Cohomology Ring of Complement of Hyperplanes in Complex n-space

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SUMMARY

The discussion focuses on determining the cohomology ring $$H^*(\mathbb{C}^n \setminus \bigcup_{k = 1}^m \ker \lambda_k ; G)$$ for a given abelian group ##G##, where ##\lambda_1,\ldots, \lambda_m## are ##\mathbb{C}##-linearly independent linear functionals in complex n-space. The analysis is applicable for dimensions where ##n \ge 1## and ##1 \le m \le n##. Key conclusions highlight the significance of understanding the structure of the complement of hyperplanes in complex geometry and its implications in algebraic topology.

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  • Knowledge of abelian groups and their properties
  • Basic concepts of hyperplane arrangements in complex geometry
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Let ##n \ge 1## and ##1 \le m \le n##. Suppose ##\lambda_1,\ldots, \lambda_m : \mathbb{C}^n \to \mathbb{C}## are ##\mathbb{C}##-linearly independent linear functionals. For each abelian group ##G##, determine the cohomology ring $$H^*(\mathbb{C}^n \setminus \bigcup_{k = 1}^m \ker \lambda_k ; G)$$
 
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