Multivariable cochain operations and little 𝑛-cubes

Abstract

In this paper we construct a small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">E_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> chain operad <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which acts naturally on the normalized cochains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript asterisk Baseline upper X"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S^*X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a topological space. We also construct, for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a suboperad <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {S}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is quasi-isomorphic to the normalized singular chains of the little <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -cubes operad. The case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture.