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nLab > Latest Changes: MUFr

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2020

    starting something, on Conner-Floyd’s (U,fr)(U,fr)-bordism theory

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2021

    In the section on the representing spectrum (here) only the component spaces had been mentioned (following Conner-Floyd 66). I have added the brief remark that the corresponding spectrum is MU/𝕊M \mathrm{U} / \mathbb{S} aka ΣMU¯\Sigma \overline {M \mathrm{U}}.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2021

    as a followup, I added the observation (here) that this realization of the representing spectrum of M(U,fr)M(\mathrm{U},fr) immediately gives the existence of the boundary cohomology operation to MFrMFr, just by stepping along the induced long cofiber sequence

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 10th 2021

    and now I used this to give a complete, abstract and quick proof (here), by inspection of one big homotopy-pasting diagram, that the boundary operation is surjective.

    diff, v6, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2021
    • (edited Jan 11th 2021)

    Have added (here) the proof of the short exact sequence

    0→Ω 2n+2 U⟶iΩ 2n+2 U,fr⟶∂Ω 2n+1 fr→0,AAAAn∈ℕ 0 \to \Omega^{\mathrm{U}}_{2n + 2} \overset{i}{\longrightarrow} \Omega^{\mathrm{U},fr}_{2n + 2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{2n + 1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N}

    namely from this long exact sequence:

    π 2d+2(𝕊)⏞puretorsion ⟶1 MU π 2d+2(MU)⏞freeabelian ⟶ π 2d+2(MU/𝕊) ⟶∂ π 2d+1(𝕊) ⟶ π 2d+1(MU)⏞trivial ↓ = ↓ = ↓ = ↓ = ↓ = Ω 2d+2 fr ⟶0 Ω 2d+2 U ⟶i Ω 2d+2 (U,fr) ⟶∂ Ω 2d+1 fr ⟶0 0 \array{ \overset{ \mathclap{ \color{darkblue} pure \; torsion } }{ \overbrace{ \pi_{2d+2} \big( \mathbb{S} \big) } } & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & \overset{ \mathclap{ \color{darkblue} free \; abelian } }{ \overbrace{ \pi_{2d+2} \big( M\mathrm{U} \big) } } & \overset{ }{\longrightarrow} & \pi_{2d+2} \big( M\mathrm{U}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{2d+1}\big(\mathbb{S}\big) &\longrightarrow& \overset{ \color{darkblue} trivial }{ \overbrace{ \pi_{2d+1}\big(M\mathrm{U}\big) } } \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{2d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{U}}_{2d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{U},fr)}_{2d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{2d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & 0 }

    diff, v10, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2021

    added pointer to

    which mentions the unit cofiber spectrum MU/𝕊MU/\mathbb{S} explicitly

    diff, v13, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2021

    also:

    diff, v13, current

1 to 7 of 7

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