nLab MString

Not to be confused with M-string.

Idea

A string spectrum is the Thom spectrum of the universal vector bundle over a string group. Their limit over the infinite string group is of particular interest since its generalized homology theory describes string bordisms.

Definition

Let $String(n)\coloneqq O(n)\langle 4\rangle=O(n)\langle 5\rangle=O(n)\langle 6\rangle=O(n)\langle 7\rangle$ be the $3$-connected cover in the Whitehead tower of the orthogonal group $O(n)$. Through the canonical projection $p\colon String(n)\twoheadrightarrow O(n)$, there is a pullback:

$$\gamma_String^n \coloneqq p^*\gamma_\mathbb{R}^n \twoheadrightarrow BString(n).$$

Its Thom spectrum is the string spectrum:

$$MString(n) \coloneqq\mathbf{Th}\left(\gamma_String^n\right) =\Sigma^{\infty-n}Th\left(\gamma_String^n\right).$$

The desuspension assures the invariance under the Whitney sum with trivial bundles, so $MString(n)=\mathbf{Th}\left(\gamma_String^n\oplus\underline{\mathbb{R}^m}\right)$. It also assures that the canonical inclusion $i\colon String(n)\rightarrow String(n+1)$, which pulls back to a canonical vector bundle homomorphism $\gamma_String^n\oplus\underline{\mathbb{R}}=i^*\gamma_String^{n+1}\rightarrow\gamma_String^{n+1}$, induces a spectrum homomorphism:

$$\begin{aligned} MString(n) &=\Sigma^{\infty-n}Th\left(\gamma_String^n\right) \cong\Sigma^{\infty-(n+1)}Th\left(\gamma_String^n\oplus\underline{\mathbb{R}}\right) \\ &\rightarrow\Sigma^{\infty-(n+1)}Th\left(\gamma_String^{n+1}\right) =MString(n+1). \end{aligned}$$

Its limit is denoted:

$$MString \coloneqq\lim_{n\rightarrow\infty}MString(n).$$

String bordism homology theory

According to Thom's theorem, there is an isomorphism to string bordism groups:

$$\Omega_n^String \cong\pi_n MString =\lim_{k\rightarrow\infty}\pi_k MString_{n+k}.$$

More general, MString defines a generalized homology theory (formally also denoted $\widetilde{MString}_*$) given by:

$$\Omega_n^String(X) \coloneqq\pi_n^stab(X_+\wedge MString) \coloneqq\lim_{k\rightarrow\infty}\pi_{n+k}(X_+\wedge MString_k)$$

for all topological spaces $X$ with the disjoint union $X_+\coloneqq X+\{*\}$. Since $\{*\}_+\cong S^0$ is the neutral element of the wedge product, one has $\Omega_n^String=\Omega_n^String(*)$. Geometrically, $\Omega_n^String(X)$ can also be described by $n$-dimensional string manifolds representing cycles and $n+1$-dimensional string bordisms representing homologous cycles, which are mapped continuous into $X$. For a detailed explanation see string bordism.

A $n$-dimensional string manifold $X$ has a string fundamental class $[X]\in\Omega_n^String(X)$. Let $i\colon X\hookrightarrow\mathbb{R}^{n+k}\hookrightarrow S^{n+k}$ be an embedding (which always exists due to the Whitney embedding theorem), then its Pontrjagin-Thom collapse map is:

$$S^{n+k}\rightarrow X_+\wedge Th(N_i X)$$

with the normal bundle $N_i X\coloneqq TS^{n+k}/i^*TX$. Since the string structure of $X$ transfers over to its stable normal bundle? ($N_i X$ for $k\rightarrow\infty$), postcomposition yields the map:

$$S^{n+k}\rightarrow X_+\wedge MString_k$$

representing the string fundamental class $[X]\in\Omega_n^String(X)$. Geometrically, it’s represented by the identity $id\colon X\rightarrow X$.

String cobordism cohomology theory

MString also defines a generalized cohomology theory given by:

$$\widetilde{MString}^n(X) \coloneqq\lim_{k\rightarrow\infty}[\Sigma^k X,MString_{n+k}]$$

for all topological spaces $X$. It can also be described geometrically with string structures.

Related concepts

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• relative bordism theories: MOFr, MUFr, MSUFr

• equivariant bordism theory: equivariant MFr, equivariant MO, equivariant MU

• global equivariant bordism theory: global equivariant mO, global equivariant mU

• algebraic: algebraic cobordism