by UNSWelearning 52,808 views
This is the Introductory lecture to a beginner's course in Algebraic Topology given by N J Wildberger of the School of Mathematics and Statistics at UNSW in 2010.
This first lecture introduces some of the topics of the course and three problems.
His YouTube site "Insights into Mathematics" at https://www.youtube.com/user/njwildberger?ob=0 under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 16,441 views
This is the first lecture of this beginner's course in Algebraic Topology (after the Introduction). In it we introduce the two basic one-dimensional objects: the line and circle. The latter has quite a few different manifestations: as a usual Euclidean circle, as the projective line of one-dimensional subspaces of a two-dimensional space, as a polygon, or as a space of orbits of a translation group on the line.
This course is given by Assoc Prof N J WIldberger of the School of Mathematics and Statistics at UNSW. See also his series on Rational Trigonometry (WildTrig), Foundations of Mathematics (MathFoundations), Universal Hyperbolic Geometry, as well as Linear Algebra (WildLinAlg) under YouTube user: njwildberger at https://www.youtube.com/user/njwildberger?ob=0
by UNSWelearning 16,993 views
This is the first video of the 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, following Lemmermeyer and as explained by S. Shirali.
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 lecture is part of a beginner's course in Algebraic Topology given by N J Wildberger at UNSW.
His YouTube site Insights into Mathematics at https://www.youtube.com/user/njwildberger?ob=0 under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry, Universal Hyperbolic Geometry, and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 9,914 views
After the plane, the two-dimensional 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 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. His YouTube site Insights into Mathematics at user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 7,603 views
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 fourth lecture of this beginner's course in Algebraic Topology given by N J Wildberger of UNSW.
His YouTube site Insights into Mathematics at user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 6,218 views
We introduce some surfaces: the cylinder, the torus or doughnut, and the n-holed 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 is the 5th lecture of this beginners course in Algebraic Topology given by Assoc Prof N J Wildberger of UNSW.
His YouTube site Insights into Mathematics at user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 5,756 views
A surface is non-orientable 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.
His YouTube site Insights into Mathematics at https://www.youtube.com/user/njwildberger?ob=0 under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
by UNSWelearning 5,206 views
The Klein bottle and the projective plane are the basic non-orientable 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.
by UNSWelearning 5,835 views
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 the 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.
by UNSWelearning 5,555 views
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 120-cell, the 600-cell and the 24-cell. 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. His other YouTube series are found at YouTube channel: njwildberger.
by UNSWelearning 5,118 views
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.
by UNSWelearning 3,461 views
This video introduces an important re-scaling of curvature, using the natural geometric unit rather than radians or degrees. We call this the turn-angle, 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.
by UNSWelearning 3,665 views
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.
by UNSWelearning 3,339 views
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.
by UNSWelearning 3,744 views
In this video we give the Borsuk Ulam theorem: a continuous map from the sphere to the plane takes equal values for some pair of antipodal points. This is then used to prove the Ham Sandwich theorem (you can slice a sandwich with three parts (bread, ham, bread) with a straight planar cut s so that each slice is cut cut in two. Also we give an application to the ontinuum: the plane is different (not homeomorphic) 3 dimensional space.
This is part of a beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of UNSW.
by UNSWelearning 2,568 views
We use our new normalization of angle called turn-angle, 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.
by UNSWelearning 2,249 views
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. This important modification to the theory is original with this lecture series!
This is the 16th lecture of this beginner's course in Algebraic Topology, given by N J Wildberger at UNSW.
by UNSWelearning 3,473 views
This lecture introduces the central theorem in Algebraic Topology: the classification of two dimensional combinatorial surfaces. We use cut and paste operations to reduce any combinatorial surface into a standard form, and also introduce an algebraic expression to encode this standard form.
by UNSWelearning 2,764 views
In this lecture we present the traditional proof of the most important theorem in Algebraic Topology: the classification of (two-dimensional) 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 either a sphere, or a number of crosscaps or 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. More of his YouTube videos may be found at his channel: njwildberger.
by UNSWelearning 2,809 views
We give a description of a variant to the proof of the Classification theorem for two dimensional combinatorial surfaces, due to John Conway and called the ZIP proof. Our approach to this is somewhat algebraic. We think about spheres with holes that are then zipped together rather than polygonal pieces which are glued together.
This is the 19th lecture in this beginner's course on Algebraic Topology, given by Assoc. Prof. N J Wildberger at UNSW.
by UNSWelearning 4,091 views
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 non-standard (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 circles---the Beltrami Poincare disk model.
This is the 20th lecture in this beginner's course on Algebraic Topology, given by N J Wildberger at UNSW.
by UNSWelearning 2,152 views
We describe how the two-holed torus and the 3-crosscaps surface can be given hyperbolic geometric structure. For the two-holed 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 3-crosscaps 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. See his YouTube channel Insights into Mathematics, at user: njwildberger.
by UNSWelearning 2,644 views
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.
by UNSWelearning 5,956 views
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.
by UNSWelearning 4,528 views
This video continues our discussion of the fundamental group of a space. We show that the homotopy classes of closed loops from a fixed point on a space actually form a group. And the important cases of the torus and the projective plane are studied in some detail.
