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# Introductory algebraic topology: review exercises

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

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

- Compute the homology of the Mobius band.
- Give an example of chains $a,b$ such that $|a+b|\neq|a\cup b|.$
- Define the cubical product of chains. Prove that it is associative.
- 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 $c-c^{\prime}\in B_{k-1}(X).$
- 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 $H_{\ast}(L)$?
- 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.