Homework 3
“Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he’s scribbled on, scrunges them all up, and throws them in the trash can.”
– John von Neumann’s housekeeper, describing the employer
This HW is due on 4th March 2025. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 4th March 2025. Please submit the HW as a single pdf file - you can scan / take a photo of your completed HW (handwritten) and convert it into a pdf file (please name your file first.last_hw3.pdf). To get credit, please provide all details and give complete reasoning for all your work. Do not consult any books, internet resources or AI. If you have questions about the problems, we can discuss them in class.
Exercise 1 (2 points) In each part below, determine if the set determines a abstract simplicial complex. Give reasons in each case for your YES/NO answer. If your answer is YES, sketch a geometric realization of the abstract simplicial complex.
- \{\{a\}, \{b\}, \{a,b\}\}
- \{\{a\}, \{b\}, \{c\}, \{d\}, \{a, b\}, \{b, c\}, \{c, d\}, \{a, d\}, \{a, c\}, \{a, b, c\}\}.
Exercise 2 (2 points) Let \mathcal{U} = \{U_0, U_1, U_2, \cdots, U_n\} such that U_i \subset U_0 for i=1, 2, \cdots, n. Explain why the nerve complex \mathcal{N} (\mathcal{U}) is contractible (homotopic to a point).
Exercise 3 (6 points) Consider the set S=\{s_1,s_2,s_3\} \subset \mathbb{R}^2, where s_1=(0,0), s_2(1,0), s_3=(0,2).
- Describe \check{C}ech (X, r), for r \in [0, \infty) (draw sketches)
- Describe Rips (X, r), for r \in [0, \infty) (draw sketches)
- Describe Alpha (X, r), for r \in [0, \infty) (draw sketches)