under construction



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


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)


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


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)


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)


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:



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.