# Topology Qualifying Exam Workshop

May 2020

Worksheet 4A

Theme: Homology: Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem.
Applications.

Part 1 - 5-10 minutes

Warm-up:

1. Give a ∆-complex structure for a standard 2-simplex ∆ and write down its chain complex.
   Explicitly describe the boundary homomorphisms.

2. Give a ∆-complex structure for $S ^ { 1 }$ ., and use that to compute the homology groups for $S ^ { 1 }$

3. What is the difference between the reduced homology and (non-reduced) homology?

4. Let X be a topological space and $A , B \subset X$ open subspaces with $X = A \cup B$ What does the Mayer-Vietoris theorem say about the homology groups of X and its subspaces?

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (May 2015) Compute the (reduced) singular homology groups of the space $X = S _ { 1 } \times ( S _ { 1 } \lor S _ { 1 } )$ , which can be thought of as two copies of $S _ { 1 } \times S _ { 1 }$ glued together along their copies of $S _ { 1 } \times \{ x _ { 0 } \}$ . [You may use your knowledge of the homology groups of $T ^ { 2 } = S _ { 1 } \times S _ { 1 }$ in your calculations.]

2. (January 2017) If X is a ∆-complex with at most one k-simplex for each $0 \leq k \leq 5 .$ show that each of the homology groups $H _ { k } ( X )$ must be cyclic for $0 \le k \le 5$ . Can all of these groups be non-trivial?
   Explain why or why not.

3. (May 2018) Let $\Delta _ { 2 }$ and $\Delta _ { 2 } ^ { \prime }$ be distinct 2-simplices, and let X be the quotient space obtained by identifying the six vertices of $\Delta _ { 2 } \cup \Delta _ { 2 } ^ { \prime }$ to a single point.
   Identify a ∆-complex structure for X and compute the simplicial homology groups $H _ { n } ^ { \Delta } ( X )$ for all n.

4. (January 2019) Recalling that $S ^ { 2 } = \{ ( x , y , z ) \in \mathbb { R } ^ { 3 } : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \}$ , define the equator of $S ^ { 2 }$ to be $\{ ( x , y , z ) \in S ^ { 2 } : z = 0 \}$ , so that the equator is homeomorphic to $S ^ { 1 }$ . Let $Z _ { 1 }$ and $Z _ { 2 }$ be disjoint copies of the 2-sphere $S ^ { 2 }$ , let f be a homeomorphism from the equator of $Z _ { 1 }$ to the equator of $Z _ { 2 }$ , and let Z be the quotient space obtained from $Z _ { 1 } \cup Z _ { 2 }$ by identifying the equator of $Z _ { 1 }$ to the equator of $Z _ { 2 }$ via f. Find a ∆-complex structure on the space $Z ,$ and compute the simplicial homology groups of Z.

5. (May 2019) For a topological space X, the cone on X, cX, is the quotient space of $X \times [ 0 , 1 ]$ under the equivalence relation $( x , s ) \sim ( y , t )$ iff either $( x , s ) = ( y , t ) { \mathrm { ~ o r ~ } } s = t = 1$ The suspension of X ΣX, is the union of two copies of cX along $X \times \{ 0 \}$ realized as the quotient space of $X \times [ 0 , 1 ]$ under the equivalence relation $( x , s ) \sim ( y , t )$ iff either (x, s) = (y, t) or s = t = 1 or s = t = 0.

(a) Show that the cone on X is contractible.

(b) Use a Mayer-Vietoris sequence to show that for every $n \geq 1$ we have that ${ \tilde { H } } _ { n } ( \Sigma X ) \cong { \tilde { H } } _ { n - 1 } ( X )$

## Part 3 - 5-10 minutes

Wrap-up discussion with everyone:

1. Which of these problems did you find easy?
   Hard?

2. Which topics in the theme do you need to review more?