This is the 25th lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J WIldberger at UNSW.
by njwildberger 11,480 views
This is a review lecture on some aspects of abstract algebra useful for algebraic topology. It provides some background on fields, rings and vector spaces for those of you who have not studied these objects before, and perhaps gives an overview for those of you who have.
Our treatment is descriptive and informal; we sketch the main ideas and give some key examples, starting with the basic number systems of elementary arithmetic: the natural numbers, integers and rational numbers.
Probably the easiest object is that of a field, with the rational numbers as the main example, although we also give an introduction to complex rational numbers, to finite fields (a non-standard approach) and to the rational numbers adjoined with an (algebraic!) square root of two.
Examples of rings include the integers, polynomials and square matrices of a certain size. Examples of vectors spaces include row vectors, also polynomials up to a certain degree, and matrices of a fixed shape only with addition and scalar multiplication.
The lecture ends with three prominent constructions familiar in almost all branches of algebra: the idea of subobjects, homomorphisms, and quotient objects.
In our next lecture we will have a closer look at groups, both commutative and non-commutative, which are perhaps the most important algebraic objects in algebraic topology.
Thanks to Nguyen Le for filming.
by njwildberger 9,545 views
This lecture gives a brief overview or introduction to group theory, concentrating on commutative groups (future lectures will talk about the non-commutative case). We generally use additive notation + for the operation in a commutative group, and 0 for the (additive) inverse. The main starting points are a set G, or in the infinite case, a type of mathematical object G, together with a binary operation * (or +) that takes two elements a,b from G and yields another element a*b (or a+b) in G. This operation must be commutative, that is a*b=b*a, and associative, that is (a*b)*c=a*(b*c), have an identity e (or 0) satisfying e*a=a*e=a and finally have the property that for any element a in G there is an element b satisfying a*b=b*a=e.
We give main examples: the cyclic groups Z_n, the integers Z under addition, the Klein 4-group, the group Q of rational numbers under addition. To explain why the Klein 4-group is associative, we introduce the idea of thinking of the elements as transformations (in this case of the four vertices of a square), with the multiplication then being composition (do one transformation, then the other). This interpretation automatically insures associativity. We use this occasion to also introduce some notation for expressing permutations.
We introduce the idea of a subgroup of a commutative group, and direct sums of commutative groups to obtain bigger groups. Finally we state the Fundamental theorem of Finite Commutative groups: every such group is isomorphic to a direct sum of cyclic groups of the form Z_n.
by njwildberger 4,602 views
We present more information on commutative groups and the fundamental structure theorem that every such group is isomorphic to a direct sum of cyclic groups Z_n. We discuss the notions of isomorphism, homomorphism, cosets of a subgroup, and the quotient of a group by a subgroup.
by njwildberger 3,661 views
Free abelian groups play an important role in algebraic topology. These are groups modelled on the additive group of integers Z, and their theory is analogous to the theory of vector spaces. We state the Fundamental Theorem of Finitely Generated Commutative Groups, which says that any such group is a direct sum of a finite number of Z (integers) together with a finite commutative group (the torsions part).
Non-commutative groups are also briefly introduced, mostly through the simplest example of S_3, the symmetric group on three objects, or alternatively the group of 3x3 permutation matrices. We illustrate that cosets of a subgroup in the commutative case are of two different types: left and right, and this makes the situation more complicated. When the left and right cosets agree we are in the situation of a normal group, and then the cosets do form a quotient group.
Thanks to Nguyen Le for filming.
by njwildberger 3,658 views
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 2-sphere 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.
by njwildberger 2,249 views
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.
by njwildberger 3,192 views
We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed 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.
by njwildberger 2,925 views
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 two-fold 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.
by njwildberger 4,876 views
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.
by njwildberger 2,930 views
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, 2-cells and so on, organized into free abelian groups called chain groups, again one for each dimension.
by njwildberger 3,057 views
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 simplix 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.
by njwildberger 3,354 views
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, 2-dim 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 in-depth 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 0-th 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 non-trivial loops in the space (roughly). The second homology group H_2 measures the number of independent non-trivial 2-dim holes in the space, and so on.
by njwildberger 2,520 views
In our last lecture, we introduced homology explicitly in the very simple cases of the circle and disk. In this lecture we tackle the 2-sphere. First we compute the homology using the model of a tetrahedron: four 2-dimensional faces, but no 3-dim 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 semi-simplicial complexes, or delta-complexes, 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.
by njwildberger 3,944 views
We continue our investigation of homology by computing the homology groups of a torus. For this we use the framework of delta-complexes, 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!