Infinitary commutativity and fundamental groups of topological monoids

Quinn谱的交换性性质

Commutativity properties of Quinn spectra

本文给出了Quinn“配边型”谱弱等价于交换对称环谱的充分条件，证明了对称签名的幺半变换性质，揭示了拓扑丛Sullivan-Ranacki定向的环映射表示，并提出了对称L理论的新描述。

The homotopy groups of a homotopy group completion

Let $M$ be a topological monoid with homotopy group completion $ΩBM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $π_k (ΩBM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, lea…

Mapping spaces from projective spaces

We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid of the homotopy type of a CW complex and $G'$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the connecting map $G\rightarrow\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(…

The Eckman-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its $Hom$-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is `the same' as a braided monoidal category. In this p…