Smith homomorphisms and mjch^{b} structures

In this article, we answer two questions of Buchanan–McKean [ <italic>KSp-characteristic classes determine Spinh cobordism</italic> , https://arxiv.org/ abs/2312.08209, 2023] about bordism for manifolds with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal s normal p normal i normal n Superscript h"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mi>h</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm {spin}^h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> structures: we establish a Smith isomorphism between the reduced <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal s normal p normal i normal n Superscript h"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm {spin}^{h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bordism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R double-struck upper P Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {RP}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal p normal i normal n Superscript h minus"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>h</mml:mi> <mml:mo> − </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm {pin}^{h-}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bordism, and we provide a geometric explanation for the isomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript 4 k Superscript normal upper S normal p normal i normal n Super Superscript c Baseline circled-times double-struck upper Z left-bracket 1 slash 2 right-bracket approximately-equals normal upper Omega Subscript 4 k Superscript normal upper S normal p normal i normal n Super Superscript h Baseline circled-times double-struck upper Z left-bracket 1 slash 2 right-bracket"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mi>c</mml:mi> </mml:msup> </mml:mrow> </mml:msubsup> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mo> ≅ </mml:mo> <mml:msubsup> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mi>h</mml:mi> </mml:msup> </mml:mrow> </mml:msubsup> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega _{4k}^{\mathrm {Spin}^c}\otimes \mathbb {Z}[1/2] \cong \Omega _{4k}^{\mathrm {Spin}^h}\otimes \mathbb {Z}[1/2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren [ <italic>The Smith fiber sequence and invertible field theories</italic> , https://arxiv.org/abs/2405.04649, 2024], specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms.