Courses:

Algebraic Topology >> Content Detail



Study Materials



Readings

This section contains documents that could not be made accessible to screen reader software. A "#" symbol is used to denote such documents.


Amazon logo Help support MIT OpenCourseWare by shopping at Amazon.com! MIT OpenCourseWare offers direct links to Amazon.com to purchase the books cited in this course. Click on the Amazon logo to the left of any citation and purchase the book from Amazon.com, and MIT OpenCourseWare will receive up to 10% of all purchases you make. Your support will enable MIT to continue offering open access to MIT courses.

The readings are assigned in the textbook for this course:

Amazon logo Hatcher, Allen. Algebraic Topology. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521795401.

The text is freely available online, but paperback copies are also available. We will be focusing on chapters 2 and 3.


WEEK #TOPICSREADINGS
1CW-complexes, delta-complexes, simplicial homology, exact sequences, diagram chasing

Chapter 0, pp. 5-8 (PDF)#

Chapter 2, Section 2.1, pp. 102-108, exact sequences from ~113 (PDF)#

2Singular homology, homotopies and chain homotopies, categories and functors, Eilenberg-Steenrod axioms

Chapter 2, Section 2.1, pp. 104-113 (PDF)#

Chapter 2, Section 2.3 (PDF)#

3Excision, computations for spheres, equivalence of simplicial and singular homology

Chapter 2, Section 2.3 (PDF)#

Chapter 2, Section 2.1, pp. 113-134 (PDF)#

4Cellular homology, Mayer-Vietoris sequences, the Mayer-Vietoris argument, homology with coefficientsChapter 2, Section 2.2, pp. 137-160 (PDF)#
5Tensor products, Tor, universal coefficient theorem for homology, products of simplices

Chapter 3, Section 3.A (PDF)#

Chapter 3, Section 3.B (PDF)#

6The Eilenberg-Zilber shuffle "product" map, diagonal approximations, the Alexander-Whitney map, method of acyclic models, Kunneth formulaNo readings in Hatcher text
7Duality, cohomology, Ext, universal coefficients for cohomologyChapter 3, Section 3.1, pp. 190-206 (PDF)#
8Projective spaces and Grassmannians, cup products, relative cup productsChapter 3, Section 3.2, pp. 206-212 (PDF)#
9Dual Kunneth formula, field coefficients, cup products in cohomology of projective spacesChapter 3, Section 3.2, pp. 212-224 (PDF)#
10Manifolds, local orientations, global orientationsChapter 3, Section 3.3, pp. 230-239 (PDF)#
11Cap products and choices of appropriate sign conventions, statement of Poincare duality, limitsChapter 3, Section 3.3, pp. 239-249 (PDF)#
12Compactly supported cohomology, proof of Poincare dualityChapter 3, Section 3.3, pp. 239-249 (PDF)#
13

Finish proof of Poincare duality

Intersection pairing and cup product

Chapter 3, Section 3.3, pp. 249-252 (PDF)#
14Lefschetz fixed point theoremNo readings in Hatcher text
15

Finish proof of Lefschetz theorem

Assorted further topics

No readings in Hatcher text

 








© 2010-2021 OpenCollege.com, All Rights Reserved.
Open College is a service mark of AmeriCareers LLC.