'. We show the algorithm, exhibit it for the trefoil and the square knot, and then discuss Euler numbers for surfaces with boundaries.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger of the School of Mathematics and Statistics, UNSW. 

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described. This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW. 

A continuation on the fundamental group of a surface, we prove that the multiplication of equivalence classes or types of loops from a base point does indeed form a group in the algebraic sense. We discuss the fundamental group of the torus and the projective plane.
This is part of a beginner's course on Algebraic Topology, given by N J Wildberger of UNSW. 

We introduce covering spaces of a space B, an idea that is naturally linked to the notion of fundamental group. The lecture starts by associating to a map between spaces, a homomorphism of fundamental groups. Then we look at the basic example of a covering space: the line covering a circle. The 2sphere covers the projective plane, and then we study helical coverings of a circle by a circle.
These can be visualized by winding a curve around a torus, giving us the notion of a torus knot. We look at some examples, and obtain the trefoil knot from a (2,3) winding around the torus. 

We illustrate the idea of a covering space by looking at the rich examples coming from a wedge of two circles. Coverings of this space are graphs with each vertex of degree four, with edges suitably labelled in a directed way with alpha's and beta's.
We also introduce the idea of a universal covering space, which is by definition simply connected, or equivalently its fundamental group is trivial, and illustrate in the case of the wedge of circles. 

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the twoholed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B. 

We begin by giving some examples of the main theorem from the last lecture: that the associated homomorphism of fundamental groups associated to a covering space p:X to B injects pi(X) as a subgroup of pi(B). We look at helical coverings of a circle, and also a twofold covering of the wedge of two circles.
So a main idea is that covering spaces of a space B are associated to subgroups of pi(B). The covering space associated to the identity subgroup is called the universal covering space of B; it has the distinguishing property that it is simply connected: any loop on it is homotopic to the constant loop. To construct the universal cover of a space B, we proceed in an indirect fashion, considering paths in B from a fixed base point b, up to homotopy. Any such path can be mapped to its endpoint: this is the covering map. The universal covering space of a sphere or projective plane is the sphere, that of the torus or Klein bottle is the Euclidean plane, while all surfaces of negative Euler characteristic, like a two holed torus, has universal cover consisting of the Hyperbolic plane. To describe this completely would be a long story, we give just an initial orientation to this important connection between geometry and topology. Finally we discuss how other covering spaces may be created from a universal covering space. 

We briefly describe the higher homotopy groups which extend the fundamental group to higher dimensions, trying to capture what it means for a space to have higher dimensional holes. Homology is a commutative theory which also deals with this issue, assigning to a space X a series of homology groups H_n(X), for n=0,1,2,3,....
In this introduction to the subject we look at a particular graph, discuss cycles and how to compute them, and introduce the first homology group, admittedly in a rather special restrictive way. We then generalize the discussion to a general graph, using the notion of a spanning tree to characterize independent cycles in terms of edges not in such a spanning tree. 

Here we carry on our introduction to homology, focussing on a particularly simple space, basically a graph and various modifications to it. We discuss cycles, boundaries, and homology as a quotient of cycles mod boundaries, one such group for each dimension.
The framework is commutative group theory, working with formal combinations of vertices, edges, 2cells and so on, organized into free abelian groups called chain groups, again one for each dimension. 

Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and convex hulls, and showing a natural hierarchy from point to line segment to triangle to tetrahedron. Each of these also has a standard representation as the convex hull of the unit basis vectors in a vector space.
Simplicial complexes are arrangements of simplices where any two are either disjoint or meet at a face, where by face we mean the convex hull of any subset of the vertices of a simplex. Each simplex has two possible orientations, corresponding to a particular ordering of the vertices. We show how an orientation of a tetrahedron induces orientations on each of its four triangular faces. The boundary of a simplex is then an alternating combination of its oriented faces of dimension one less. A key theorem is that if we apply the boundary to a boundary, we always get zero. This is the basis of the definition of homology. 

The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each stage, or dimension.
To make this more understandable, we give in this lecture an indepth look at some examples. Here we start with the simplest ones: the circle and the disk. For each space it is necessary to look at each dimension separately. The 0th homology group H_0(X) measures the connectivity of the space X, for a connected space it is the infinite cyclic group Z of the integers. The first homology group H_1 measures the number of independent nontrivial loops in the space (roughly). The second homology group H_2 measures the number of independent nontrivial 2dim holes in the space, and so on. 

