MATH 527, Spring 2026
Course Title: Topology
Instructor: Nigel Higson. My office is 228 McAllister Building.
Meeting Times: Tuesdays and Thursdays, 9:05-10:20, in 106 McAllister Building.
Office Hours: Wednesdays 2:00-4:00 (starting January 21) or by appointment. Contact me in class or send me an email. My PSU ID is ndh2, from which you can figure out my PSU email address.
Prerequisites: The main requirements will be familiarity with the basic concepts of general topology (topological spaces, continuity, campactness, connectedness and so on) as well as familiarity with basic abstract algebra (groups, homomorphisms, quotient groups) and linear algebra. Some experience with manifolds will be helpful, but this is not required.
Content, Goals, Objectives, Expectations: The main topic of the course will be the homology and cohomology of topological spaces: I shall present the basic theory, and give applications to manifolds and fixed point theory. Along the way I shall also discuss covering spaces and the fundamental group. I hope that by the end of the class, participants will have an understanding of, and an appreciation for, some of the ways in which algebra may be used to proved purely topological theorems, among them, for instance, the Jordan curve theorem. I also hope that by the end of the class, those participants who intend to take the topology qualifying exam will be adequately prepared for it. For those who wish to reach these goals, regular attendance of classes is very strongly encouraged, and steady work on problems, beyond the required homework problems, will be essential.
Texts: There are plenty of decent books out there, but for homology theory I shall mostly use Vick - Homology theory, which I recommend to you, but don’t require. If you’re a Penn State student you should be able to follow the link and (legally} download the book at no charge for class use. The same applies to the book Singer and Thorpe - Lecture notes on elementary topology and geometry, which is excellent for covering space theory and many other things, including prerequisite material for the class. Here are some other standard books, also available at no charge: Fulton - Algebraic topology, a first course, Hatcher - Algebraic topology, Rotman - An introduction to algebraic topology, Weintraub - The fundamentals of algebraic topology. In addition, I shall probably recommend some other sources during the lectures, and also hand out lecture notes for the homology part of the course.
Class page: I will keep a record of events in class, and post homeworks and various supplementary materials here.
Homework: I shall hand out regular homework problems. I shall ask you to hand in some, but not all of the homework, so that I can monitor your progress in learning how to present mathematical arguments clearly. I shall try to devote class periods once in a while to discussions of homework problems.
Exams: There will be none. But many of you will be preparing for the qualifying exam in topology, and I will perhaps create one or two practice exams based on previous qualifying exams and other sources.
Grading Policy: Grades will be assessed on the basis of class participation (25%) and homework (75%).
Contacting you: If I need to contact you in a hurry (because of a class cancellation, for instance) I will do so by email, to your psu.edu addresses. Please monitor your email!
Changes to the syllabus: This syllabus is subject to change. If any change is made, then I will notify registered class participants of it by email.
And now, the boilerplate …
Academic Integrity: Academic integrity is the pursuit of scholarly activity in an open, honest, and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle.
According to Penn State policy G-9: Academic Integrity, an academic integrity violation is “an intentional, unintentional, or attempted violation of course or assessment policies to gain an academic advantage or to advantage or disadvantage another student academically.” Unless your instructor tells you otherwise, you must complete all course work entirely on your own, using only sources that have been permitted by your instructor, and you may not assist other students with papers, quizzes, exams, or other assessments. If your instructor allows you to use ideas, images, or word phrases created by another person (e.g., from Course Hero or Chegg) or by generative technology, such as ChatGPT, you must identify their source. You may not submit false or fabricated information, use the same academic work for credit in multiple courses, or share instructional content. Students with questions about academic integrity should ask their instructor before submitting work.
Students facing allegations of academic misconduct may not drop/withdraw from the affected course unless they are cleared of wrongdoing (see G-9: Academic Integrity). Attempted drops will be prevented or reversed, and students will be expected to complete course work and meet course deadlines. Students who are found responsible for academic integrity violations face academic outcomes, which can be severe, and put themselves at jeopardy for other outcomes which may include ineligibility for Dean’s List, pass/fail elections, and grade forgiveness. Students may also face consequences from their home/major program and/or The Schreyer Honors College.
Education Equity and Reporting of Bias: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.
Disability Accommodation: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website.
In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations request with them as early as possible. You must follow this process for every semester that you request accommodations.
Students with disabilities participating in internship, practicum, student teaching, or other experiential learning opportunities as part of their degree requirements may also be eligible for reasonable accommodations to ensure equal access and opportunity. These accommodations are determined through an interactive process involving the student, their University supervisor, and the site supervisor. Student Disability Resources can assist students with identifying potential barriers, facilitating accommodation requests, and coordinating with University supervisors to promote inclusive learning experiences.
Counseling and Psychological Services: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation.
Counseling and Psychological Services at University Park (CAPS): 814-863-0395
Penn State Crisis Line (24 hours/7 days/week): 877-229-6400
Crisis Text Line (24 hours/7 days/week): Text LIONS to 741741
Instructor: Nigel Higson. My office is 228 McAllister Building.
Meeting Times: Tuesdays and Thursdays, 9:05-10:20, in 106 McAllister Building.
Office Hours: Wednesdays 2:00-4:00 (starting January 21) or by appointment. Contact me in class or send me an email. My PSU ID is ndh2, from which you can figure out my PSU email address.
Prerequisites: The main requirements will be familiarity with the basic concepts of general topology (topological spaces, continuity, campactness, connectedness and so on) as well as familiarity with basic abstract algebra (groups, homomorphisms, quotient groups) and linear algebra. Some experience with manifolds will be helpful, but this is not required.
