Algebraic Topology MTH 477 For Master of Mathematics By

1. Algebraic Topology • MTH 477 • For • Master of Mathematics • By Dr. SOHAIL IQBAL Assistant Professor Department of Mathematics, CIIT Islamabad MTH 477-Algebraic Topology by Dr SohailIqbal

2. Lecture # 3 • Homotopy • Topology, some basics • Homotopy MTH 477-Algebraic Topology by Dr Sohail Iqbal

3. Topology Topology began with the investigations of certain questions in geometry. In topology we are concerned with the properties of the spaces which are preserved under “continuous deformations” including stretching and bending, but not tearing or gluing. Topological spaces appear in almost every branch of mathematics, which makes it one of unifying ideas of mathematics. Topology has many fields but in this course we will be interested only “algebraic topology” where we connect topological spaces with some special algebraic structures, for example, homology and homotopy groups. The term “topology” also refers to a structure that we provide to a set, the structure is carefully chosen so that it takes care of certain properties under transformation. MTH 477-Algebraic Topology by Dr Sohail Iqbal

4. Topology Definition(Topology): Let be a set and be a collection of subsets of which satisfies the following conditions: • The intersection of two members of is in • The arbitrary union of members of is in The collection with the above properties is called a topology for and the set is called as topological space, denoted by or . The members of are called open sets. A subset of topological space is said to be closed if and only if is open. MTH 477-Algebraic Topology by Dr Sohail Iqbal

5. Topology Definition(Topology): Let be a set and be a collection of subsets of which satisfies the following conditions: • The intersection of two members of is in • The arbitrary union of members of is in The collection with the above properties is called a topology for and the set is called as topological space, denoted by or . The members of are called open sets. A subset of topological space is said to be closed if and only if is open. Example MTH 477-Algebraic Topology by Dr Sohail Iqbal

6. Topology Definition(Closure of a set): Let be a topological space and The smallest closed set containing is called the closure of and is denoted by . Thus where is the family of all closed sets containing . MTH 477-Algebraic Topology by Dr Sohail Iqbal

7. Topology Definition(Induced topology): Let be a topological space and let be a subset. Then the topology on induced by the topology of consists of family of sets of the form , where is open in . Hence the open sets of the induced topology on are given by The induced topology is sometimes called relative topology. If has the induced topology then is called a subspace of . MTH 477-Algebraic Topology by Dr Sohail Iqbal

8. Topology Definition(Open cover): Let Then a cover of is a collection of subsets of such that If each is open then the cover is called open cover. If the indexing set is finite then the cover is called an finite cover. MTH 477-Algebraic Topology by Dr Sohail Iqbal

9. Topology Definition(Continuous function): Let and be topological spaces. A mapping is continuous if for every open set of , is open in . MTH 477-Algebraic Topology by Dr Sohail Iqbal

10. Topology Definition(Continuous function): Let and be topological spaces. A mapping is continuous if for every open set of , is open in . MTH 477-Algebraic Topology by Dr Sohail Iqbal

11. Topology Equivalent criteria for continuity The following statements are equivalent • is continuous. • Let and be topological spaces. A mapping is continuous if and only if for every closed set of , is closed in . • For each and each neighborhood in , there exists a neighbourhood in such that • whenever . • , whenever . MTH 477-Algebraic Topology by Dr Sohail Iqbal

12. Topology Definition (Homeomorphism): A homeomorphism (or a topological mapping) between two topological spaces is functions that satisfies the following: • is continuous. • is bijective. • is continuous. MTH 477-Algebraic Topology by Dr Sohail Iqbal

13. Topology To show that two spaces are homeomorphic we need to construct a homeomorphism between them. Example: Any two open intervals on are homeomorphic. Let and be two open intervals. Consider the following function defined by It is a simple exercise to show that the above function is a homeomorphism. MTH 477-Algebraic Topology by Dr Sohail Iqbal

14. Topology Example: Any two triangles in plane are homeomorphic. In fact for two fixed triangles there is a unique “affine transformation” of the plane which maps the first triangle to the second. Roughly speaking an affine transformation is composite of a linear transformation and a translation. We will discuss affine transformation later in the course. MTH 477-Algebraic Topology by Dr Sohail Iqbal

15. Topology Definition (Compact): A subset of is called compact if every open cover has a finite open subcover. MTH 477-Algebraic Topology by Dr Sohail Iqbal

16. Topology Some important results on compactness • A closed subset of a compact space is compact. • If and are topological spaces then and are compact if and only if is compact. • A subset of is compact if and only if it is closed and bounded [Heine-Borel theorem]. Theorem: Continuous image of a compact topological space is compact. From the above result we can say the compactness is a topological invariant. MTH 477-Algebraic Topology by Dr Sohail Iqbal

