nLab pin⁺ bordism

Idea

Pin⁺ bordism is the B-bordism theory for manifolds with pin⁺ tangential structure ((B,f)-structure). This is defined to be a lift of the principal $O_n$-bundle of frames across the double cover $\mathrm{Pin}_n^+\to O_n$. This can also be characterized using Stiefel-Whitney classes as follows:

Lemma

(Stolz 1988, Corollary 6.4) A vector bundle $V\to X$ has a pin⁺ structure if and only if $w_2(V) = 0$ in $H^2(X;\mathbb{Z}/2)$.

Non-additivity

Lemma and the Whitney sum formula? imply that, in general, the direct sum of two vector bundles with pin⁺ structures is not pin⁺, due to the presence of $w_1$ cross terms in the formula for $w_2$ of a direct sum. This has a few consequences.

• The direct product of two pin⁺ manifolds need not have a pin⁺ structure: $\mathbb{RP}^4$ is pin⁺, but $\mathbb{RP}^4\times\mathbb{RP}^4$ is not. Usually bordism rings are defined using the direct product, so the pin⁺ bordism groups $\Omega_*^{\mathrm{Pin}^+}$ cannot be made into a graded ring in the usual way. (In fact, they cannot be made into a graded ring at all; see below.)
• If $V\to X$ is a pin⁺ vector bundle, the virtual bundle $-V\to X$ does not in general admit a pin⁺ structure, but rather a pin⁻ structure. Thus, a manifold admits a pin⁺ structure on its tangent bundle if and only if its stable normal bundle? admits a pin⁻ structure! See Giambalvo, §1.

Analogous facts are true for generic $(B, f)$-structures, but with the exception of pin⁺ and pin⁻ structures, the most commonly studied tangential structures do not have this issue.

The inequivalence of pin⁺ structures on the tangent and stable normal bundles has an important consequence for bordism theory. The Pontryagin-Thom theorem identifying bordism groups with homotopy groups of the corresponding Thom spectrum uses $(B, f)$-structures on the stable normal bundle, not the tangent bundle, but pin⁺ structures on manifolds are defined on the tangent bundle. As stated above, a pin⁺ structure on the tangent bundle is equivalent to a pin⁻ structure on the stable normal bundle, so the Thom spectrum whose homotopy groups are the pin⁺ bordism groups is MPin⁻!

It is common in the literature to, given $f\colon B\to BO$, to define the Madsen-Tillmann spectrum MTB to be the Thom spectrum of $f$ precomposed with the map $-1\colon BO\to BO$. Thus the homotopy groups of MTB are naturally isomorphic to the bordism groups of manifolds with $(B, f)$-structures on their tangent bundles.

With this notation, MTPin⁺ = MPin⁻.

Pin⁺ bordism groups

Giambalvo computed the pin⁺ bordism groups. Kirby-Taylor calculate through $\Omega_4$ using arguments from geometric topology.

• $\Omega_0^{Pin^+}\cong \mathbb{Z}/2$
• $\Omega_1^{Pin^+}\cong 0$
• $\Omega_2^{Pin^+}\cong\mathbb{Z}/2$
• $\Omega_3^{Pin^+}\cong\mathbb{Z}/2$
• $\Omega_4^{Pin^+}\cong\mathbb{Z}/16$
• $\Omega_5^{Pin^+}\cong 0$
• $\Omega_6^{Pin^+}\cong 0$
• $\Omega_7^{Pin^+}\cong 0$
• $\Omega_8^{Pin^+} \cong \mathbb{Z}/32 \oplus \mathbb{Z}/2$.

As a corollary, $\Omega_*^{Pin^+}$ cannot be a graded ring: in any graded ring, the degree-$n$ elements are a module over the degree-$0$ ones, but $\Omega_4^{Pin^+}$ is not a $\mathbb{Z}/2$-module.

Properties

As discussed above, the direct sum of two pin⁺ vector bundles does not in general have a pin⁺ structure. However, the direct sum of a pin⁺ vector bundle and a spin vector bundle has a canonical pin⁺ structure. This implies the following:

Proposition

$\Omega_*^{Pin^+}$ is a module over the spin bordism ring $\Omega^Spin$.

Proposition

(Giambalvo, Theorem 3.4 and Corollary 3.5)

1. For all $n\ge 0$, there is a $\mathbb{Z}/2^{4n+1}$ summand of $\Omega_{8n}^{Pin^+}$, generated by the bordism class of $\mathbb{RP}^{8n}$ with either of its two pin⁺ structures.
2. For all $n\ge 0$, there is a $\mathbb{Z}/2^{4n+4}$ summand of $\Omega_{8n+4}^{Pin^+}$, generated by the bordism class of $\mathbb{RP}^{8n+4}$ with either of its two pin⁺ structures.
3. Modulo the $\Omega_*^{\mathrm{Spin}}$-submodule of $\Omega_*^{Pin^+}$ generated by the above classes, the pin⁺ bordism groups are $\mathbb{Z}/2$-vector spaces.

Related entries

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

References

• Vince Giambalvo?: Pin and Pin’ cobordism, Proc. Amer. Math. Soc. 39 (2), 1973, [doi:10.2307/2039653]

• Stephan Stolz: Exotic structures on 4-manifolds detected by spectral invariants, Invent. Math. 94 (1988) 147–162 [doi:10.1007/BF01394348, pdf]

• Robion Kirby, Laurence Taylor: A calculation of $Pin^+$ bordism groups, Commentarii Mathematici Helvetici 65 (1990) 434–447 [doi:10.1007/BF02566617, pdf]

• Robion Kirby, Laurence Taylor: Pin structures on low-dimensional manifolds. In Geometry of Low-Dimensional Manifolds 2: Symplectic Manifolds and Jones-Witten Theory, Cambridge University Press, 1991. [doi:10.1017/CBO9780511629341.015, pdf]