Question 1: $$0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0$$ is an exact short sequence,in order to prove $$\cdots \to H^q (A)\mathop \to \limits^{f^* } H^q (B)\mathop \to \limits^{g^* } H^q (C)\mathop \to \limits^{d^* } H^{q + 1} (A) \to \cdots$$ is an exact long cohomology sequence,we need to prove$${\mathop{\rm Im}\nolimits} f^* = \ker g^* $$,I can prove $$(adsbygoogle = window.adsbygoogle || []).push({});

{\mathop{\rm Im}\nolimits} f^* \subset \ker g^* $$ because $${\mathop{\rm Im}\nolimits} f = \ker g$$,but how to prove $

{\mathop{\rm Im}\nolimits} f^* \supset \ker g^* $? \\

\\

\\

Question 2: What makes the cohomology groups different by digging 2 points of $R^2$ from that of $R^2$ ?How does the closed and exact differential forms change?

Thank you!

the visualized questions are also in the attach files

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# 2 questions on coholomogy groups

**Physics Forums | Science Articles, Homework Help, Discussion**