Abstract
By results of Löffler and Comezaña, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S1×⋯×S1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.
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Hanke, B. Geometric versus homotopy theoretic equivariant bordism. Math. Ann. 332, 677–696 (2005). https://doi.org/10.1007/s00208-005-0648-0
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DOI: https://doi.org/10.1007/s00208-005-0648-0