Plan for AlgTopI 2015

This plan may change during the course
Week Hatcher Topics Exercises Presentations
 1 Chp 0 Homotopy theory 0.1, 0.2, 0.3, 0.5, 0.6, 0.9, 0.12, 0.11  
 2  Chp 0


0.10, 0.14, 0.17, 0.20, 0.21, 0.23 0.10, 0.11, 0.23
 3 Chp 2 Homology of Δ-sets 0.21 (finish), 2.1.1, 2.1.4, 2.1.5, 2.1.7, 2.1.9 2.1.1, 2.1.4, 2.1.5
4 Chp 2 Homological algebra
Singular homology
Homotopy invariance
2.1.11, 2.1.14, 2.1.16, 2.1.8, 2.1.30 2.1.11, 2.1.16, 2.1.8

Chp 2

Excision 2.1.20, 2.1.29, 2.2.2, 2.2.4, 2.2.6, backlog exercises 2.1.20, 2.2.2, 2.2.6
 6 Chp 2 Cellular homology
2.2.8, 2.2.9, 2.2.11, 2.2.14, 2.2.28 2.2.9(a,b), 2.2.28, 2.2.11
 7 Chp 2B-2C Applications of homology 2.2.43, 2.2.29, 2.2.41, backlog exercises, discussion of exam questions  

Exercises in boldface

If you hand in your written answer to the exercise in boldface to Dieter you will get very valuable feedback in return! You will boost your competence in written formulation of mathematics!


If you prepare a short oral expostion about these exercises you will be much better prepared for the oral exam! You will boost your competence in oral formulation of mathematics!

Comments to exercises

What Hatcher calls a Δ-complex we'll later call "the realization of a Δ-set".
What Hatcher means is this: Take the 2-dimensional Δ-set consistsing of all subsets of 2+. This Δ-set has 3 elements in degree 0, 3 in degree 1, and 1 in degree 2. Construct a quotient Δ-set S by identifying all three elements in degree 0 to a single element. Find the homology groups of S.
What Hatcher means is this: Compute the homology groups of the 2-dimensional Δ-set KB given by KB0 = {p}, KB1 = {a,b,c}, KB2 = {U,L}. The face maps are d0U=b, d1U=c, d2U=a, d0L=a, d1L=b, d2L=c. Remember your row and column operations or look at the Smith normal form .
Let Δ[3+] be the Δ-set of all subsets of 3. Let S be the quotient Δ-set of Δ[3+] obtained by identifying d0Δ3 with d1Δ3 and d2Δ3 with d3Δ3. S is a Δ-set with one 3-simplex, two 2-simplices, three 1-simplices, and two 0-simplices. What is the realization of the 2-skeleton of S? You can use this to show that the realization of S is homotopy equivalent to S3. In fact, it is homeomorphic to S3. (Cut the 3-simplex into two halfs each homemorphic with a solid torus.) This exercise is challenging and you may want to ignore it in the first round.
Again, Hatcher wants you to compute the homology groups of a Δ-set. This is similar to 2.1.4.
This exercise is more involved than it may look!
The universal covering space of the torus is the plane. The universal covering space of the wedge of spheres is the infinite tree shown on p 59 (or p 79) with a 2-sphere joined at one point at each node of the graph.
Every self-map of a real projective space is covered by a self-map of the corresponding sphere.
You may try with a self-map of the n-sphere that goes through the wedge of two n-spheres.
Alternatively, show that any self-map of any path-connected space is homotopic to a self-map with a fixed point under some mild hypothesis.
(a): You may view this space as a the quotient of a CW-complex by its 0-skeleton. (We have met this space before.) (b): This space is a quotient space of the disjoint union of two tori. (c) and (d): These spaces are 2-dimensional CW-complexes.
See Exercise 1.2.14
Week 6
For some of these exercises you need the cellular chain complex. Read about it in Hatcher's book in case I haven't yet covered it at the lectures.

Jesper Michael Møller
Last modified: Mon Oct 18 09:44:07 CEST 2010