nLab MOFr

under construction

Idea

cobordism theory for unoriented manifolds with stably framed boundaries, thus unifying MO with MFr.

Definition

Consider the cofiber sequence of the unit morphism of the ring spectrum MO

(1)$$\array{ \mathbb{S} & \overset{ 1^{M\mathrm{O}} }{ \longrightarrow } & M \mathrm{O} \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast &\longrightarrow& M \mathrm{O}/ \mathbb{S} }$$

The homotopy cofiber

$$M(\mathrm{O},fr) \;\coloneqq\; M \mathrm{O} / \mathbb{S} \,,$$

has stable homotopy groups the cobordism ring of unoriented bordisms with stably framed boundaries

(2)$$\Omega^{\mathrm{O},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{O}/\mathbb{S} \big)$$

Properties

Boundary morphism to $MFr$

The realization (1) makes it manifest that there is a cohomology operation to MFr of the form

(3)$$\array{ M(\mathrm{O},fr) \;= & M \mathrm{O}/\mathbb{S} & \overset{ \;\;\; \partial \;\;\; }{\longrightarrow} & \Sigma \mathbb{S} & =\; \Sigma Mfr \\ \pi_{2d+2}\big( M(\mathrm{O},fr) \big) && \longrightarrow && \pi_{2d+1}\big( Mfr \big) } \,.$$

Namely, $\partial$ is the second next step in the long homotopy cofiber-sequence starting with $1^{M \mathrm{O}}$. In terms of the pasting law:

(4)$$\array{ \mathbb{S} & \overset{ 1^{M\mathrm{O}} }{ \longrightarrow } & M \mathrm{O} & \longrightarrow & \ast \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast & \longrightarrow & M \mathrm{O}/ \mathbb{S} & \underset{ \partial }{ \longrightarrow } & \Sigma \mathbb{S} }$$

Relation to $M \mathrm{O}$ and $M Fr$

Proposition

The unit morphism of MO is trivial on stable homotopy groups in positive degree:

$$\array{ \pi_{n+1} \big( \mathbb{S} \big) & \overset{ 1^{M \mathrm{O}} }{\longrightarrow} & \pi_{n + 1} \big( M \mathrm{O} \big) \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{n + 1} &\underset{i}{\longrightarrow}& \Omega^{\mathrm{O}}_{n + 1} } \phantom{AAAAAAA} n \in \mathbb{N}$$

(Stong 68, p. 102-103)

Proposition

In positive degree, the underling abelian groups of the bordism rings for MO, MFr and $MOFr$ (2) sit in short exact sequences of this form:

(5)$$0 \to \Omega^{\mathrm{O}}_{n+2} \overset{i}{\longrightarrow} \Omega^{\mathrm{O},fr}_{n+2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{n+1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N} \,,$$

where $i$ is the evident inclusion, while $\partial$ is the boundary homomorphism from above.

(Stong 68, p. 102-103)

Proof

We have the long exact sequence of homotopy groups obtained from the cofiber sequence $\mathbb{S} \overset{1^{M\mathrm{O}}}{\longrightarrow} M \mathrm{O} \to M \mathrm{O}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}$ (4), the relevant part of which looks as follows:

(6)$$\array{ \pi_{d+2} \big( \mathbb{S} \big) & \overset{ 1^{M\mathrm{O}} }{ \longrightarrow } & \pi_{d+2} \big( M\mathrm{O} \big) & \overset{ }{\longrightarrow} & \pi_{d+2} \big( M\mathrm{O}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{d+1}\big(\mathbb{S}\big) &\longrightarrow& \pi_{d+1}\big(M\mathrm{O}\big) \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{O}}_{d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{O},fr)}_{d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & \Omega^{\mathrm{O}}_{d+1} }$$

Here the two outermost morphisms shown are zero morphisms, by Prop. , and hence the claim follows.

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, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

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

• Robert Stong, p. 102-106 of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)

Analogous discussion for MO-bordism with MSO-boundaries:

• G. E. Mitchell, Bordism of Manifolds with Oriented Boundaries, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 (doi:10.2307/2040234)