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# Advanced Topology: exercises

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Jump to navigationJump to search- List all antiderivatives of $1/x$ and prove, from the definition, that they are continuous.
- Show that the set of differential forms is a vector space.
- Prove that $\Omega ^k({\bf R})$ is infinite dimensional.
- Form the axiomatic definition of differential form, prove that $\Omega^1(p)=span\{dx,dy\}.$
- Prove that if $\varphi \in \Omega^k$ and $k$ is odd, then $\varphi \wedge \varphi = 0$.
- Compute (a) $x^2dx^1$ on $<1,2,3>$ at $(3,2,1)$; (b) $dx^1 \wedge dx^3 + x^1dx^2 \wedge dx^4$ on the pair $<-1,0,1,1>, <0,-1,0,1>$ at $ (1,0,0,0)$.
- Simplify: $$2dx^1 \wedge dx^3 \wedge dx^2 + 3dx^2 \wedge dx^1 \wedge dx^2 - dx^2 \wedge dx^3 \wedge dx^1 .$$
- Expand and simplify: $$(x^1 dx^2 + x^2 dx^1) \wedge (x^3 dx^1 \wedge dx^2 + x^2 dx^1 \wedge dx^3 + x^1 dx^2 \wedge dx^3).$$
- Show that a form of even degree commutes with any other form.
- Find the dimension of the space of differential forms of a point, $\dim \Omega ^k (p) = ?$
- Provide explicit formulas of the basic forms in $\Omega ^2({\bf R}^3)$, i.e., $dxdy,dydz,dzdx$ in terms of the axiomatic definition. Suggest a formula for the general case of $\Omega ^k({\bf R}^n)$.
- Compute the exterior derivative of $xydx \wedge dy + yz dy \wedge dz + dz \wedge dx$.
- To solve the differential equation $$\frac{dy}{dx} = \frac{f(x)}{g(y)}$$ one uses "separation of variables" $$g(y)dy = f(x)dx.$$ Explain the relation between these two equations.
- Compute $df$, where $f=x^1+2x^2+...+nx^n$, on $<1,-1,...,(-1)^{n-1}>$ at $(1,2,...,1)$.
- Prove that in ${\bf R}^n$, $$df^1 \wedge ... \wedge df^n(x) = \det \frac{\partial f^i}{\partial x^j}(x)dx^1 \wedge ... \wedge dx^n.$$
- Write the form $df$, where $f(x) = (x^1) + (x^2)^2 + ... + (x^n)^n$, as a combination of $dx^1,...,dx^n$.
- Suppose $\alpha, \beta, \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$.
- A set $Y \subset {\bf R}^n$ is called star-shaped if there is $a\in Y$ such that for any $x \in Y$ the segment from $x$ to $a$ lies entirely within $Y$. Prove that any two continuous functions $f,g:\rightarrow Y$ are homotopic.
- Given that the 2nd de Rham cohomology of the sphere is $1$-dimensional: $\Omega^2_{dR}({\bf S}^2)={\bf R}$, list the cohomology classes of the sphere. No proof required.
- Prove that the boundary of a boundary is zero, in dimension $2$.
- Show that any $(k−1)$-dimensional face of a $(k+1)$-dimensional cube $Q$ is a common face of exactly two $k$-dimensional faces of $Q$.
- Two graphs $(V_1, E_1)$ and $(V_2, E_2)$ are equivalent if there exists a bijection $f : V_1 \rightarrow V_2$ such that, for any $u, v \in V_1$, we have $(u, v) \in E_1$ if and only if $(f(u), f(v)) \in E_2$.
- Prove that any graph that is a tree can be represented as a one-dimensional cubical complex $X$ in the sense that it is equivalent to the graph $(K_0(X), E(X))$, where $E(X)$ are pairs of vertices of edges in $K_0(X)$. Give an example of a graph that cannot be represented by a one-dimensional cubical complex.
- Let $Q_d$ be a cube and let $X$ be its one-dimensional skeleton, that is, the union of all edges of $Q_d$. For $d = 2, 3, 4, 5, 6$, determine the number of vertices of $Q_d$ and the number of edges of $Q_d$ (note that $X$ has the same vertices and edges as $Q_d$).
- Use discrete differential forms to describe population growth.
- For discrete forms in ${\bf R}^3$ compute: (a) $dx \wedge dy$; (b) $dx \wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
- Compute the exterior derivative of the following form:
- Simplify: $(a)\, 2dxdydz+3dydxdz-dydxdz,(b)(x^1dx^2+x^2dx^2)(x^2dx^1dx^2+x^2dx^1dx^3+x^1dx^2dx^3).$
- Prove that $dx_{i_1}...dx_{i_{k}}=0$ if and only if $i_{s}=i_{r}$ for some $s\ne r$.
- (a) Define discrete differential forms for the triangular grid on the plane. (b) Define the exterior derivative of $0$- and $1$-forms and prove that $dd=0$.
