In generalization to how the complex cobordism ring Ω 2k U\Omega^U_{2k} is represented by homotopy classes of maps into the Thom spectrum MU, so Ω 2k U,fr\Omega^{\mathrm{U},fr}_{2k} is represented by maps into the quotient spaces MU 2k/S 2kMU_{2k}/S^{2k} (for S 2k=Th(ℂ k)→Th(ℂ k× U(k)EU(k))=MU 2kS^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{\mathrm{U}(k)} E \mathrm{U}(k) ) = M \mathrm{U}_{2k} the canonical inclusion):

(1)Ω • (U,fr)=π •+2k(MU 2k/S 2k),for any2k≥•+2. \Omega^{(\mathrm{U},fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,.

(Conner-Floyd 66, p. 97)

Hence the representing spectrum M(U,fr)M(\mathrm{U},fr) is the homotopy cofiber of the ring spectrum unit 1 MU:𝕊⟶MU1^{M\mathrm{U}} \;\colon\; \mathbb{S} \longrightarrow M \mathrm{U} out of the sphere spectrum (Conner-Smith 69, p. 156 (41 of 106), Smith 71) which deserves to be denoted

M(U,fr)≔MU/𝕊, M(\mathrm{U},fr) \;\coloneqq\; M \mathrm{U} / \mathbb{S} \,,

but which in notation common around the Adams spectral sequence would be “ΣMU¯\Sigma \overline {M \mathrm{U}}” (as in Adams 74, theorem 15.1 page 319) or just “MU¯\overline{ M \mathrm{U} }” (e.g. Hopkins 99, Cor. 5.3):

(2)𝕊 ⟶1 MU MU ↓ (po) ↓ * ⟶ MU/𝕊 \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast &\longrightarrow& M \mathrm{U}/ \mathbb{S} }

So in terms of stable homotopy groups of this spectrum we have the (U,fr)(\mathrm{U},fr)-cobordism ring

(3)Ω • U,fr≔π •(MU/𝕊) \Omega^{\mathrm{U},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{U}/\mathbb{S} \big)

Boundary morphism to MFrMFr

The realization (2) makes it manifest (this is left implicit in Conner-Floyd 66, p. 99) that there is a cohomology operation to MFr of the form

(4)M(U,fr)= MU/𝕊 ⟶∂ Σ𝕊 =ΣMfr π 2d+2(M(U,fr)) ⟶ π 2d+1(Mfr). \array{ M(\mathrm{U},fr) \;= & M \mathrm{U}/\mathbb{S} & \overset{ \;\;\; \partial \;\;\; }{\longrightarrow} & \Sigma \mathbb{S} & =\; \Sigma Mfr \\ \pi_{2d+2}\big( M(\mathrm{U},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 MU1^{M \mathrm{U}}. In terms of the pasting law:

(5)𝕊 ⟶1 MU MU ⟶ * ↓ (po) ↓ (po) ↓ * ⟶ MU/𝕊 ⟶∂ Σ𝕊 \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} & \longrightarrow & \ast \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast & \longrightarrow & M \mathrm{U}/ \mathbb{S} & \underset{ \partial }{ \longrightarrow } & \Sigma \mathbb{S} }

Relation to MUMU and MFrMFr

Proposition

In positive degree, the underlying abelian groups of the bordism rings for MU, MFr and MUFrMUFr (3) sit in short exact sequences of this form:

(6)0→Ω 2n+2 U⟶iΩ 2n+2 U,fr⟶∂Ω 2n+1 fr→0,AAAAn∈ℕ, 0 \to \Omega^{\mathrm{U}}_{2n + 2} \overset{i}{\longrightarrow} \Omega^{\mathrm{U},fr}_{2n + 2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{2n + 1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N} \,,

where ii is the evident inclusion, while ∂\partial is the boundary homomorphism from above.

This is stated without comment in Conner-Floyd 66, p. 99. The beginning of an argument appears inside the proof of CF66, Thm. 16.2 (p. 100), attributed there to Peter Landweber (see Remark below). The idea for how to complete the argument is a little more explicit in Stong 68, p. 102.

