Real Projective Spaces

1. Show that \(\mathbb {R}P^2\) has a 2-fold cover which is \(S^2\).
2. Show that the Klein bottle has a 2-fold cover which is a torus.

Construct a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to calculate the cohomology groups \(H^i(\mathbb {R}P^2;\mathbb {Z}_2)\) and the cup product structure.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute the homology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}/2\).

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). For each cell define its attaching map.

Let \(X\) be a connected cell complex and \(f:\mathbb {R}P^{2n}\to X\) a covering map from the real projective space of dimension \(2n\). Show that \(f\) is a homeo- morphism.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). For each cell define its attaching map.

State the homotopy lifting property of covering maps and use it to prove that the covering map \(S^n\to \mathbb {R}\mathrm {P}^n\) from the \(n\)-sphere to the real projective \(n\)-space is not nullhomotopic for any \(n\ge 1\).

Prove that \(\mathbb {R}\mathrm {P}^2\) is not orientable.

Let \(X\) be the wedge sum of two copies of \(\mathbb {R}\mathrm {P}^2\). Show that there is only one connected degree three cover of \(X\) and determine if it is regular or irregular.

Construct a \(\Delta \)-complex structure for \(\mathbb {R}\mathrm {P}^2\) and use it to calculate the homology, cohomology and cup product structure for \(\mathbb {R}\mathrm {P}^2\) with \(\mathbb {Z}_2\)-coefficients.

Let \(a:S^2\to S^2\) be the antipodal map \(a(x)=-x\) and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a two form on \(\mathbb {R}\mathrm {P}^2\).

1. Prove that \(\int _{S^2}p^*\omega =0\). [Hint: \(p^*\omega \) is \(a\)-invariant.]
2. Prove that there is a one form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(d\eta =\omega \). You are allowed to use the fact that if \(\beta \) is a two-form on \(S^2\) and \(\int _{S^2}\beta =0\) then \(\beta \) is exact, but you are not allowed to use the de Rham Theorem.
3. Prove that \(\mathbb {R}\mathrm {P}^2\) is not orientable.

Let \(n\ge 1\) be a positive integer.

1. Give an elementary description of the universal cover \(X\to \mathbb {R}\mathrm {P}^n\) of \(\mathbb {R}\mathrm {P}^n\).
2. Describe a CW structure on \(\mathbb {R}\mathrm {P}^n\).
3. Compute the cohomology ring \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\) and the map (from (a)) \(H^*(\mathbb {R}\mathrm {P}^n;\mathbb {Z}/2)\to H^*(X;\mathbb {Z}/2)\).

Let \(\mathbf {P}^n(\mathbb {R})\) be \(n\)-dimensional real projective space. Use an explicit collection of charts to prove that \(\mathbf {P}^n(\mathbb {R})\) is a smooth manifold.

Let \(n,k\in \N \) with \(n\ge 2\). Suppose \(f:\mathbb {R}\mathrm {P}^n\to T^k\) is continuous. Determine \[ f_*:H_m(\mathbb {R}\mathrm {P}^n)\to H_m(T^k) \] for all \(m\).

Construct a \(\Delta \)-complex structure on \(\mathbb {R}\mathrm {P}^3\) and use it to compute the cohomology rings of \(\mathbb {R}\mathrm {P}^3\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Let \(a:S^2\to S^2\) be the antipodal map on the 2-sphere: \(a(x)=-x\), and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a 2-form on \(\mathbb {R}\mathrm {P}^2\).

1. Prove that \(\int _{S^2}p^*\omega =0\). Hint: \(p^*\omega \) is \(a\)-invariant.
2. Prove that there is a 1-form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(\omega =d\eta \). You are allowed to use the fact that if \(\zeta \) is a 2-form on \(S^2\) and \(\int _{S^2}\zeta =0\) then \(\zeta \) is exact, but you are not allowed to use de Rham’s theorem.

What is the universal cover of \(\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\)?

