Topology

Let $A$ and $B$ be connected subspaces of a topological space $X$, such that $A \cap \overline{B} \neq \emptyset$. Prove that $A \cup B$ is connected. If $A$ and $B$ are path connected, need $A \cup B$ be path connected?

Suppose that $A$ and $B$ are compact subspaces of spaces $X$ and $Y$ respectively, and that $N$ is an open neighborhood of

Prove that there are open sets $U \subset X$ and $V \subset Y$ such that

Let $X$ be a locally compact Hausdorff space. Let $A$ and $B$ be disjoint subsets of $X$, with $A$ compact and $B$ closed. Does there exist a continuous function $f : X \to [0,1]$ such that $f|_{A} = 0$ and $f|_{B} = 1$?

No proofs, only the correct answers to the question asked, are required for this problem.

If $X$ and $Y$ are topological spaces, the join of $X$ and $Y$ is the quotient space

where

Let $f : X \to Y$ be a continuous function from a space $X$ to a Hausdorff space $Y$. Let $C$ be a closed subspace of $Y$, and let $U$ be an open neighborhood of $f^{-1}(C)$ in $X$.

Let $X$ be a normal topological space and $A \subset X$ a closed subspace.

Let $p : E \to B$ be a covering map with $E$ locally path connected and simply connected. Let $X$ be a connected space, let $f : X \to B$ be a continuous map, and $f_{1}, f_{2} : X \to E$ be two lifts of $f$. Prove that there is a deck transformation $g : E \to E$ such that $f_{2} = g f_{1}$.

Let $p : \tilde{X} \to X$ be an $n$-sheeted covering projection, $n < \infty$. Suppose that $X$ is compact. Prove that $\tilde{X}$ is compact.

Let $\mathcal{T}$ and $\mathcal{U}$ be two different topology on $X$ such that $X$ is compact and Hausdorff with respect to both. Prove that $\mathcal{T} \not\subset \mathcal{U}$.

(Recall that $\mathcal{T} \subset \mathcal{U}$ means that every set in the topology $\mathcal{T}$ is contained in $\mathcal{U}$.)

Let $X$ be a topological space and let $A \subset X$. Show that if $C$ is a connected subset of $X$ that intersects both $A$ and $X - A$, then $C$ intersects $\mathrm{Bd}A$. (Recall that $\mathrm{Bd}A = \overline{A} \cap \overline{(X-A)}$.)

Let $(X,d)$ be a metric space. For any subspace $A \subset X$ and real number $\varepsilon > 0$, let

Suppose that $X$ is a dense subspace of a topological space $Y$. Prove or give counterexamples to the following assertions:

Let $X$ and $Y$ be topological spaces, $X = U \cup V$, and $f : X \to Y$ be a function so that $f|_{U}$ and $f|_{V}$ are continuous.

Let $q : X \to Y$ be a quotient space projection from a topological space $X$ to a connected topological space $Y$. Assume that $q^{-1}(y)$ is connected for each $y \in Y$.

Let $X$ and $Y$ be topological spaces, with $Y$ compact. Let $p : X \times Y \to X$ be the usual projection onto the first factor. Show that $p$ is a closed map.

Let $Y$ be a connected subset of the topological space, and let $Z$ be a set such that $Y$ is a subset of $Z$ and $Z$ is a subset of the closure of $Y$. Prove that $Z$ is connected.

Let $A$ be a connected subspace of a connected set $X$. If $C$ is a component of $X \setminus A$, show that $X \setminus C$ is connected.

A metric space $X$ has property $S$ if for every $\varepsilon > 0$ there is a cover of $X$ by connected sets each of which has diameter $< \varepsilon$.

Let $X$ be the topologist's sine curve defined by

Let $T = S^{1} \times S^{1}$ denote the torus.

Let $\mathcal{U} = \left\{ U_{\alpha }\right\}_{\alpha \in J}$ be an open cover of the space $X$.

Let $S$ be a set and let $F$ be a family of real valued functions on $S$ such that $f(s_{1}) = f(s_{2})$ for all $f \in F$ implies $s_{1} = s_{2}$. Prove that there exists a weakest topology in $S$ amongst all those for which all members of $F$ are continuous. Show further that the resulting topological space satisfies the Hausdorff separation axiom.

