SIMPLICIAL HOMOLOGY

MATH 180, SPRING 2020

Your task as a group, is to research the topics and questions below, write up clear notes as a group explaining these topics and the answers to the questions, and then make a video (20-30 minutes) presenting your findings. Your video and notes will be presented to the class to teach them your findings. Make sure that in your notes and video you give examples and intuition, along with formal definitions, theorems, proofs, or calculations. Make sure that you point out what the is most important take away message, and what aspects may be tricky or confusing to understand at first. You will need to work together as a group. People working on answering later questions will need to understand the earlier questions. Each member of the group should complete at least one full example from questions2 and4. Other than that, you may decide how to split up the work, but make sure that students who work out the answers to the theory questions1 and3 explain these answers to the whole group early on so the rest of the group can proceed on the later questions.

1. Resources The primary resource for this project is Algebraic Topology by Allen Hatcher which can be found freely available on Allen Hatcher’s website at https://pi.math.cornell. edu/~hatcher/AT/AT.pdf, Chapter 2.1 page 102-107. For some intuition about what the homology of a space is, you may want to read the introduction to Chapter 2 of Hatcher pages 97-101. You may also look at other references you find online, or in Lee’s book Chapter 13.

2. Topics and Questions As you research, you may find more examples, definitions, and questions, which you defi- nitely should feel free to include in your notes and/or video, but make sure you at least go through the following discussion and questions. (1) What is an n-dimensional simplex? What is the boundary of an n-dimensional sim- plex? Draw a 0-simplex, 1-simplex, 2-simplex, and 3-simplex and explain what their boundaries are. What is a ∆-complex? (2) Construct ∆ complexes which build the manifolds (or manifolds with boundary) in the following examples (we are always working up to homeomorphism equivalence): (a) The circle S1 (b) The 2-dimensional sphere S2 1 2 MATH 180, SPRING 2020

(c) The 2-dimensional torus T 2 (d) The 3-dimensional sphere S3 (e) The real projective plane RP 2 (f) The Klein bottle K (g) The interval I = [0, 1] (h) The cylinder (with boundary) C (i) The Mobius band (with boundary) M (j) The closed 2-dimensional disk (with boundary) D2

(3) What is the module of n-chains ∆n(X) and the boundary map ∂n? Prove that the th image of ∂n+1 is in the kernel of δn. What is the definition of the n homology group

of this chain complex (∆n(X), ∂n)? (4) Using the ∆ complexes constructed in Question2, determine the chain complexes

(∆n(X), ∂n) and the homology groups for each of the spaces in the above examples. Among the orientable examples of manifolds without boundary, if the manifold is k-dimensional what is the rank of its kth homology group? How does this differ from the non-orientable manifolds or manifolds with boundary? Are there any examples in the list which have isomorphic homology groups (which ones)?