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Title consistent with @DavidRoberts's edits
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LSpice
LSpice
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$String $\mathrm{String}/CP^\mathbb{CP}^{\infty}=Spin$=\mathrm{Spin}$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ ... \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to ... $$$$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

  • So $\mathbb{CP}^{\infty}$ is a normal subgroup of $\mathrm{String}$, so $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $\mathrm{Spin}$ is a quotient group of $\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

$String/CP^{\infty}=Spin$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ ... \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

  • So $\mathbb{CP}^{\infty}$ is a normal subgroup of $\mathrm{String}$, so $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $\mathrm{Spin}$ is a quotient group of $\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

  • So $\mathbb{CP}^{\infty}$ is a normal subgroup of $\mathrm{String}$, so $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $\mathrm{Spin}$ is a quotient group of $\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

formatting
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David Roberts ♦
David Roberts ♦
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We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^4 Z \to ... $$$$ ... \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 Z = B S^1= CP^{\infty} \to String \to Spin \to 1? $$$$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 Z = B S^1= CP^{\infty}$$B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $CP^{\infty}$$\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

  • So $CP^{\infty}$$\mathbb{CP}^{\infty}$ is a normal subgroup of $String$$\mathrm{String}$, so $String/CP^{\infty}=Spin$$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $Spin$$\mathrm{Spin}$ is a quotient group of $String$$\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^4 Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 Z = B S^1= CP^{\infty} \to String \to Spin \to 1? $$

If so, does the String group contain a normal subgroup $B^2 Z = B S^1= CP^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $CP^{\infty}$ an abelian group or nonabelian group?

  • So $CP^{\infty}$ is a normal subgroup of $String$, so $String/CP^{\infty}=Spin$ where $Spin$ is a quotient group of $String$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

We know that there is a fiber sequence: $$ ... \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty} \to \mathrm{String} \to \mathrm{Spin} \to 1? $$

If so, does the String group contain a normal subgroup $B^2 \mathbb Z = B S^1= \mathbb{CP}^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $\mathbb{CP}^{\infty}$ an abelian group or nonabelian group?

  • So $\mathbb{CP}^{\infty}$ is a normal subgroup of $\mathrm{String}$, so $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ where $\mathrm{Spin}$ is a quotient group of $\mathrm{String}$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

edited body
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zeta
zeta
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We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ... $$$$ ... \to B^3 Z \to B String \to B Spin \to B^4 Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 Z = B S^1= CP^{\infty} \to String \to Spin \to 1? $$

If so, does the String group contain a normal subgroup $B^2 Z = B S^1= CP^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $CP^{\infty}$ an abelian group or nonabelian group?

  • So $CP^{\infty}$ is a normal subgroup of $String$, so $String/CP^{\infty}=Spin$ where $Spin$ is a quotient group of $String$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 Z = B S^1= CP^{\infty} \to String \to Spin \to 1? $$

If so, does the String group contain a normal subgroup $B^2 Z = B S^1= CP^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $CP^{\infty}$ an abelian group or nonabelian group?

  • So $CP^{\infty}$ is a normal subgroup of $String$, so $String/CP^{\infty}=Spin$ where $Spin$ is a quotient group of $String$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

We know that there is a fiber sequence: $$ ... \to B^3 Z \to B String \to B Spin \to B^4 Z \to ... $$

Is this fiber sequence induced from a short exact sequence?

  • If so, is that $$ 1 \to B^2 Z = B S^1= CP^{\infty} \to String \to Spin \to 1? $$

If so, does the String group contain a normal subgroup $B^2 Z = B S^1= CP^{\infty}$?

  • Is the classifying space $B S^1$ of the abelian group $S^1$ also a group? Is $CP^{\infty}$ an abelian group or nonabelian group?

  • So $CP^{\infty}$ is a normal subgroup of $String$, so $String/CP^{\infty}=Spin$ where $Spin$ is a quotient group of $String$ group?

Please kindly correct me if I said anything wrong or stupid! Many thanks(giving)!

edited body
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Dmitri Pavlov
Dmitri Pavlov
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zeta
zeta
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