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It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds: $$ \Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = …

Consider the extension $$1\to SU(2)\to X\to O\to1,$$ there are 4 possibilities for $X$: $X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\mathb …

This is an attempt to make my relation between bordism invariants in $d$ and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific …

Are there some relationship between mapping the bordism invairants of eq.1 and eq.2? $$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$ and $$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2 …

Let $M^d$ be a nontrivial bordism generator for the bordism group $$ \Omega_d^G= \mathbb{Z}_n, $$ suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}_ …

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ Th …

I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement: (1) Both $H$ and $Q$ are connected topological groups or Lie groups …