An Additivity Theorem for the Interchange of E_n Structures
An Additivity Theorem for the Interchange of E_n Structures
复制标题
E_n结构互换的加性定理
DOI：
10.1016/j.aim.2014.10.020
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
R. Vogt
中科院分区：
文献类型：
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作者：
Z. Fiedorowicz;R. Vogt
设A和B是运算子，X是具有A-代数和B-代数结构的对象.如果A-结构的每个运算α：Xn → X是关于B-结构的同态，反之亦然，则这些结构被称为互换。在这种情况下，组合结构由两个操作数的张量积A B编码。关于A和B，一般来说知之甚少，因为张量积的分析需要解决一个棘手的文字问题。直觉上，人们可能会期望一个Ek-操作数与一个El-操作数（分别编码k重循环空间和l重循环空间的乘法结构）的张量积应该是一个Ek + l-操作数。然而，对于这个天真的猜想，有一些简单的反例。在本文中，我们基本上解决了字的零元，一元和二元操作的张量积任意拓扑运算的问题，并表明，张量积的一个coherant E k-操作与coherant E l-操作是一个E k+ l-操作。因此，如果A i是Ek i操作数，其中i= 1，2，...，n，则A 1 <$...<$An至少是一个Ek 1+...+ k n操作数，即存在一个Ek 1+...+ k n-操作数C和一个操作数C→ A 1 <$...<$An的映射。
Let A and B be operads and let X be an object with an A-algebra and a B-algebra structure. These structures are said to interchange if each operation α: X n→ X of the A-structure is a homomorphism with respect to the B-structure and vice versa. In this case the combined structure is codified by the tensor product A⊗ B of the two operads. There is not much known about A⊗ B in general, because the analysis of the tensor product requires the solution of a tricky word problem. Intuitively one might expect that the tensor product of an E k-operad with an E l-operad (which encode the multiplicative structures of k-fold, respectively l-fold loop spaces) ought to be an E k+ l-operad. However, there are easy counterexamples to this naive conjecture. In this paper we essentially solve the word problem for the nullary, unary, and binary operations of the tensor product of arbitrary topological operads and show that the tensor product of a cofibrant E k-operad with a cofibrant E l-operad is an E k+ l-operad. It follows that if A i are E k i operads for i= 1, 2,…, n, then A 1⊗…⊗ A n is at least an E k 1+…+ k n operad, ie there is an E k 1+…+ k n-operad C and a map of operads C→ A 1⊗…⊗ A n.