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Advanced Topology -- Spring 2013 -- final exam

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Name:_________________________

7 problems, 10 points each

  • Show enough work to justify your answers. Refer to theorems whenever necessary.
  • One page per problem.
  1. 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 $X$.
  2. How do you construct a cochain from a continuous form?
  3. Construct the dual cubical complex of the cubical complex of the figure 8 (the one with 7 edges).
  4. 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.
  5. Using row operations compute the homology (or cohomology) of the one pixel complex: CubicalDisk.png.
  6. 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?
  7. Provide a cubical complex equipped with a metric tensor -- for a hollow cube.

With comments:

  1. 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 $X$. >>>>> The proof is the same as for a convex set -- show that every map is homotopic to the constant equal to $a$.
  2. How do you construct a cochain from a continuous form? >>>>>>> The de Rham map...
  3. Construct the dual cubical complex of the cubical complex of the figure 8 (the one with 7 edges). >>>>>>>> There will be 6 faces, 6 edges, and 2 vertices, and then you'll need to add the boundary cells...
  4. 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. >>>>>>>> (a) will give you $1$s on the horizontal faces and $0$s elsewhere (it's $dxdy$!). (b) is a bit trickier but you only need to compute a few wedge products of the cell involved.
  5. Using row operations compute the homology (or cohomology) of the one pixel complex: CubicalDisk.png. >>>>>>> A similar example: Cohomology of figure 8.
  6. 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? >>>>>> $C_1(K),C_1(L)$ are the chain complexes of these two simple cubical complexes. The computation is then trivial. If you think of this in terms of quotients, you'll see the meaning of the result as a certain kind of homology...
  7. Provide a cubical complex equipped with a metric tensor -- for a hollow cube. >>>>>> The complex is simple: $6$ faces, $12$ edges, and $8$ vertices. The metric tensor will be just numbers assigned to the cells (all $1$s), and to the corners (all $\pi /2$)...