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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 6th 2020

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTime6 days ago

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 $M \mathrm{U} / \mathbb{S}$ aka $\Sigma \overline {M \mathrm{U}}$.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTime6 days ago

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTime6 days ago

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTime5 days ago
• (edited 5 days ago)

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

$0 \to \Omega^{\mathrm{U}}_{2n + 2} \overset{i}{\longrightarrow} \Omega^{\mathrm{U},fr}_{2n + 2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{2n + 1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N}$

namely from this long exact sequence:

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