Let $K$ be a compact subset of $\mathbb {R}^{n}$ and $\left\{ B_{j}\right\}$ a sequence of open balls that covers $K$. Prove that there is a positive number $\varepsilon$ such that each $\varepsilon$-ball centered at a point of $K$ is contained in one of the ball $B_{j}$.

Let $(X_{1}, d_{1})$ and $(X_{2}, d_{2})$ be metric spaces and $f : X_{1} \to X_{2}$ a continuous surjective map such that $d_{1}(p,q) \leq d_{2}(f(p), f(q))$ for every pair of points $p, q$ in $X_{1}$.

1. If $X_{1}$ is complete, must $X_{2}$ be complete? Give a proof or a counterexample.

2. If $X_{2}$ is complete, must $X_{1}$ be complete? Give a proof or a counterexample.

Let $(M,d)$ be a nonempty complete metric space. Let $S$ map $M$ into $M$, and write $S^{2}$ for $S \circ S$, that is, $S^{2}(x) = S(S(x))$. Suppose that $S^{2}$ is a strict contraction, that is, there is a constant $\lambda < 1$ such that for all points $x, y \in M$, $d(S^{2}(x), S^{2}(y)) \leq \lambda d(x,y)$. Show that $S$ has a unique fixed point in $M$.

Let $K$ be a continuous function on the unit square $0 \leq x, y \leq 1$ satisfying $\left|K(x,y)\right| \leq 1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have

Can there be more than one such function $f$?

Let $A$ be a connected, closed subspace of a compact Hausdorff space $X$, and suppose $f : A \to A$ is a continuous map. For each positive integer $n$ let $f^{n}(A) = \underbrace{f \circ f \circ \cdots \circ f}_{n \text{ times}}(A)$.

Let $\mathbb {R}^{P^{n}}$ be the real projective $n$-space and $T^{m}$ be the $m$-torus $S^{1} \times \cdots \times S^{1}$ ($m$ factors). Prove that any continuous map $\mathbb {R}^{P^{n}} \to T^{m}$ is null-homotopic.

Let $X$ be the quotient space obtained by collapsing $\left\{ \mathrm{pt.}\right\} \times S^{1} \subset S^{1} \times S^{1}$ to a point.

Compute $\pi_{1}(X)$ and $H_{*}(X)$.

A continuous map of topological spaces $p : E \to B$ is called a fibration if it has the homotopy lifting property --- that is, for any pair of continuous maps $G : X \times I \to B$ and $h : X \times \left\{ 0\right\} \to E$ such that $ph = G|_{X \times \left\{ 0\right\} }$ there exists a continuous map $H : X \times I \to E$ such that $H|_{X \times \left\{ 0\right\} } = h$ and $pH = G$. Let $p : E \to B$ be a fibration, $b_{0} \in B$ be a base point, $F = p^{-1}(b_{0})$ (the ``fiber''), and $e_{0} \in F$. Let $i : F \to E$ denote the inclusion map.

Is the canonical map $q : S^{2} \to \mathbb {R}^{P^{2}}$ (which identifies antipodal points of $S^{2}$) nullhomotopic? Why or why not?

Let $p : E \to B$ be a universal cover with $E$ and $B$ path-connected and locally path-connected. Let $T : B \to B$ be a map so that $T^{n} = \mathrm{Id}$ and so that $T(b) = b$ for some $b \in B$. (Here $T^{n} = T \circ T \circ \cdots \circ T$, $n$ times.) Show that there is a map $\tilde{T} : E \to E$ so that $p \circ \tilde{T} = T \circ p$ and $\tilde{T}^{n} = \mathrm{Id}$.

Let $T^{2} = S^{1} \times S^{1}$, and let $X \subset T^{2}$ be the subset $S^{1} \times \left\{ 1\right\} \cup \left\{ 1\right\} \times S^{1}$. Prove that there is no retraction of $T^{2}$ to $X$.

Let $A \subset \mathbb {R}^{3}$ be the union of the $x$ and $y$-axis,

and let $p = (0,0,1)$.

