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Let $n=p^k$ be a prime power. When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and ...

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...

I’m reading Pengelley’s paper “The mod 2 homology of $MSO$ and $MSU$ as $\mathfrak A^*$ comodule algebras, and the cobordism ring”. He has chosen very special generators $z_n \in H_n(MO; \mathbb F_2)$...

Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...

There is a well-known formula for the cup product of an $i$-cochain $A$ and $j$-cochain $B$ in simplicial homology given by $$(A\cup B)(0\ldots i+j) = A(0\ldots i) B(0\ldots j)\;.$$ This formula ...

Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big(...