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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 6th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something, on Conner-Floyd's $(U,fr)$-bordism theory <a href="https://ncatlab.org/nlab/revision/MUFr/1">v1</a>, <a href="https://ncatlab.org/nlab/show/MUFr">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTime6 days ago
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn the section on the representing spectrum ([here](https://ncatlab.org/nlab/show/MUFr#RepresentingSpectrum)) 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 $\Sigma \overline {M \mathrm{U}}$. <a href="https://ncatlab.org/nlab/revision/diff/MUFr/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/MUFr/5">v5</a>, <a href="https://ncatlab.org/nlab/show/MUFr">current</a>

   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 $\Sigma \overline {M \mathrm{U}}$.

   diff, v5, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTime6 days ago
   • PermaLink
   Author: Urs
   Format: MarkdownItexas a followup, I added the observation ([here](https://ncatlab.org/nlab/show/MUFr#BoundaryMorphism)) 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 <a href="https://ncatlab.org/nlab/revision/diff/MUFr/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/MUFr/6">v6</a>, <a href="https://ncatlab.org/nlab/show/MUFr">current</a>

   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

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTime6 days ago
   • PermaLink
   Author: Urs
   Format: MarkdownItexand now I used this to give a complete, abstract and quick proof ([here](https://ncatlab.org/nlab/show/MUFr#BoundaryOperationFromFrBordToUFrBordIsSurjective)), by inspection of one big homotopy-pasting diagram, that the boundary operation is surjective. <a href="https://ncatlab.org/nlab/revision/diff/MUFr/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/MUFr/6">v6</a>, <a href="https://ncatlab.org/nlab/show/MUFr">current</a>

   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

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTime5 days ago
   • (edited 5 days ago)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave added ([here](https://ncatlab.org/nlab/show/MUFr#AShortExactSequenceOfUFrBordismRings)) 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: $$ \array{ \overset{ \mathclap{ \color{darkblue} pure \; torsion } }{ \overbrace{ \pi_{2d+2} \big( \mathbb{S} \big) } } & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & \overset{ \mathclap{ \color{darkblue} free \; abelian } }{ \overbrace{ \pi_{2d+2} \big( M\mathrm{U} \big) } } & \overset{ }{\longrightarrow} & \pi_{2d+2} \big( M\mathrm{U}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{2d+1}\big(\mathbb{S}\big) &\longrightarrow& \overset{ \color{darkblue} trivial }{ \overbrace{ \pi_{2d+1}\big(M\mathrm{U}\big) } } \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{2d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{U}}_{2d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{U},fr)}_{2d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{2d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & 0 } $$ <a href="https://ncatlab.org/nlab/revision/diff/MUFr/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/MUFr/10">v10</a>, <a href="https://ncatlab.org/nlab/show/MUFr">current</a>

   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:

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

   diff, v10, current