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Math4202 Topology II Exam 1 Practice

In the following, please provide complete proof of the statements and the answers you give. The total score is 25 points.

Problem 1

• (2 points) State the definition of a topological manifold.

A topological manifold is a topological space that satisfies the following:

1. It is Hausdorff
2. It has a countable basis
3. Each point of x of X has a neighborhood that is homeomorphic to an open subset of \mathbb{R}^m.

• (2 points) Prove that real projective space \mathbb{R}P^2 is a manifold.

Let \mathbb{R}P^2=\mathbb{R}^3/\sim where (x,y,z)\sim(x',y',z') if \lambda(x,y,z)=(x',y',z') for some \lambda\in \mathbb{R},\lambda\neq 0.

1. It is Hausdorff since \mathbb{R}^3 is Hausdorff, subspace of Hausdorff space is Hausdorff.
2. It has a countable basis since \mathbb{R}^3 has a countable basis, subspace of countable basis has countable basis.
3. Each point of x of RP^2 has a neighborhood that is homeomorphic to an open subset of \mathbb{R}^3. Let p be an arbitrary point in RP^2, Consider the projection on to the tangent plane of p defined as \mathbb{R}P^2\to \mathbb{R}^2.

Solution on class

Consider \mathbb{R} P^n be the lines in \mathbb{R}^{n+1} through the origin.

\mathbb{R}P^n=\{v\neq 0|v\in \mathbb{R}^{n+1}\}/\sim

where a\sim b if there exists \lambda\in \mathbb{R},\lambda\neq 0 such that \lambda a=b.

S^n=\{v\in \mathbb{R}^{n+1}|||v||=1\}

First we test the local euclidean structure.

Consider the hemisphere cap U_{1,+}=\{(x_1,\dots,x_{n+1})|x_1>0\}, note that this cap induce a quotient mapping to some open set of \mathbb{R}P^n

Note that the cap U_{1,+} is local euclidean by the bijective projection map to \mathbb{R}^n (x_1,\dots,x_{n+1})\mapsto(x_2,\dots,x_{n+1}).

And with U_{1,-},U_{2,+},U_{2,-},\dots,U_{n,+},U_{n,-} we can construct a open cover of \mathbb{R}P^n. Since for any of the point in \mathbb{R} P^n we can have some non-zero coordinates that projects to S^n and we can build such cap.

Second we show the second countability.

Take the cap with rational coordinates, and this creates a countable basis.

Third we prove the Hausdorff property.

Consider x=(x_1,\dots,x_{n+1})\in \mathbb{R}P^n, y=(y_1,\dots,y_{n+1})\in \mathbb{R}P^n.

• (2 points) Find a 2-1 covering space of RP^2.

Take \mathbb{R}P^2\to S^2 with quotient topology where v\sim -v.

Problem 2

• (2 points) State the definition of a CW complex.

Let X_0 be arbitrary set of points, and X_n be a CW complex defined by X_n=\{(e_\alpha^n,\varphi_\alpha)|\varphi_\alpha: \partial e_\alpha^n\to X_{n-1}\}=(\sqcup_{\alpha\in A}e_\alpha^n)\sqcup X_{n-1}

• (4 points) Describe a CW complex homeomorphic to the 2-torus.

Take two points a,b, connect a,b with two lines, and add a with a circle connecting to itself, b with a circle connecting to itself. Then wrap a 2-cell on that.

Problem 3

• (2 points) State the definition of the fundamental group of a topological space X relative to x_0 \in X.

The fundamental group of X relative to x_0 is the group of all continuous paths from x_0 to x_0 under path homotopy equivalence.

• (4 points) Compute the fundamental group of R^n relative to the origin.

The fundamental group of R^n relative to the origin is the trivial group.

Problem 4

• (2 points) Give a pair of spaces that are homotopic equivalent, but not homeomorphic.

\mathbb{R} and one point set is homotopic equivalent, (using contraction), but not homeomorphic.

• (4 points) Let A be a subspace of R^n, and h : (A, a_0) \to (Y, y_0). Show that if h is extendable to a continuous map of R^n into Y, then h_* : \pi_1(A, a_0) \to \pi_1(Y, y_0) is the trivial homomorphism (the homomorphism that maps everything to the identity element).

Since h is extendable to a continuous map of \R^n into Y, consider the continuous function H:(\R^n, x_0)\to (Y,y_0), with H|_{A}(f)=h(f).

Note that the inclusion map i:(A,x_0)\to (\R^n,x_0) induces i_* gives a homomorphism, therefore H\circ i=h is a homomorphism. Then h_*=H_*\circ i_*. where \pi_1(\R^n,x_0) is trivial since \R^n is contractible.

Thus H_* is the trivial homomorphism. Therefore h_* is the trivial homomorphism.