The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a Hopfkbc-valued Brown functor where Hopfkbc is the category of bicommutative Hopf algebras over a field k : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary Hopfkbc-valued homology theory.

中文翻译：

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由布朗函子诱导的一对同伦理论版本的 TQFT

本文的目的是研究由同伦理论版本的射影 TQFT（简称射影 HTQFT）的构造所引发的一些障碍类别。射影 HTQFT 由对称单曲面射影函子给出，其域是指有限 CW 空间的 cospan 范畴而不是 cobordism 范畴。我们构造一对射影 HTQFT，从H○pFķbC值布朗函子，其中H○pFķbC是域上的双交换 Hopf 代数的范畴ķ：布朗函子的空间路径积分和空间路径积分。它们通过与投影表示相关的第二上同调类的类似物来诱导障碍类。在本文中，我们推导了这些障碍类别的一些公式。我们应用这些公式来证明空间和空间路径积分的降维被提升到 HTQFT 的。在另一个应用程序中，我们从普通的H○pFķbC值同调理论。