Let $p : (\tilde{Y}, \tilde{y}) \to (Y,y)$ be a regular covering space; that is, $p_{*}(\pi_{1}(\tilde{Y}, \tilde{y}))$ is a normal subgroup of $\pi_{1}(Y,y)$. Suppose that $f : X \to Y$ is a continuous function from the path-connected space $X$ to $Y$ with $f(x_{0}) = f(x_{1}) = y$, and that there is a lifting of $f$, $\tilde{f}_{0} : (X,x_{0}) \to (\tilde{Y}, \tilde{y})$. Show that there is a second lifting, $\tilde{f}_{1} : (X,x_{1}) \to (\tilde{Y}, \tilde{y})$. ($\tilde{f}_{1}$ need not be distinct from $\tilde{f}_{0}$.)

Let $X$ be the identification space obtained from a unit 2-disk by identifying points on its boundary if the arc distance between them on the boundary circle is $\tfrac {2\pi }{3}$. Compute the fundamental group of $X$.

Sketch a proof of the Fundamental Theorem of Algebra (every nonconstant polynomial with complex coefficients has a complex zero) using techniques of algebraic topology.

Let $X$ denote the subspace of $\mathbb {R}^{3}$ that is the union of the unit sphere $S^{2}$, the unit disk $D^{2}$ in the $x-y$ plane, and the portion, call it $A$, of the $z$ axis lying within $S^{2}$.

Let $X$ be a path-connected space, $f : X \to Y$ a continuous function, and $x_{0}, x_{1} \in X$. Suppose that the induced homomorphism

is surjective. Show that

is also surjective.

Let $B$ denote the ``figure eight space''. Let $p : X \to B$ and $q : Y \to B$ be 2-fold covering maps, where both $X$ and $Y$ are connected. Prove that $X$ and $Y$ are homotopy equivalent, but not necessarily homeomorphic.

Calculate the fundamental group of the space $\mathbb {R}^{P^{2}} \times S^{2}$.

Let $Z$ denote the figure 8 space, $Z = X \vee Y$, $X$ and $Y$ circles. Let $x, y \in \pi_{1}(Z, *)$ be the elements in $Z$ defined by $X, Y$, where $*$ denotes the vertex.

Let $X$ be the result of attaching a 2-cell $D^{2}$ to the circle $S^{1}$ by the map $f : S^{1} \to S^{1}$ given in terms of complex numbers by $z \to z^{6}$.

If $X$ is any topological space and $S^{1}$ denotes the unit circle in the complex plane with its usual topology as a topological group with multiplication given by the multiplication of complex numbers, then it is known that the set $[X, S^{1}]$ of homotopy classes of maps from $X$ to $S^{1}$ inherits a natural group structure.

If $X$ is a path-connected space whose universal cover is compact, show that $\pi_{1}(X, x_{0})$ is finite.

Prove that if $X$ is locally path connected and simply connected then every map $X \to S^{1}$ is homotopic to a constant. What can you say if we just assume that $X$ is path connected, locally path connected and the fundamental group of $X$ is finite?

Assume that a 6-element group $\Gamma$, isomorphic to the group of permutation on three letters $S_{3}$, acts on $X = S^{3}$ freely. Compute $\pi_{1}(Y)$ and $H_{1}(Y, \mathbb {Z})$, where $Y = X/\Gamma$.

Let $T^{2}$ be the two-dimensional torus and let $\varphi : S^{2} \to T^{2}$ be smooth map. Show that for any top de Rham cohomology class $[v] \in H^{2}(T^{2})$, we have $\varphi^{*}[v] = 0$.

A closed surface is a compact, connected metrizable 2-manifold without boundary.

Prove the $3 \times 3$ Lemma.

Consider the following commutative diagram of abelian groups

If all 3 columns and the first two rows are short exact, then the last row is also short exact.

Prove that if $M$ is a compact manifold of odd dimension, then $\chi (M) = 0$. Show examples of compact 4-manifolds with $\chi = 0, 1, 2, 3, 4$.

Let $X$ be a topological space. The suspension $\Sigma X$ of $X$ is defined to be the identification space obtained from $X \times [-1,1]$ by identifying $X \times \left\{ -1\right\}$ to a point and $X \times \left\{ 1\right\}$ to another point. For example the sphere $S^{n}$ is the suspension of $S^{n-1}$ with the north and south poles corresponding to the two identification points.

Compute the homology of $\Sigma X$ in terms of the homology of $X$.

