## Tesi etd-07032009-204654 |

Thesis type

Tesi di laurea specialistica

Author

D'ANTONIO, GIACOMO

URN

etd-07032009-204654

Title

Orlik-Solomon algebras and Hyperplane Arrangements

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Gaiffi, Giovanni

Parole chiave

- configuration spaces
- braids
- braid arrangement
- extended action
- representations symmetric group
- orlik-solomon algebra
- hyperplane arrangements

Data inizio appello

24/07/2009;

Consultabilità

Completa

Riassunto analitico

This work fits into the topic of hyperplane arrangements; this is a widely studied subject involving many different areas of mathematics (combinatorics, commutative algebra, topology, group theory, representation theory, etc..).

To each hyperplane arrangement are associated certain combinatorial data; it turns out that the cohomology algebra of the complement of a complex hyperplane arrangement is uniquely determined by these combinatorial data. A famous isomorphic model for this algebra is the so-called Orlik-Solomon algebra. In the first part of the thesis

(chapters 1-3) we study the topology of the complement of a complex hyperplane arrangement; in chapter 1 we review in detail the general theory of hyperplane arrangements, in chapter 2 we specialize to the braid arrangement and prove

some explicit results and in chapter 3 we prove the isomorphism between the Orlik-Solomon algebra and the cohomology algebra of the complement.

In the second part of the thesis (chapters 4 and 5) we study some cohomology representations of the symmetric group $S_n$. In chapter 4 we study the action of $S_n$ on the complement of the braid arrangement (the so-called pure

braid space); we introduce an extended action of the symmetric group $S_{n+1}$ on $n+1$ elements that allows for a simple computation of the character of the action of $S_n$. We also study some properties of $H^*(M(B_n); C)$ as graded $S_n$-module. In chapter 5 we study the action of $S_n$ on the configuration space of $n$ points in $R^d$; it turns out that an argument similar to the case of the pure braid space applies. In particular we build an extended $S_{n+1}$-action and use it to study in detail the action of $S_n$.

In the appendices we prove some fundamental results in algebraic topology and group actions that are important for the preceding discussion: the thom isomorphism and the theorem of transfer.

To each hyperplane arrangement are associated certain combinatorial data; it turns out that the cohomology algebra of the complement of a complex hyperplane arrangement is uniquely determined by these combinatorial data. A famous isomorphic model for this algebra is the so-called Orlik-Solomon algebra. In the first part of the thesis

(chapters 1-3) we study the topology of the complement of a complex hyperplane arrangement; in chapter 1 we review in detail the general theory of hyperplane arrangements, in chapter 2 we specialize to the braid arrangement and prove

some explicit results and in chapter 3 we prove the isomorphism between the Orlik-Solomon algebra and the cohomology algebra of the complement.

In the second part of the thesis (chapters 4 and 5) we study some cohomology representations of the symmetric group $S_n$. In chapter 4 we study the action of $S_n$ on the complement of the braid arrangement (the so-called pure

braid space); we introduce an extended action of the symmetric group $S_{n+1}$ on $n+1$ elements that allows for a simple computation of the character of the action of $S_n$. We also study some properties of $H^*(M(B_n); C)$ as graded $S_n$-module. In chapter 5 we study the action of $S_n$ on the configuration space of $n$ points in $R^d$; it turns out that an argument similar to the case of the pure braid space applies. In particular we build an extended $S_{n+1}$-action and use it to study in detail the action of $S_n$.

In the appendices we prove some fundamental results in algebraic topology and group actions that are important for the preceding discussion: the thom isomorphism and the theorem of transfer.

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