17. Topology Definition(Connectedness): A topological space is connected if is not union of non-empty disjoint open sets. Example: The interval is connected. Theorem: If and are topological spaces then is connected if and only if and are connected. A continuous image should not tear a topological space into pieces, that is, continuous image of connected space should be connected. Theorem: Continuous image of a connected topological space is connected. So we can say that connectedness is a topological invariant. MTH 477-Algebraic Topology by Dr Sohail Iqbal

18. Topology The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connected components of the space. MTH 477-Algebraic Topology by Dr Sohail Iqbal

19. Topology To show that two spaces , and are homeomorphic we need to find a homeomorphism , such that is continuous, bijective, and is continuous. But in order to show that two spaces are NOT homeomorphicwe need some topological invariant. In fact we show that some carefully chosen topological invariant does not remain same in two spaces. Example: Show that the following two spaces are not homeomorphic. We know that “If is a homeomorphism then is homeomorphic to ”. If we remove from the space then we have four connected components. But removal of no point from gives us more than three components. MTH 477-Algebraic Topology by Dr Sohail Iqbal

20. Topology Example: Let , and be an interval. Define map given by The function is continuous and is bijective as well. But the inverse of is not continuous. In fact and are not homeomorphic. Since we know from the properties of compactness that is not compact but circle is a compact space. We also know that “continuous image of a compact space is compact”, so there does not exist a homeomorphism between and . We can also use connectedness to show that these spaces are not homeomorphic. MTH 477-Algebraic Topology by Dr Sohail Iqbal

21. Topology Example: Using connectedness arguments we can see that is not homeomorphic to for . MTH 477-Algebraic Topology by Dr Sohail Iqbal

22. Topology Topological invariance of dimension Theorem:In general is not homeomorphic to for . The proof of above requires machinery of algebraic topology. MTH 477-Algebraic Topology by Dr Sohail Iqbal

23. Topology Example: The sphere is not homeomorphic to torus. The Jordan curve theorem: Let be a simple closed curve in the sphere, that is is subset of which is homeomorphic to a circle. Then the complement of has exactly two connected components. MTH 477-Algebraic Topology by Dr Sohail Iqbal

24. Homotopy Main idea of the homotopy is to replace a continuous function with another continuous function which is obtained after continuous deformation. Definition(Homotopy): If and are spaces and if , are continuous maps from to then is homotopic to , denoted by , if there is a continuous map with and for all . Such a map is called homotopy, One often writes in one wished to display a homotopy. MTH 477-Algebraic Topology by Dr Sohail Iqbal

25. Homotopy In fact for fixed , Is a path joining and . Moreover consider the functions , defined by for . Think of as the deformation at time . Then the one parameter family of functions is a path of functions joining with , such that and . MTH 477-Algebraic Topology by Dr Sohail Iqbal

26. Homotopy Example: Let be a function defined by . Let be another function defined by . Then and are homotopic, since there exist , defined by It can be seen that , and Also is a continuous map. MTH 477-Algebraic Topology by Dr Sohail Iqbal

27. Homotopy For our next example we need definition of convex set. Definition(Convex set): A subset of is convex if, for each pair of points , the line segment joining and is contained in . In other words, if , then Example: The n-sphere (of radius 1 and center the origin) is a subset of defined as So is a circle in , and is a sphere in . We can notice that circle and sphere are not convex. Example: The n-disk (or n-ball) is a set of points of satisfying Observe that . The set is a convex set for every . MTH 477-Algebraic Topology by Dr Sohail Iqbal

28. Homotopy Let us consider another example of homotopy. Example: Let be a convex set. Let be two continuous functions, here is any space. Then , and are homotopic. In fact there exist a function defined by Observe that and Also is a continuous function. MTH 477-Algebraic Topology by Dr Sohail Iqbal

29. Homotopy Example: Let be two continuous functions. Suppose that that is, and are never antipodal points. Then and are homotopic. Define given by Since , so . Now , since . Similarly In fact any two distinct points , and on define a great circle. The function deforms to along the shortest segment of the great circle obtained from and . MTH 477-Algebraic Topology by Dr Sohail Iqbal

30. Homotopy Theorem: Homotopy is an equivalence relation on the set of all continuous maps . Proof: MTH 477-Algebraic Topology by Dr Sohail Iqbal

31. Homotopy Theorem: Homotopy is an equivalence relation on the set of all continuous maps . Proof: MTH 477-Algebraic Topology by Dr Sohail Iqbal

32. Next time Homotopy Continued MTH 477-Algebraic Topology by Dr Sohail Iqbal

33. End of the lecture MTH 477-Algebraic Topology by Dr Sohail Iqbal