Describe a cell structure on real projective space \(\mathbb {R}\mathrm {P}^n\) and use it to compute all the cohomology groups \(H^i(\mathbb {R}\mathrm {P}^n,\mathbb {Z})\) and \(H^i(\mathbb {R}\mathrm {P}^n,\mathbb {Z}/2)\).

Find all homology and cohomology groups (with integer coefficients) for the following three spaces: \(\mathbb {R}\mathrm {P}^3\), \(\mathbb {C}\mathrm {P}^3\), and \(S^1\vee S^3\).

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute \(H_i(\mathbb {R}P^n;\mathbb {Z}/2)\) for \(i\ge 0\).

Show that the degree of every map \(\mathbb {C}P^2\to \mathbb {C}P^2\) of the complex projective plane to itself is nonnegative.

Prove that there is no map \(h:\mathbb {R}\mathrm {P}^3\to \mathbb {R}\mathrm {P}^2\) that induces an isomorphism between fundamental groups. Hint: use cup products with \(\mathbb {Z}/2\mathbb {Z}\) coefficients. State all theorems you are using.

Let \(X=\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\). Give an example of an irregular covering space \(\widetilde {X}\to X\). You should draw the picture of the 1-skeleton of \(\widetilde {X}\) (make sure 2-cells lift).

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give a proof by induction on \(n\) using (a). State any theorems you are using.
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give an inductive proof. You may use Poincaré duality.

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n)\) (integral coefficients)? Give a proof by induction on \(n\) using (a).
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n)\)? Give an inductive proof. You may use Poincaré duality.

Show that \(S^3\) (the 3-sphere) is the covering space of a manifold \(M\) which is not diffeomorphic to \(\mathbb {R}\mathrm {P}^3\). (Hint: find an appropriate finite group action on \(S^3\).)

Construct a \(\Delta \)-complex structure on \(\mathbb {R}\mathrm {P}^2\) and use it to calculate the homology and cohomology of of \(\mathbb {R}\mathrm {P}^2\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Let \(f:\mathbb {R}P^3\to \mathbb {R}P^3\) be defined by \[ f([x_0:x_1:x_2:x_3])=[x_0^2:x_1^2:x_2^2:x_3^2]. \] Compute the degree and the Lefschetz number of \(f\).

Find a perturbation of the identity map \(\mathbb {R}P^3\to \mathbb {R}P^3\) which is a Lefschetz map and compute its Lefschetz number.

Give a proof that \(\mathbb {R}P^2\) is nonorientable.

Define carefully a CW structure on \(\mathbb {R}P^n\) (you don’t have to prove it here), and use it to compute \(H_*(\mathbb {R}P^n;\mathbb {Q})\) and \(H_*(\mathbb {R}P^n;\mathbb {Z}/2)\).

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point.

1. Let \(a:S^n\to S^n\) be the antipodal map. Suppose that \(\omega \) is a smooth form on \(S^n\) such that \(a^*\omega =\omega \). Prove that if \(\omega \) is exact, then there is a smooth form \(\eta \) with \(\omega =d\eta \) and \(a^*\eta =\eta \).
2. Use (a) and the fact that \(H^k_{\mathrm {DR}}(S^n)=0\) for \(0<k<n\) to deduce that \(H^k_{\mathrm {DR}}(\mathbb {R}P^n)=0\) for \(0<k<n\).

Is there a smooth embedding of \(\mathbb {R}P^2\) into an orientable 3-manifold? Provide a proof or an example.

Carefully construct a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to compute the homology groups of \(\mathbb {R}P^2\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Prove that \(\mathbb {R}P^3\) and \(\mathbb {R}P^2\vee S^3\) are not homotopy equivalent. (Hint: Cup products.)

1. Let \(X\) be a CW-complex and \(A\) a sub-complex. Assume that all cells of \(X-A\) have dimension \(>n\). If \(X\) retracts to \(A\) show that \(\pi _n(X)=\pi _n(A)\).
2. Show that \(\mathbb {R}P^n\) does not retract to \(\mathbb {R}P^k\) if \(k<n\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point. (Hint: Work in local coordinates, in standard charts \(U_0,U_1,U_2\) where \(U_0=\{x\ne 0\}\) etc.)

