Difference between revisions of "Algebraic Topology II - 2019"

Revision as of 08:18, 26 August 2019

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 \(H_c^i(X \times \mathbb{R};G) = H_c^{i-1}(X;G) \).

@@ Line 3: / Line 3: @@
 . Define Cohomology with compact supports.
-. Calculate Cohomology with compact supports of R^n.
+. Calculate \( H_c^*(\mathbb{R}^n;G) \).
-. Does a map f:X -> Y induce a map on the cohomologies with cpt support?
+. Does a map \(f:X \rightarrow Y\) induce a map \(f^* : H_c^*(Y;G) \rightarrow H_c^*(Y;G)\) ?
 . Do homotopic spaces have the same cohomologies with cpt support?
@@ Line 13: / Line 13: @@
 . State the Poincaré duality theorem for (not neccesarrily cpt) manifolds.
-. Show that H_c^i(X x R;G)=H_c^i-1(X;G)
+. Show that \(H_c^i(X \times \mathbb{R};G) = H_c^{i-1}(X;G) \).