Questions handed in by students on 2017-01-23.

Lecture -

1. We have said f_# \from C_k \to D_k does not change dimension, but under the conditions given we can take f to be a function that continuously collapses our space X to a lower dimensional subspace (say a point).  Does this not take k-dimensional chains to lower-dimensional chains?

A: It does not.  The dimension of a singular simplex \sigma \from \Delta^n \to X is determined by the dimension of \Delta^n (here n) and not by the "dimension" of the image (which we have not defined). 

2. When seeing elements of a homology as cycles/boundaries are we considering boundaries of all dimensions?

A: No.  Recall that H_k = Z_k/B_k.  Both of Z_k and B_k are subgroups of C_k - so everything here is k-dimensional.

3. When we use f_* or f_k?

A: The symbol f_* will (almost!) always denote an induced homomorphism on homology.  The symbol f_k will (almost!) always denote either the k-th homomorphism in a chain map f_# or the k-th induced map on homology.  The only way to be 100% sure is to pay attention to context. 

4. What is nice about reduced homology other than \tilde{H}_k(pt) = 0 "Just feels right"?

A: Reduced homology makes the "exact triangle for a pair" be, in fact, exact.  We will cover this next week. 

Homework -

1. Hint for the \Delta-complex Hausdoff exercise from Assignment Sheet 2?

A: The problem includes references to pages in Hatcher (the book).

2. Could we get a sketch outline for non-assessed exercises?

A: No.  Please see the lecture, the model solutions, and the book for many detailed examples of how to write proofs and how to compute.  If you need help with a particular exercise please look (a) in your personal brain (b) in the book (c) in Google's brain (d) in the brains of other students (e) in the brains of the TAs (f) in my brain (g) in Professor Mond's brain (h) in Professor Hatcher's brain -- more or less in that order. 

General -

1. I've forgotten where we're going with all this, what are interesting qus. that these tools allow us to answer?

A: We have already mentioned the Brouwer fixed point theorem and also the problem of deciding if S^2 \cross S^2 is homeomorphic to S^4.  As another application, we will give a mathematically precise definition of the "orientation" of a manifold, and thus understand why the Mobius band does not embed in the torus (say).

Exploration -

1. Any obvious spaces that aren't singular or delta complexes?

A: The short answer is "no".  If you want a better answer, you need to define words "obvious" and "singular".  You may be interested in the following article: https://www.quantamagazine.org/20150113-a-proof-that-some-spaces-cant-be-cut/