nLab
MOFr

Contents

under construction

Contents

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)𝕊 1 MO MO (po) * MO/𝕊 \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(O,fr)MO/𝕊, 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)Ω O,frπ (MO/𝕊) \Omega^{\mathrm{O},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{O}/\mathbb{S} \big)

Properties

Boundary morphism to MFrMFr

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

(3)M(O,fr)= MO/𝕊 Σ𝕊 =ΣMfr π 2d+2(M(O,fr)) π 2d+1(Mfr). \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 MO1^{M \mathrm{O}}. In terms of the pasting law:

(4)𝕊 1 MO MO * (po) (po) * MO/𝕊 Σ𝕊 \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 MOM \mathrm{O} and MFrM Fr

Proposition

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

π n+1(𝕊) 1 MO π n+1(MO) = = Ω n+1 fr i Ω n+1 OAAAAAAAn \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 MOFrMOFr (2) sit in short exact sequences of this form:

(5)0Ω n+2 OiΩ n+2 O,frΩ n+1 fr0,AAAAn, 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 ii 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 𝕊1 MOMOMO/𝕊Σ𝕊\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)π d+2(𝕊) 1 MO π d+2(MO) π d+2(MO/𝕊) π d+1(𝕊) π d+1(MO) = = = = = Ω d+2 fr 0 Ω d+2 O i Ω d+2 (O,fr) Ω d+1 fr 0 Ω d+1 O \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.

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

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

relative bordism theories:

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

References

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)

Last revised on January 18, 2021 at 10:32:55. See the history of this page for a list of all contributions to it.