Highly structured orientations from equivariant Thom spectra

We report significant progress toward establishing an obstruction theory of equivariant Thom spectra with multiplicative structures arising from maps of $E_V$ spaces. Focusing on the Fujii-Landweber Real bordism spectrum, we explain an argument to show that a homotopy ring Real orientation of an $E_{\rho}$-spectrum satisfying strong equivariant evenness and mild additional commutativity condition lifts to an $E_{\rho}$ map.First, we address the foundational issues. We discuss model categories for equivariant $C_2$-spectra indexed on Real inner product spaces and comparisons among them. We explain which foundations are needed for our project and describe the parts that would be original even non-equivariantly.Next, we construct equivalences between several $E_V$ operads, including little disks and Steiner operads. We show that algebras over $E_{V \oplus \Rbb}$ operads can be strictified to monoids in the category of $E_V$-algebras.To compute obstruction groups for maps of $E_V$ algebras, we analyze the spectra of the derived indecomposibles of augmented $E_V$-algebras. We determine that such spectra are $V$-fold desuspensions of a lift of the May delooping machine to the augmented algebras.Next, we discuss the induced map on cohomology for a map between spaces whose $RO(C_2)$-graded cohomology carries obstruction classes for homotopy ring and $E_\rho$ ring maps out of the Real bordism spectrum.Finally, we review the strategies for the proofs of the goal results of our project, explaining how to use the technique of climbing the slice tower of the target to construct highly structured orientations out of Real bordism.