Formality of the Little $N$-disks Operad

The little img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png" -disks operad, img style="width:11px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0042.png" , along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png" -dimensional disks inside the standard unit disk in img style="width:12px;height:11px;margin-right:0.018em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/AMS/Regular/120/0052.png" img style="width:11px;height:9px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/085/004E.png"   and it was initially conceived for detecting and understanding img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png" -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 img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png" -disks operad over the field of real numbers. More precisely, one can consider the singular chains img style="width:12px;height:12px;margin-right:0.059em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0043.png" img style="width:6px;height:6px;margin-right:0.068em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/085/2217.png" img style="width:6px;height:18px;vertical-align:-4px;margin-right:0.059em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png"img style="width:11px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0042.png"img style="width:4px;height:12px;vertical-align:-4px;margin-right:0.08em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/003B.png"img style="width:12px;height:11px;margin-right:0.018em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/AMS/Regular/120/0052.png"img style="width:5px;height:18px;vertical-align:-4px;margin-right:0.099em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png"  on img style="width:11px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0042.png"  as well as the singular homology img style="width:13px;height:12px;margin-right:0.027em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0048.png" img style="width:6px;height:6px;margin-right:0.068em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/085/2217.png" img style="width:6px;height:18px;vertical-align:-4px;margin-right:0.059em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0028.png"img style="width:11px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0042.png"img style="width:4px;height:12px;vertical-align:-4px;margin-right:0.08em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/003B.png"img style="width:12px;height:11px;margin-right:0.018em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/AMS/Regular/120/0052.png"img style="width:5px;height:18px;vertical-align:-4px;margin-right:0.099em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0029.png"  of img style="width:11px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0042.png" . 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 img style="width:15px;height:7px;margin-right:0.022em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/006D.png" -disks operad in the little img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png" -disks operad when img style="width:15px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/004E.png"img style="width:12px;height:14px;vertical-align:-3px;margin-right:0.088em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/2265.png"img style="width:8px;height:12px;margin-right:0.053em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0032.png"img style="width:15px;height:7px;margin-right:0.022em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/006D.png"img style="width:12px;height:12px;vertical-align:-1px;margin-right:0.059em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png"img style="width:8px;height:12px;margin-right:0.076em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png" .