2
$\begingroup$

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)!

$\endgroup$
2
  • $\begingroup$ The projective unitary group $PU$ of a separable Hilbert space is weakly(?) homotopy equivalent to $\mathbb CP^\infty$. It is not abelian. $\endgroup$ Nov 24, 2023 at 8:57
  • 1
    $\begingroup$ There exists a model for the classifying space $BG$ of a topological group $G$, such that when $G$ is abelian, $BG$ is again an abelian group. This is Segal's construction, namely, the geometric realization of the groupoid with a single object with automorphism group $G$. The same is true for $EG$. $\endgroup$ Nov 24, 2023 at 13:37

2 Answers 2

3
$\begingroup$

These are sequences not of groups, but of ∞-groups, which can be modeled as simplicial groups or topological groups, equipped with a class of weak equivalences induced from simplicial sets or topological spaces via the obvious forgetful functor.

In particular, the homotopy fiber of the morphism of ∞-groups $\def\String{{\sf String}}\def\Spin{{\sf Spin}}\def\B{{\sf B}} \String→\Spin$ is the ∞-group $\def\Z{{\bf Z}}\B^2\Z$.

Normal ∞-subgroups can be defined as monomorphisms of ∞-groups that are also homotopy fibers, so the answer to the stated question about normal subgroups is positive, although the notion of a normal ∞-subgroup is rarely used in such a context.

The ∞-group $\B^2\Z$ is indeed an ∞-group. In fact, much more generally, every connective chain complex of abelian groups corresponds to a simplicial abelian group (and therefore to an abelian ∞-group), using the Dold–Kan correspondence. The ∞-group $\B^2\Z$ is obtained from the chain complex $\Z[2]$.

Finally, in the most obvious model for ∞-groups $\String$ and $\Spin$ (as simplicial groups, say), the canonical map $\String→\Spin$ is surjective in every simplicial degree, so its kernel $K$ can be computed degreewise. If we take $K$ as our model for $\B^2\Z$, then indeed $\Spin$ is isomorphic to the quotient (i.e., degreewise cokernel) $\String/K$, which in this case computes the homotopy cofiber of the map $K→\String$.

$\endgroup$
3
  • $\begingroup$ I believe $PU$ is a model for $B^2\mathbf Z$. Is it possible to represent String by an honest group containing $PU$ such that the quotient is Spin? This might not be the natural thing to do, but is it possible? $\endgroup$ Nov 24, 2023 at 8:55
  • 1
    $\begingroup$ @SebastianGoette: not that I know. The group $Gau(P)$ from my answer below maps via evaluation at some point of $P$ to $PU(H)$, establishing the homotopy equivalence. The total space of $P$ is homotopy equivalent to $String$ but does not seem to allow a group structure. $\endgroup$ Nov 24, 2023 at 13:33
  • 1
    $\begingroup$ @SebastianGoette: I am not aware of any written accounts. Here is my guess how it could work. Working in the setting of presheaves of simplicial groups on the site of smooth manifolds, we can construct String(H) as an extension of Spin by the smooth ∞-group U(H)//U(1), where // denotes the stacky quotient (i.e., homotopy colimit). Then we can take π_0 of the resulting fiber sequence of presheaves of simplicial groups, which produces an exact sequence PU(H)→π_0(String(H))→Spin of sheaves of groups on the site of smooth manifolds. $\endgroup$ Nov 24, 2023 at 18:39
2
$\begingroup$

The topological groups $String$ and $BS^1$ are - a priori - only defined up to homotopy equivalence. In that setting, it makes sense to talk about fibre sequences, but the question for a group extension, as well as the question of whether or not a group is abelian, depend on models.

In Stephan Stolz's original work about the string group [1], reviewed in [2], he realizes String as an extension of topological groups $$ Gau(P) \to String \to Spin. $$ The group $Gau(P)$ is the group of gauge transformations in a principal $PU(H)$-bundle $P$ over $Spin$, topologized in some appropriate way. It is normal in $String$, but it is not abelian. It also is not isomorphic as topological groups to $PU(H)$, $\mathbb{CP}^\infty$, or $BS^1$, but all these groups are weakly homotopy equivalent.

When working with (infinte-dimensional, strict) 2-groups and strictly exact sequences, the work of Baez et al. [3] gives a concrete model for a $String$-2-group, fitting into an exact sequence of 2-groups $$ \mathcal{L}Spin \to String \to Spin $$ (where $Spin$ is regarded as a 2-group with only identity morphisms). The kernel 2-group $\mathcal{L}Spin$ has objects $\Omega Spin$, the based loop group, and morphisms a semi-direct product of $\Omega Spin$ and its universal central extension. By geometric realization, one obtains another group extension of topological groups, $$ |\mathcal{L}Spin| \to String \to Spin $$ with some other group $\mathcal{L}Spin$ whose homotopy type is again a $K(\mathbb Z,2)$.

Allowing non-strict (homotopy-)exact sequences of strict 2-groups, any model of the string 2-group is a central extension $$ BS^1 \to String \to Spin $$ by the abelian 2-group $BS^1$. This is telling us (at least me) that this is a good setting. A very simple such model was described lately in [4]. Note that geometric realization will not result into an exact sequence of topological groups.

[1] Stolz, Stephan, A conjecture concerning positive Ricci curvature and the Witten genus, Math. Ann. 304, No. 4, 785-800 (1996). ZBL0856.53033.

[2] Nikolaus, Thomas; Sachse, Christoph; Wockel, Christoph, A smooth model for the string group, Int. Math. Res. Not. 2013, No. 16, 3678-3721 (2013). ZBL1339.22009.

[3] Baez, John C.; Stevenson, Danny; Crans, Alissa S.; Schreiber, Urs, From loop groups to 2-groups, Homology Homotopy Appl. 9, No. 2, 101-135 (2007). ZBL1122.22003.

[4] Ludewig, Matthias; Waldorf, Konrad, Lie 2-groups from loop group extensions

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.