Introductory algebraic topology: review exercises

Grading

These are exercises for Introductory algebraic topology: course.

Overview

• Are intervals homeomorphic?
• Can a set to be both open and closed?
• Intersection of any collection of closed sets is closed
• Is Mobius strip homeomorphic to the cylinder?
• Is a closed subset of a compact space always compact?
• Is a restriction of a continuous function always continuous?
• Is the intersection of any collection of open sets always open?
• Is the inverse of a continuous function always continuous?
• Is the max of two continuous functions continuous?
• Is the union of any collection of closed sets always closed?
• Addition is continuous
• Prove that the composition of continuous functions is continuous
• Prove that two closed intervals are homeomorphic
• Why "preimage" in the definition of continuity?

Part 1

1. Can a discontinuous function satisfy the Intermediate Value Theorem? 2. Sketch the realization of the following cubical complex:

• $0$-cells: $A,B,...,H;$
• $1$-cells: $a,b,...,l.$
• $∂a = A+B,∂b=B+C,∂c=C+D,∂d=D+A,$
• $∂e = B+F,∂f=C+G,∂g=D+H,∂h=A+E,$
• $∂i = E+F,∂j=F+G,∂k=G+H,∂l=H+E.$

3. Evaluate the homology of:

4.Prove that the cubical complex $K$ given below:

• $0$-cells: $0,1,...,n;$
• $1$-cells: $(0,1),(1,2),...,(n-1,n),$

satisfies: $|H₀(K)|=1$.

5. Prove that the homology of a cubical complex is an equivalence relation.

6. Prove that neighborhoods are open.

7. Is the union of a collection of closed sets always closed?

8. Prove that the frontier is closed.

Part 2

1. Suppose A is a subset of a topological space X and τ is the topology of X. Define a collection of subsets of A as $τ_A = \{W∩A: W∈τ\}$. Prove that the union of any subcollection of $τ_{A}$ belongs to $τ_{A}$.

2. Prove that $f(x) = x²$ is continuous at $x=0$.

3. (a) Prove that the projection $p: R² → R$ is continuous. (b) Prove that the projection of a (filled) square on one of its sides is continuous.

4. Prove that a function is continuous if and only if the preimage of any closed set is closed.

5. Suppose the chain maps of continuous functions $f: S¹ → S¹$ are recorded by means of cubical complexes K and L containing 4 edges each. Present the chain map of the $90$ degree rotation.

6. Find cubical complexes and a chain map to represent a surjective map $f: S¹ → [0,1]$.

7. (a) Prove that all open intervals of finite length are homeomorphic. (b) Are $(0,1)$ and $(0,∞)$ homeomorphic?

First half

1. True or false?

• a. If the homologies of $X$ and $Y$ coincide then $X$ and $Y$ are homeomorphic.
• b. If $X×Y$ is path-connected then so is $X$ and $Y$.
• c. In $Rⁿ$, an unbounded set is not compact.
• d. The sum of two cycles is a cycle.
• e. The empty set is compact.

2. Give an example of:

• a. a figure with non-trivial $1$- and $2$-homology which isn't the torus,
• b. a continuous function $f:X→X×Y$ which isn't constant,
• c. non-compact, bounded subset of $R²$ which isn't open,
• d. a projection $p:X×Y→X$ which isn't continuous,
• e. a non-Hausdorff topology on $R²$ which isn't anti-discrete.

3. (a) Give the definition of the boundary operator $∂$ on a cubical complex. (b) Prove that $∂∂(a)=0$ for any $1$-chain $a$. (c) Prove that $∂∂(θ)=0$ for any $2$-chain $θ$.

4. (a) Find a cubical complex representation of the figure $8$. (b) Compute the homology of the complex. (c) Describe the homology classes of the sphere and the torus.

5. (a) Give the definition of a Hausdorff space. (b) Prove that a subspace of a Hausdorff space is Hausdorff. (c) State the theorem about homeomorphisms of Hausdorff spaces.

6. (a) Give the definition of a chain map. (b) Present cubical complexes and a chain map for the gluing function of a segment that turns it into a circle, $f:[0,1]→S¹$. (c) Describe what happens to the homology classes under $f:S¹→S²$.

7. (a) Define the product of topological spaces. (b) Prove that the product of two path-connected spaces is path-connected. (c) State the theorem about products of compact spaces.

8. Compute the homology of a "wire-frame" pyramid.

Part 3

1. Suppose K is the unit disk in the plane: $$K = \{(x,y)∈R²: x² + y² ≤ 1\}.$$ Suppose an equivalence relation on $K$ is given by:

• $(x,y) \sim (-x,y)$ for all $x,y$, except
• $(0,1)\sim (0,-1)$.

Sketch $K/ _{\sim} $. Is it a surface, a surface with boundary, or neither?

2. Represent the sphere as a cell complex with two $2$-cells, list all cells, and describe/sketch the gluing maps.

3. Cut a hole in the middle of the projective plane and represent the result as a cell complex with a single $2$-cell (standard diagram for surfaces). What is it?