Find all connected covering spaces of \(\mathbb {R}P^2\vee S^1\) of degree less than three. Which of these coverings are regular?

Show that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point.

By \(a:S^n\to S^n\) denote the antipodal map. Then the space \(A^p(S^n)\) of smooth \(p\)-forms on \(S^n\) breaks up into the \(+1\) and the \(-1\) eigenspace of \(a^*\), i.e. \[ A^p(S^n)=A^p_+(S^n)\oplus A^p_-(S^n) \] where \[ A^p_\epsilon (S^n)=\{\omega \in A^p(S^n)\mid a^*(\omega )=\epsilon \omega \}. \] Show that if \(\omega \) is an exact \(p\)-form and belongs to the \(+1\)-eigenspace, then \(\omega =d\eta \) for some \((p-1)\)-form \(\eta \) in the \(+1\)-eigenspace. What can you conclude about DeRham cohomology \(H^i(\mathbb {R}P^n)\) from this?

Prove that \(\mathbb {R}P^3\) and \(\mathbb {R}P^2\vee S^3\) are not homotopy equivalent. (Hint: Cup products.)

Construct carefully a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to compute homology groups of \(\mathbb {R}P^2\) with \(\mathbb {Z}\) and with \(\mathbb {Z}_2\) coefficients.

A. Find an explicit CW complex structure for the complex projective space \(\mathbb {C}P^3\), and compute the singular homology groups (over \(\mathbb {Z}\)) using cellular homology.

B. Find an explicit CW complex structure for \(S^2\times S^4\), and compute the singular homology groups using cellular homology.

A. Compute the DeRham cohomology (or if you prefer, the singular homology) of \((S^2\times S^2)\#\mathbb {C}P^2\), where, \(M\#N\) denotes the connected sum of the manifolds \(M\) and \(N\).

B. Let \(p\) be a positive integer. Compute the singular homology with coefficients in \(\mathbb {Z}_p\) of real projective 3-space (\(\mathbb {Z}_p\) being the cyclic group of order \(p\)). That is, find \(H_k(\mathbb {R}P^3,\mathbb {Z}_p)\) for \(k\ge 0\).

Classify all connected covering spaces of \(\mathbb {R}P^2\vee \mathbb {R}P^2\) up to equivalence.

Find the singular homology groups \(H_*(\mathbb {R}P^5;\mathbb {Z})\).

Show that \(\pi _1(\mathbb {R}P^2\#\mathbb {R}P^2\#\mathbb {R}P^2)\) is infinite.

True or false: There is no orientable smooth 3-dimensional manifold \(M\) and a smooth embedding \(\mathbb {R}P^2\to M\). Justify your answer.

True or false: Any closed differential form of degree \(\ge 1\) on \(\mathbb {R}P^2\times \mathbb {R}P^2\) is exact. Justify your answer.

a. Let \(P:M\to N\) be a regular covering map between connected differentiable manifolds \(M,N\) and let \(M\) be orientable. Prove that \(N\) is orientable if and only if every deck transformation (covering space automorphism) preserves the orientation of \(M\).

b. Prove that \(\mathbb {R}P^n\) is orientable if and only if \(n\) is odd.

Denote by \([X,Y]\) the set of free homotopy classes of maps \(X\to Y\). Show that \([S^2,\mathbb {R}P^2]\) is infinite. In other words, construct a sequence of maps \(f_i:S^2\to \mathbb {R}P^2\), \(i=1,2,3,\ldots \), such that no two are homotopic to each other.

Calculate \(\pi _2(\mathbb {R}P^3)\).

II. Pick four of the following seven questions to answer

Let \(\Sigma \) denote the surface obtained from an annulus by identifying antipodal points of the inner circle and by identifying antipodal points of the outer circle. Compute the fundamental group of \(\Sigma \). Can you identify \(\Sigma \) as a connected sum of tori and projective planes?