cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
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:
global equivariant bordism theory:
algebraic:
In joint generalization of the cobordism cohomology theories MSU and MFr of closed $SU$-manifolds and of $Fr$-manifolds, respectively, an $(SU,fr)$-manifold (Conner-Floyd 66, Section 16, p. 103 onwards) is a compact manifold with boundary equipped with special unitary group-tangential structure on its stable tangent bundle and equipped with a trivialization (stable framing) of that over the boundary.
The corresponding bordism classes form a bordism ring denoted $\Omega^{SU,fr}_\bullet$.
In generalization to how $\Omega^{SU}_{2k}$ is represented by homotopy classes of maps into the Thom spectrum MSU, so $\Omega^{SU,fr}_{2k}$ is represented by maps into the quotient spaces $MSU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):
In every eighth degree, the bordism rings for MSU, $MSUFr$ and MFr sit in a short exact sequence of the form (Conner-Floyd 66, p. 104):
where $i$ is the evident inclusion, while $\partial$ is restriction to the boundary.
In particular, this means that $\partial$ is surjective, hence that every $Fr$-manifold of dimension $8k + 3$ is the boundary of a $(U,fr)$-manifold.
In refinement of how the complex e-invariant is the Todd class of cobounding (U,fr)-manifolds we have for special unitary group-structure instead of unitary group-structure and in dimensions $8\bullet + 4$:
Since on $(8 \bullet + 4)$-dimensional $SU$-manifolds the Todd class is divisible by 2 Conner-Floyd 66, Prop. 16.4, we have (Conner-Floyd 66, p. 104) the following short exact sequence of MSUFr-bordism rings:
This produces $e_{\mathbb{R}}$, the Adams e-invariant with respect to KO-theory instead of KU (Adams 66, p. 39), which, in degrees $8k + 3$, is indeed half of the e-invariant $e_{\mathbb{C}}$ for $KU$ (by Adams 66, Prop. 7.14).
In fact, for $k = 0$ we have:
(Adams 66, Example 7.17 and p. 46)
In degree 3, the KO-theoretic e-invariant $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ and hence reflects the full third stable homotopy group of spheres:
while $e_{\mathbb{C}}$ sees only “half” of it (by Adams 66, Prop. 7.14).
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:
global equivariant bordism theory:
algebraic:
The concept of $(SU,fr)$-bordism theory and its relation to the e-invariant originates with:
Created on December 18, 2020 at 10:54:42. See the history of this page for a list of all contributions to it.