Consider the chain complex

Compute $H_{n}(C)$ for all $n$.

Construct a CW complex which has the following $\mathbb {Z}$-homology groups:

Compute $H_{p}(S^{n} \vee S^{n} \vee \cdots \vee S^{n})$, $n \geq 0$, for all $p$.

Build a CW complex $X$ by adding two 2-cells to $S^{1}$, one by the map $z \to z^{4}$ and the other by the map $z \to z^{6}$. What is the homology of this space?

Let

Let the real projective plane $\mathbb {R}^{P^{2}}$ be embedded in the standard way in the real projective 5-space $\mathbb {R}^{P^{5}}$. Compute $H_{q}(\mathbb {R}^{P^{5}}/\mathbb {R}^{P^{2}})$, where $\mathbb {R}^{P^{5}}/\mathbb {R}^{P^{2}}$ is the space obtained from $\mathbb {R}^{P^{5}}$ by identifying $\mathbb {R}^{P^{2}}$ to a point.

Let

Prove that $Y \times \mathbb {R}$ is not homeomorphic to $\mathbb {R}^{2}$.

Let $A$ be a non-empty subset of $X$ and $X \cup CA$ be the union of $X$ with the cone of $A$; that is, $X \cup CA$ is obtained from the subset $X \times \left\{ 0\right\} \cup A \times [0,1]$ of $X \times [0,1]$ by identifying $A \times \left\{ 1\right\}$ to a point.

Prove that the reduced homology group $\tilde{H}_{n}(X \cup CA)$ is isomorphic to $H_{n}(X, A)$, for every $n$.

For any topological space $X$ let $\Sigma X$ denote its (unreduced) suspension. ($\Sigma X$ is the quotient space $(X \times [0,1])/{\sim }$, where $\sim$ denotes the equivalence relation generated by requiring that $(x,t) \sim (y,s)$ if $s = t = 0$ or $s = t = 1$.) If $f : X \to Y$ is a map, let $\Sigma f : \Sigma X \to \Sigma Y$ be the map of suspensions induced by the map $(x,t) \to (f(x), t)$ of $X \times [0,1]$.

Consider the following commutative diagram of abelian groups in which the row and column are exact sequences.

Suppose that $\gamma$ and $\beta_{0}$ are surjective. Prove that $\alpha_{3}$ is injective.

Suppose that a topological space $X$ is expressed as the union $U \cup V$ of two open path-connected subspaces such that $U \cap V$ is path connected, $H_{1}(U) = 0$, and the inclusion $U \cap V \to V$ induces a surjective homomorphism $H_{1}(U \cap V) \to H_{1}(V)$. (Here $H$ denotes singular homology.)

Let

denote the standard unit ball in Euclidean space $\mathbb {R}^{n}$, and

denote the standard $(n-1)$-sphere. Suppose that $f : D^{n} \to D^{n}$ is a continuous map such that the restriction of $f$ to $S^{n-1}$ is a homeomorphism from $S^{n-1}$ to $S^{n-1}$. Prove that $f$ is surjective.

Let $X$ be a connected CW complex with two 0-cells, three 1-cells, three 2-cells, and no higher-dimensional cells. Assume $H_{1}(X) \approx \mathbb {Z} \oplus \mathbb {Z}/3$. Compute the Euler characteristic of $X$ and determine (with proof) all possibilities for $H_{2}(X)$.

Let $X$ be a CW complex with exactly one $k{+}1$-cell. Prove that if $H_{k}(X; \mathbb {Z})$ is a non-trivial finite group then $H_{k+1}(X; \mathbb {Z}) = 0$.

Let

and

Let $X = A \cup B$. Compute the relative homology groups $H_{i}(X, X - (0,0,0,0))$ for all $i$.

Let $X$ be a path connected and locally path connected space and let $x \in X$. Let $Y = S^{1} \times S^{1} \times S^{2}$. Show that if $\pi_{1}(X, x)$ is finite, then any continuous map $f : X \to Y$ induces the trivial map $f_{*} : H_{i}(X, \mathbb {Z}) \to H_{i}(Y, \mathbb {Z})$ for all $i$ different from 0 and 2.

Call a commutative diagram of abelian groups

``exact'' if the sequence of groups and homomorphisms