Difference between revisions of "Algebraic Topology II - 2019"
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1. Define Cohomology with compact supports. | 1. Define Cohomology with compact supports. | ||
− | 2. Calculate | + | 2. Calculate \( H_c^*(\mathbb{R}^n;G) \). |
− | 3. Does a map f:X | + | 3. Does a map \(f:X \rightarrow Y\) induce a map \(f^* : H_c^*(Y;G) \rightarrow H_c^*(Y;G)\) ? |
4. Do homotopic spaces have the same cohomologies with cpt support? | 4. Do homotopic spaces have the same cohomologies with cpt support? | ||
Line 13: | Line 13: | ||
6. State the Poincaré duality theorem for (not neccesarrily cpt) manifolds. | 6. State the Poincaré duality theorem for (not neccesarrily cpt) manifolds. | ||
− | 7. Show that H_c^i(X | + | 7. Show that \(H_c^i(X \times \mathbb{R};G) = H_c^{i-1}(X;G) \). |
Revision as of 08:18, 26 August 2019
Joel, 26.08.2019 9:30-10:00
1. Define Cohomology with compact supports.
2. Calculate \( H_c^*(\mathbb{R}^n;G) \).
3. Does a map \(f:X \rightarrow Y\) induce a map \(f^* : H_c^*(Y;G) \rightarrow H_c^*(Y;G)\) ?
4. Do homotopic spaces have the same cohomologies with cpt support?
5. Is the cup product well defined on cohomology with cpt support?
6. State the Poincaré duality theorem for (not neccesarrily cpt) manifolds.
7. Show that \(H_c^i(X \times \mathbb{R};G) = H_c^{i-1}(X;G) \).