Homework 4

“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 15th April 2025. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 15th April 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_hw4.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+2+6=10 points) Consider the simplicial complex \mathcal{K}=\{\langle a,b,c \rangle, \langle a,b,d \rangle, \langle a,c,d \rangle, \langle b,c, d \rangle, \langle b,e \rangle, \langle c, e \rangle\}.

1. Sketch a geometric realization of \mathcal{K}
2. Calculate the Euler Characterictic of \mathcal{K}
3. Write down the matrices for the boundary maps \partial_1 and \partial_2, find their ranks, and then find the Betti numbers \beta_0, \beta_1 and \beta_2.