Dyer-Lashof operations and bordism

We introduce external geometric operations in unoriented bordism and examine their behavior. In particular they satisfy a Cartan type formula. We then show that these operations cover the Dyer-Lashof operations in homology with z₂ coefficients. The bordism operations are used with standard characteristic number techniques to derive a formula for the action of the Dyer-Lashof operations on H*(BO;Z₂).
In the third chapter we answer a question posed by Iberkleid in Splitting the Tangent Bundle, Trans. Amer. Math. Soc., 191 (1974), pages 53-59. Specifically, we exhibit examples of odd-dimensional, orientable, non-hounding, line-element parallelizable manifolds which represent indecomposable elements in the torsion part of the oriented bordism ring.