Formality of the Little $N$-disks Operad (eBook)

Pascal Lambrechts (Autor)

eBook Download: PDF

116 Seiten
American Mathematical Society (Verlag)
978-1-4704-1669-0 (ISBN)

Infos

The little $N$-disks operad, $/mathcal B$, along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint $N$-dimensional disks inside the standard unit disk in $/mathbb{R}^N$ and it was initially conceived for detecting and understanding $N$-fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics. In this paper, the authors develop the details of Kontsevich's proof of the formality of little $N$-disks operad over the field of real numbers. More precisely, one can consider the singular chains $/operatorname{C}_*(/mathcal B; /mathbb{R})$ on $/mathcal B$ as well as the singular homology $/operatorname{H}_*(/mathcal B; /mathbb{R})$ of $/mathcal B$. These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. The authors additionally prove a relative version of the formality for the inclusion of the little $m$-disks operad in the little $N$-disks operad when $N/geq2m+1$.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 1-4704-1669-7 / 1470416697
• ISBN-13: 978-1-4704-1669-0 / 9781470416690