4. Represent the cylinder as an abstract simplicial complex.

5. Find an open cover of the sphere $S²$ the nerve of which is homeomorphic to $S²$.

6. Suppose $χ(X) = χ(Y)$, does it mean that $X$ and $Y$ are homeomorphic? (a) $X$ and $Y$ are graphs, (b) $X$ and $Y$ are orientable surfaces (without boundary). Explain.

7. For two finite $2$-dimensional cell complexes $K$ and $L$, prove that $$χ(K×L) = χ(K)χ(L).$$

Part 4

1. Compute the cycle group $Z₁(K)$, by solving a system of linear equations, of the square frame $K$ with a diagonal ($5$ edges).

2. Prove that the boundary group is a vector space.

3. Represent the Klein bottle as a cell complex $K$ with two $2$-cells, find the matrix of the boundary operator $∂₂:C₂(K)→C₁(K)$.

4. Represent the projective plane as a cell complex $K$ with two $2$-cells, compute the homology group $H₁(K)$.

5. Triangulate the torus and show that it can be compatibly oriented.

6. What are the Betti numbers of these spaces (no computations required)? (a) the sphere with a circle attached to the north pole; (b) the complement in $R³$ of the unit sphere; (c) the disjoint union of the $n$-ball and the $n$-sphere, for $n=1,2,...$.

Second half

1. Compute the cycle group $Z₁(K)$, by solving a system of linear equations, of the triangulated square $K$ with a diagonal added ($5$ edges).

2. Prove that the boundary group $B_{k}(K)$ is a vector space.

3. Let X be the closure in $R³$ of the unit ball with a smaller ball taken out: $$X=Cl\left(B(0,1) - B(0,1/2)\right).$$ Represent $X$ as a cell complex $K$. Compute $B₂(K)$.

4. Represent the Klein bottle as a cell complex K with two $2$-cells, find the matrix of the boundary operator $∂₂:C₂(K)→C₁(K)$.

5. Represent the projective plane as a cell complex $K$ with two $2$-cells, compute the homology group $H₁(K)$.

6. Triangulate the torus and show that it can be compatibly oriented.

7. What are the Betti numbers of these spaces (no computations required)? (a) the sphere with a circle attached to the north pole; (b) the complement of set $X$ is problem 3; (c) the disjoint union of the $n$-ball and the $n$-sphere, for $n=1,2,...$.

More

1. Represent the Mobius band as a simplicial complex, list all the cells, their boundaries, find its Euler characteristic.
2. Provide a diagrammatic proof that $\mathbf{P}^{2}\#\mathbf{P}^{2}=\mathbf{K}^{2}.$
3. Prove that the Euler characteristic of a tree is 1.
4. (a) State the Classification Theorem for Surfaces; classify the following surfaces: (b) $\mathbf{T}^{2}\#\mathbf{K}^{2}$; (c) $\mathbf{K}^{2}\#\mathbf{K}^{2}.$
5. Compute the homology of the figure eight.
6. Compute the homology of the sphere with two whiskers.
7. Compute the homology of the Klein bottle.
8. Provide a diagrammatic proof of the fact $\mathbf{P}^{2}\#\mathbf{S}^{2}=\mathbf{P}^{2}.$
9. Is the following simplicial complex a surface: ABD, BCD, ACD, ABE, BCE, ACE?
10. Identify the surface which has planar diagram with outer edges labeled as follows: $aabcb^{-1}c^{-1}.$
11. Compute the Euler characteristic of the union of two touching spheres.
12. Compute the homology of the projective plane.
13. Give the definition of the homology of a simplicial function. State the relevant theorems.
14. The boundary of the Mobius band $M$ is a circle. Let $f:\mathbf{S}^{1}\rightarrow M$ be the function that wraps the circle onto this boundary. Compute the homology of $f.$
15. Suppose $f:K\rightarrow L$ is a deformation retraction. Prove that $f_{\ast}:H_{\ast}(K)\rightarrow H_{\ast}(L)$ is an isomorphism.
16. Compute the homology groups of $\mathbf{R}^{3}\backslash(x$-axis $\cup$ $y$-axis $\cup$ $z$-axis$).$
17. Prove the Brouwer Fixed Point Theorem.
18. What is the relation between the Lefschetz number and the Euler characteristic?

1. Compute the homology of the Mobius band.
2. Give an example of chains $a,b$ such that $|a+b|\neq|a\cup b|.$
3. Define the cubical product of chains. Prove that it is associative.
4. Suppose $X$ is a cubical set and $X^{\prime}$ is obtained from $X$ via an elementary collapse of $P\in\mathcal{K}_{k}(X)$ by $Q\in\mathcal{K}_{k-1}(X)$. Prove that for every $c\in C_{k-1}(X)$ there exists $c^{\prime}\in C_{k-1}(X^{\prime})$ such that $c-c^{\prime}\in B_{k-1}(X).$
5. Just the answers... (a) What is the homology of $\mathbf{R}^{3}\backslash(x$-axis $\cup$ $y$-axis $\cup$ $z$-axis$)?$ (b) Suppose $L$ is a retract of $K$. What can you say about $H_{\ast}(K)$ and $H_{\ast}(L)$?
6. For the set $X$ consisting of the points $(0,0),(1,0),(2,0),(1,1)$ draw the Delauney triangulation. Compute the (distance based) persistence of the alpha-complex of this set.