The following is the complete and quick proof using the formulation (5) via abstract homotopy above:

Proof

We have the long exact sequence of homotopy groups (long exact sequence in generalized cohomology on spheres) obtained from the cofiber sequence 𝕊⟶1 MUMU→MU/𝕊→∂Σ𝕊\mathbb{S} \overset{1^{M\mathrm{U}}}{\longrightarrow} M \mathrm{U} \to M \mathrm{U}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S} (5), the relevant part of which looks as follows:

(7)π 2d+2(𝕊)⏞puretorsion ⟶1 MU π 2d+2(MU)⏞freeabelian ⟶ π 2d+2(MU/𝕊) ⟶∂ π 2d+1(𝕊) ⟶ π 2d+1(MU)⏞trivial ↓ = ↓ = ↓ = ↓ = ↓ = Ω 2d+2 fr ⟶0 Ω 2d+2 U ⟶i Ω 2d+2 (U,fr) ⟶∂ Ω 2d+1 fr ⟶0 0 \array{ \overset{ \mathclap{ \color{darkblue} pure \; torsion } }{ \overbrace{ \pi_{2d+2} \big( \mathbb{S} \big) } } & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & \overset{ \mathclap{ \color{darkblue} free \; abelian } }{ \overbrace{ \pi_{2d+2} \big( M\mathrm{U} \big) } } & \overset{ }{\longrightarrow} & \pi_{2d+2} \big( M\mathrm{U}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{2d+1}\big(\mathbb{S}\big) &\longrightarrow& \overset{ \color{darkblue} trivial }{ \overbrace{ \pi_{2d+1}\big(M\mathrm{U}\big) } } \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{2d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{U}}_{2d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{U},fr)}_{2d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{2d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & 0 }

Observing now that the stable homotopy groups of MUM\mathrm{U} are free abelian groups concentrated in even degrees (by this theorem at MU) it follows that:

the rightmost morphism shown in (7) is the zero morphism since its codomain is zero;
the leftmost morphism shown in (7) is the zero morphism, since the stable homotopy groups of spheres are all pure torsion groups in positive degrees (by the Serre finiteness theorem), and the only morphism from a torsion group to a free abelian group is the zero morphism.

Relation to Todd classes and the e-invariant

Proposition

(e-invariant is Todd class of cobounding (U,fr)-manifold)

Evaluation of the Todd class on (U,fr)(U,fr)-manifolds yields rational numbers which are integers on actual UU-manifolds. It follows with the short exact sequence (6) that assigning to FrFr-manifolds the Todd class of any of their cobounding (U,fr)(U,fr)-manifolds yields a well-defined element in Q/Z.

Under the Pontrjagin-Thom isomorphism between the framed bordism ring and the stable homotopy group of spheres π • s\pi^s_\bullet, this assignment coincides with the Adams e-invariant in its Q/Z-incarnation:

(8)0→ Ω •+1 U ⟶i Ω •+1 U,fr ⟶∂ Ω • fr ≃ π • s ↓ Td ↓ Td ↓ ↓ e 0→ ℤ ↪ ℚ ⟶ ℚ/ℤ = ℚ/ℤ, \array{ 0 \to & \Omega^{\mathrm{U}}_{\bullet+1} & \overset{i}{\longrightarrow} & \Omega^{\mathrm{U},fr}_{\bullet+1} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_\bullet & \simeq & \pi^s_\bullet \\ & \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,,

(Conner-Floyd 66, Theorem 16.2)

The first step in the proof of (8) is the observation (Conner-Floyd 66, p. 100-101) that the representing map (1) for a (U,fr)(U,fr)-manifold M 2kM^{2k} cobounding a FrFr-manifold represented by a map ff is given by the following homotopy pasting diagram (see also at Hopf invariant – In generalized cohomology):

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:

MOFr, MUFr, MSUFr

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

algebraic cobordism

References

The concept of (U,fr)(U,fr)-bordism theory and its relation to the e-invariant originates with:

Pierre Conner, Edwin Floyd, Section 16 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)
Pierre Conner, Larry Smith, Section 6 of: On the complex bordism of finite complexes, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (numdam:PMIHES_1969__37__117_0)

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)

Analogous discussion for MOFr is in

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