Content, Goals, Objectives, Expectations: The main topic of the course will be the homology and cohomology of topological spaces: I shall present the basic theory, and give applications to manifolds and fixed point theory. Along the way I shall also discuss covering spaces and the fundamental group. I hope that by the end of the class, participants will have an understanding of, and an appreciation for, some of the ways in which algebra may be used to proved purely topological theorems, among them, for instance, the Jordan curve theorem. I also hope that by the end of the class, those participants who intend to take the topology qualifying exam will be adequately prepared for it. For those who wish to reach these goals, regular attendance of classes is very strongly encouraged, and steady work on problems, beyond the required homework problems, will be essential.
Texts: There are plenty of decent books out there, but for homology theory I shall mostly use Vick - Homology theory, which I recommend to you, but don’t require. If you’re a Penn State student you should be able to follow the link and (legally} download the book at no charge for class use. The same applies to the book Singer and Thorpe - Lecture notes on elementary topology and geometry, which is excellent for covering space theory and many other things, including prerequisite material for the class. Here are some other standard books, also available at no charge: Fulton - Algebraic topology, a first course, Hatcher - Algebraic topology, Rotman - An introduction to algebraic topology, Weintraub - The fundamentals of algebraic topology. In addition, I shall probably recommend some other sources during the lectures, and also hand out lecture notes for the homology part of the course.
Class page: I will keep a record of events in class, and post homeworks and various supplementary materials here.
Homework: I shall hand out regular homework problems. I shall ask you to hand in some, but not all of the homework, so that I can monitor your progress in learning how to present mathematical arguments clearly. I shall try to devote class periods once in a while to discussions of homework problems.
Exams: There will be none. But many of you will be preparing for the qualifying exam in topology, and I will perhaps create one or two practice exams based on previous qualifying exams and other sources.
Grading Policy: Grades will be assessed on the basis of class participation (25%) and homework (75%).
Contacting you: If I need to contact you in a hurry (because of a class cancellation, for instance) I will do so by email, to your psu.edu addresses. Please monitor your email!
Changes to the syllabus: This syllabus is subject to change. If any change is made, then I will notify registered class participants of it by email.
And now, the boilerplate …
Academic Integrity: Academic integrity is the pursuit of scholarly activity in an open, honest, and responsible manner. Academic integrity is a basic guiding principle for all academic activity at The Pennsylvania State University, and all members of the University community are expected to act in accordance with this principle.
According to Penn State policy G-9: Academic Integrity, an academic integrity violation is “an intentional, unintentional, or attempted violation of course or assessment policies to gain an academic advantage or to advantage or disadvantage another student academically.” Unless your instructor tells you otherwise, you must complete all course work entirely on your own, using only sources that have been permitted by your instructor, and you may not assist other students with papers, quizzes, exams, or other assessments. If your instructor allows you to use ideas, images, or word phrases created by another person (e.g., from Course Hero or Chegg) or by generative technology, such as ChatGPT, you must identify their source. You may not submit false or fabricated information, use the same academic work for credit in multiple courses, or share instructional content. Students with questions about academic integrity should ask their instructor before submitting work.
Students facing allegations of academic misconduct may not drop/withdraw from the affected course unless they are cleared of wrongdoing (see G-9: Academic Integrity). Attempted drops will be prevented or reversed, and students will be expected to complete course work and meet course deadlines. Students who are found responsible for academic integrity violations face academic outcomes, which can be severe, and put themselves at jeopardy for other outcomes which may include ineligibility for Dean’s List, pass/fail elections, and grade forgiveness. Students may also face consequences from their home/major program and/or The Schreyer Honors College.
Education Equity and Reporting of Bias: Consistent with University Policy AD29, students who believe they have experienced or observed a hate crime, an act of intolerance, discrimination, or harassment that occurs at Penn State are urged to report these incidents as outlined on the University’s Report Bias webpage.
Disability Accommodation: Penn State welcomes students with disabilities into the University’s educational programs. Every Penn State campus has an office for students with disabilities. Student Disability Resources (SDR) website provides contact information for every Penn State campus. For further information, please visit the Student Disability Resources website.
In order to receive consideration for reasonable accommodations, you must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: See documentation guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with an accommodation letter. Please share this letter with your instructors and discuss the accommodations request with them as early as possible. You must follow this process for every semester that you request accommodations.
Students with disabilities participating in internship, practicum, student teaching, or other experiential learning opportunities as part of their degree requirements may also be eligible for reasonable accommodations to ensure equal access and opportunity. These accommodations are determined through an interactive process involving the student, their University supervisor, and the site supervisor. Student Disability Resources can assist students with identifying potential barriers, facilitating accommodation requests, and coordinating with University supervisors to promote inclusive learning experiences.
Counseling and Psychological Services: Many students at Penn State face personal challenges or have psychological needs that may interfere with their academic progress, social development, or emotional wellbeing. The university offers a variety of confidential services to help you through difficult times, including individual and group counseling, crisis intervention, consultations, online chats, and mental health screenings. These services are provided by staff who welcome all students and embrace a philosophy respectful of clients’ cultural and religious backgrounds, and sensitive to differences in race, ability, gender identity and sexual orientation.
Counseling and Psychological Services at University Park (CAPS): 814-863-0395
Penn State Crisis Line (24 hours/7 days/week): 877-229-6400
Crisis Text Line (24 hours/7 days/week): Text LIONS to 741741