Questions handed in by students on 2017-02-07.

Lecture -

1. Could you please explain the "bungalow analogy" in writing, with what each map does?

Ans: The "short prism operator" aka the "bungalow operator" $T$ is defined recursively via $T[v] = [v, v]$ for zero simplices and $T \sigma = b_\sigma(1 - T \partial) \sigma$ in higher dimensions.

Suppose that $\sigma$ is an $n$-simplex.  Then we take $\sigma$ (the "floor") and add $-T \partial \sigma$ (the "walls").  This is again an $n$-chain.  We cone this to obtain $b_\sigma(\sigma - T \partial \sigma)$ which is an $n+1$-chain.  A computation shows that $\partial T \sigma$ (the boundary of the bungalow) is equal to $\sigma - T \partial \sigma - S\sigma$ (that is, the floor, minus the walls, minus the ceiling, in that order).

2. How many 3-simplices are formed by applying $T$ to the 2-simplex as in the second picture drawn?

Ans: The one-chain $T[v_0]$ is just a single edge, the two-chain $T[v_0, v_1]$ has three triangles, and the three-chain $T[v_0, v_1, v_2]$ has ten tetrahedra.

Exercise: Determine the number of $n+1$-simplices in $T[v_0, \ldots, v_n]$.

3. Could you recap all of (or as much as possible) the key important ideas and motivation for subdivision and how it relates to what we're doing    Please :)

Ans: I think you are asking about excision, and the answer there is that subdivision is the first key idea.  The second is understanding how to patch together subdivided simplices, to get a subordinate chain with homotopic boundary.  The motivation is proving Proposition 2.21.  This is needed to prove Excision, which is needed to understand relative homology of good pairs, which is needed to prove Theorem 2.27, which is needed to prove the applications mentioned in the first lecture. 

Exam -

1. Do we need operators like $P, S, b_\sigma, T$ outside of these proofs? (i.e. the exam) or are they just needed for class?

Ans: The material covered in lecture and in the example sheets is examinable.  Obviously the proof of excision is massive (three lectures!) and contains many cool ideas... so if it is to be examined, I'll break it down a bit - eg ask you questions along the lines of Exercise 5.8.