- (a) Describe the vector spaces of discrete differential forms for the complex:
- Evaluate the cohomology of the complex above (just the answer).
- (a) Given a smooth function $F:{\bf R}\rightarrow{\bf R}$, what happens to the forms $1,dx$ under $F$? Hint: you can think of $F$ as a change of variables. (b) Given a smooth function $F:{\bf R}^2\rightarrow{\bf R}^2$, what happens to the forms $1,dx,dy,dxdy$ under $F$? (c) Given a function $F:R_1\rightarrow R_2$, what kind of function does $F$ generate between $\Omega^{k}(R_1)$ and $\Omega^{k}(R_2)$?
- Prove that the following is or is not a (smooth) $2$-manifold: (a) point, (b) circle, (c) torus, (d) ${\bf R}^3$.
- Evaluate $\int_{C}dx+dy$, where $C$ is the upper half of the unit circle oriented counter-clockwise.
- Suppose $d:\Omega ^{k}({\bf R}^2)\rightarrow \Omega ^{k+1}({\bf R}^2)$ is the exterior derivative. Suppose $D:\Omega ^{k}(R^2)\rightarrow\Omega ^{k+1}({\bf R}^2)$ is an "alternative" exterior derivative; it satisfies the following: (a) the linearity, (b) the Product Rule, and (c) $DD=0$. Assume that $D=d$ on $\Omega^0({\bf R}^2)$ and prove that $D=d$ on $\Omega^1({\bf R}^2)$.
- Define wedge product of forms. Give examples. State and prove the skew commutativity property.
- Prove $dd=0$ for smooth $0$-forms.
- Define de Rham cohomology. Give examples.
- Define simple connectedness. Give examples. State the theorem about the closed and exact forms on a simply connected region in the plane.
- Define discrete forms in the plane. Give $dx,dy,dxdy.$
- Define the Hodge $\star$ operator for discrete forms on the plane. Give examples.
- Compute $d(Fdx+Gdy)$ for discrete forms in the plane.
- Define smooth manifolds. Give examples and non-examples.
- Define orientation of $0$- and $1$-manifolds.
- How do you construct a cochain from a smooth form?
- Prove that $\int _{-C} \psi = \int _C \psi$ for $1$-forms.
- State the general Stokes theorem. Give applications.
- Draw the Mobius band in the 3D grid.
- Using row operations compute the cohomology of the following complexes:
- Show that the set of differential forms is a vector space.
- Compute (a) $x^2dx^1$ on $<1,2,3>$ at $(3,2,1)$; (b) $dx^1 \wedge dx^3 + x^1dx^2 \wedge dx^4$ on the pair $<-1,0,1,1>, <0,-1,0,1>$ at $ (1,0,0,0)$.
- Simplify: $$2dx^1 \wedge dx^3 \wedge dx^2 + 3dx^2 \wedge dx^1 \wedge dx^2 - dx^2 \wedge dx^3 \wedge dx^1 .$$
- Compute the exterior derivative of $xydx \wedge dy + yz dy \wedge dz + dz \wedge dx$.
- Suppose $\alpha, \beta, \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$.
- Given that the 2nd de Rham cohomology of $3$-space with a point removed is $1$-dimensional: $\Omega^2_{dR}({\bf R}^3-\{0\})={\bf R}$, list its cohomology classes. No proof required.
- Prove that the boundary of a boundary is zero, in dimension $2$ (for cubical complexes).
- For discrete forms in ${\bf R}^3$ compute: (a) $dx \wedge dy$; (b) $dx \wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
- Compute the exterior derivative of the following form:
- (a) Describe the vector spaces of discrete differential forms for the complex: (b) Evaluate the cohomology of the complex above (just the answer).
- A set $Y \subset {\bf R}^n$ is called star-shaped if there is $a\in Y$ such that for any $x \in Y$ the segment from $x$ to $a$ lies entirely within $Y$. Prove that any two continuous functions $f,g:X\rightarrow Y$ are homotopic, for any $X$.
- How do you construct a cochain from a continuous form?
- Construct the dual cubical complex of the cubical complex of the figure 8 (the one with 7 edges).
- For discrete forms in ${\bf R}^3$ compute: (a) $dx \wedge dy$; (b) $dx \wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
- Using row operations compute the homology (or cohomology) of the one pixel complex: .
- Let $K$ be the cubical complex of $[0,1]$ and let $L$ be the complex of $\{0,1\}$. Compute $C_1(K)/C_1(L)$. What is the meaning of what you've found?
- Provide a cubical complex equipped with a metric tensor -- for a hollow cube.
- State the continuous Poincare Lemma along with all necessary definitions.
- Give examples of forms that are closed but not exact, both continuous and discrete.
- Explain the Fundamental Correspondence (between forms and functions/vector fields).
- Assuming that the Hodge duality of cells is already given define Hodge duality of forms.