Questions handed in by students on 2017-01-16.

Lecture -

1a. Why consider only finite sums for definition of free abelian group ZZ[X]? 

A: Infinite sums require more care [cf infinite sums in analysis].  In this module we are only dealing with the simplest varients of homology, as appropriate for an introduction.

1b. Why did you take a formal sum of some groupy things, and then draw their complexes joined together? (Top Middle Board)

A: There is a difference between elements/subgroups of a group and points/subspaces of a space.  However homology does its best to transform the latter into the former. 

2. Why does \bdy_{k-1} \circ \bdy_k = 0 => B^\Delta_k(X) \subset Z^\Delta_k(X)?

A: This is an important exercise. 

3. Do any two \Delta-cplx structures on the same space give isomorphic simplicial homology groups?  What if "same space" is "homeomorphic spaces"?

A: Yes and yes. The former follows from Theorem 2.27.  The latter is an important exercise. 

4a. Why is \Delta^3 the tetrahedron [figure]? and \Delta^4 this weird thing [figure]?  Surely \Delta^3 lives in \RR^{3 + 1} = \RR^4 and \Delta^4 lives in \RR^5, and so should be difficult to draw.

A: \Delta^3 lives in \RR^4, but it can be projected to \RR^3 and it there gives a regular tetrahedron.  Exercise: Find such a projection and compute the image of \Delta^3 explicitly.  As for \Delta^4: yes, this is difficult to draw.  See https://en.wikipedia.org/wiki/5-cell for various attempts and further discussion. 

4b. What would [v_0 \ldots \hat{v_i} \ldots v_k] actually look like in comparison to [v_0 \ldots v_k]?

A: It is the maximal (aka "codimension one") face which, uniquely, does not contain v_i.

5. Doesn't a set of vectors to be linearly independent imply affine independence? (And vice versa!)

A: Linear independence implies affine independence, but the converse is false.

6. Could we see more examples of \Delta complex structures and use of Notation C^\Delta_k, Z^\Delta_k etc. 

A: Yes!

7. Could we do an example of working out H^\Delta_k(X)

A: This is done in detail in the solutions to Exercise 2.8 and 2.10.
I also asked the TAs to do this in support class in week 2.

Homework -

1. How can we compute the cokernel of a transformation given as a matrix or given that we know the image and kernel?  Other than elements of the image, how do we subdivide the space we are mapping into elements of the cokernel?  i.e. given T \from A \to B why is \Coker(T) not just \Im(T) and \not \Im(T)?

A: Suppose that we are in the special case where A and B are both free abelian groups.  Then the cokernel can be computed using Smith normal form (Algebra I).

Exploration -

1a. Why do we care about boundary operators?

A: It is crucial in the definition of homology groups.

1b. Why do we need homology?  I was quite happy with the fundamental group & homotopy classes. 

A: Exercise: Show S^2 \cross S^2 \nothomeo S^3. 

2a. Could we see some more examples of things that look like \Delta-complexes but aren't?

A: Yes - I did this in lecture.

2b. Do all spaces have \Delta-complex structure?

A: No.  For example, the Cantor set has no \Delta-complex structure. 

3. Can you give non-metrizable examples please.

A: Yes - I did this in lecture. 

Future -

1. is \{\Delta-complexes\} \subseteq \{cell complexes\}, how do these concepts relate?

A: Yes, every \Delta-complex is a (particularly nice) cell complex.

2. Will we cover diagram chasing, as I hear it's a useful method for this module?

A: Yes. 

Administrative -

1. Will the lectures be recorded?

A: I have filled out the lecture capture forms.