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Virtual vector bundles and graded Thom spectra. (English) Zbl 1429.55010

The paper’s primary aim is to develop a framework to construct and analyse graded Thom spectra. The starting point is the (topological) category \(\mathcal{O}\) of the inner product spaces \(\mathbb{R}^n\) for \(n \geqslant 0\) and the linear isometric isomorphisms. The authors apply Quillen’s localisation construction to obtain \(\mathcal{W} = \mathcal{O}^{-1} \mathcal{O}\). The objects of \(\mathcal{W}\) are pairs \((n_1, n_2)\) with morphism spaces given by \[ \mathcal{W}\big( (m_1, m_2), (n_1, n_2) \big) = \big( \mathcal{O}(m_1 + m, n_1) \times \mathcal{O}(m_2 + m, n_2) \big)/ O(m) \] provided there exists a natural number \(m\) such that \(n_1 - m_1 = m = n_2-m_2\). Otherwise, the space is empty.
The category \(\text{Top}^{\mathcal{W}}\) has a monoidal model structure with weak equivalences given by maps which induce weak homotopy equivalences on homotopy colimits. The monoidal product is given by the convolution product using the permutative structure on \(\mathcal{W}\). There is a Quillen equivalence \[ \text{Top}^{\mathcal{W}} \simeq \text{Top}/Gr_{h \mathcal{V}} \] where \(Gr_{h \mathcal{V}}\) is a model for \(BO \times \mathbb{Z}\) constructed from the theory. Hence, \(\text{Top}^{\mathcal{W}}\) is a suitable model for virtual vector bundles. The classical Stiefel manifolds appear as a part of a commutative \(\mathcal{W}\)-space monoid \(V\).
The model category \(\mathcal{W}\) has a useful strong symmetric monoidal functor \(\mathbb{S}\) to orthogonal spectra \(\text{Sp}^{\mathcal{O}}\). The image of \(V\) under this functor is a commutative orthogonal ring spectrum \(MOP\), which is a model of the periodic unoriented cobordism spectrum. The graded Thom space construction is a functor \[ T \colon \text{Top}^{\mathcal{W}}/V \longrightarrow \text{Sp}^{\mathcal{O}}/MOP. \] This construction has good multiplicative properties. That is, for an operad \(\mathcal{D}\) (augmented over the Barratt-Eccles operad) the construction \(T\) gives a functor of \(\mathcal{D}\)-algebras \[ \text{Top}[\mathcal{D}]/Gr_{h \mathcal{V}} \longrightarrow \text{Sp}^{\mathcal{O}}[\mathcal{D}]/MOP. \]
This theory enables the authors to establish a theory of orientations and graded Thom isomorphisms with good multiplicative properties. The paper ends with an application to the analysis of logarithmic structures on commutative ring spectra.

MSC:

55P42 Stable homotopy theory, spectra
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55R25 Sphere bundles and vector bundles in algebraic topology
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