In our last lecture, we introduced homology explicitly in the very simple cases of the circle and disk. In this lecture we tackle the 2sphere. First we compute the homology using the model of a tetrahedron: four 2dimensional faces, but no 3dim solid. This illustrates how linear algebra naturally arises in this kind of problem.
We then provide a much simpler alternative calculation using the more flexible framework of semisimplicial complexes, or deltacomplexes, where only two triangular faces are needed, and the calculation is much simplified, however still giving the same final result (which by the way is that H_0 (S^2)=Z, H_1 (S^2)=0 and H_2 (S^2)=Z, with all higher homology groups being 0. 

We continue our investigation of homology by computing the homology groups of a torus. For this we use the framework of deltacomplexes, a somewhat general and flexible approach to simplicial complexes that allows us to use just two triangles in the standard square planar representation of a torus with opposite edges identified. While this means that all three corners of each triangle is actually identified with one point, if we go through the algebraic formalities of computing the homology groups, we get the same answer as with a more complicated geometrical simplical subdivision.
So we actually make the computation, discuss some generalities on subgroups of Z plus Z, and then move to the computation of the homology of the projective plane, for which a new phenomenon appears called torsion: a homology group which is actually a finite commutative group, not a free abelian group. In fact H_1 (P) is just Z/2Z or Z_2, the group with two elements. This corresponds to a cycle which is not a boundary, but which has the property that twice that cycle is a boundary! Then we introduce the Betti numbers b_0, b_1, b_2 etc. which are the ranks of the homology groups H_0, H_1, H_2 etc, and mention the celebrated result that the alternating sum of Betti numbers gives us the Euler characteristic of the space. 

Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and convex hulls, and showing a natural hierarchy from point to line segment to triangle to tetrahedron. Each of these also has a standard representation as the convex hull of the unit basis vectors in a vector space.
Simplicial complexes are arrangements of simplices where any two are either disjoint or meet at a face, where by face we mean the convex hull of any subset of the vertices of a simplex. Each simplex has two possible orientations, corresponding to a particular ordering of the vertices. We show how an orientation of a tetrahedron induces orientations on each of its four triangular faces. The boundary of a simplex is then an alternating combination of its oriented faces of dimension one less. A key theorem is that if we apply the boundary to a boundary, we always get zero. This is the basis of the definition of homology. 

The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each stage, or dimension.
To make this more understandable, we give in this lecture an indepth look at some examples. Here we start with the simplest ones: the circle and the disk. For each space it is necessary to look at each dimension separately. The 0th homology group H_0(X) measures the connectivity of the space X, for a connected space it is the infinite cyclic group Z of the integers. The first homology group H_1 measures the number of independent nontrivial loops in the space (roughly). The second homology group H_2 measures the number of independent nontrivial 2dim holes in the space, and so on. 

This video continues to discuss difficulties with angles. It takes a historical approach, and emphasizes that with angles, imprecision is unaaIn our last lecture, we introduced homology explicitly in the very simple cases of the circle and disk. In this lecture we tackle the 2sphere. First we compute the homology using the model of a tetrahedron: four 2dimensional faces, but no 3dim solid. This illustrates how linear algebra naturally arises in this kind of problem.
We then provide a much simpler alternative calculation using the more flexible framework of semisimplicial complexes, or deltacomplexes, where only two triangular faces are needed, and the calculation is much simplified, however still giving the same final result (which by the way is that H_0 (S^2)=Z, H_1 (S^2)=0 and H_2 (S^2)=Z, with all higher homology groups being 0. 

We continue our investigation of homology by computing the homology groups of a torus. For this we use the framework of deltacomplexes, a somewhat general and flexible approach to simplicial complexes that allows us to use just two triangles in the standard square planar representation of a torus with opposite edges identified. While this means that all three corners of each triangle is actually identified with one point, if we go through the algebraic formalities of computing the homology groups, we get the same answer as with a more complicated geometrical simplical subdivision.
So we actually make the computation, discuss some generalities on subgroups of Z plus Z, and then move to the computation of the homology of the projective plane, for which a new phenomenon appears called torsion: a homology group which is actually a finite commutative group, not a free abelian group. In fact H_1 (P) is just Z/2Z or Z_2, the group with two elements. This corresponds to a cycle which is not a boundary, but which has the property that twice that cycle is a boundary! Then we introduce the Betti numbers b_0, b_1, b_2 etc. which are the ranks of the homology groups H_0, H_1, H_2 etc, and mention the celebrated result that the alternating sum of Betti numbers gives us the Euler characteristic of the space. This is the last lecture in this series for a while! 
