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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2018
    • (edited Mar 18th 2018)

    I have corrected and expanded my note (at 4-sphere: here) of the result of Roig-Saralegi 00, p. 2 on minimal rational dg-models of the following maps over S 3S^3

    S 4 S 3,AAS 4//S 1 S 3,AAS 0 S 3 \array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 }

    induced from the “suspended Hopf action” of S 1S^1 on S 4S^4.

    My aim in extracting this is to rename the generators given in Roig-Saralegi 00, p. 2 such as to make their degrees and their pattern more manifest. I hope I got it right now:

    fibration vector space underlying minimal dg-model differential on minimal dg-model S 4 S 3 Sym h 3ω 2pdeg=2p,f 2p+4deg=2p+4|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 4 0 f 2p+6 h 3f 2p+4 S 0 S 3 Sym h 3ω 2pdeg=2p,f 2pdeg=2p|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 0 0 f 2p+2 h 3f 2p S 4//S 1 S 3 Sym h 3,f 2ω 2pdeg=2p,f 2p+4deg=2p+4|p d:{ω 0 0 ω 2p+2 h 3ω 2p f 2 0 f 2p+4 h 3f 2p+2 \array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_4 & \mapsto 0 \\ f_{2p+6} & \mapsto h_3 \wedge f_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ f_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_0 & \mapsto 0 \\ f_{2p+2} &\mapsto h_3 \wedge f_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3 , f_2 \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_2 & \mapsto 0 \\ f_{2p+4} & \mapsto h_3 \wedge f_{2p + 2} \end{aligned} \right. }
    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 17th 2018

    Typos:

    norma

    canonically homotopy types over $S^3

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2018

    Thanks, fixed now.

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