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For the wedge sum of a countably infinite number of $k$-spheres the $k$-th cohomology group with $\mathbb Z$ coefficients is the direct product of a countably infinite number of copies of $\mathbb Z$, … In fact there is no space whose only nonvanishing cohomology group is the direct sum of a countably infinite number of copies of $\mathbb Z$. …

There is a problem with your definition of the boundary map $\partial : C_q(K)\to C_{q-1}(K)$. The formula $\partial _i[v_0,\cdots,v_q]=[v_0,\cdots,v_{i-1},v_{i+1},\cdots,v_q]$ depends on an orderin …

The naturality condition 1'' reduces the question of whether $v_1=w_1$ to the special case of the tautological bundles $\gamma_n$. In this case both $v_1$ and $w_1$ lie in $H^1(BO(n);{\mathbb Z}_2)$. …

Zvengrowski called "The cohomology ring of a class of Seifert manifolds" in Top. Appl. 105 (2000), 123-156. … Additively this cohomology ring is the same as $$ H^*({\mathbb R}P^3\#{\mathbb R}P^3;{\mathbb Z}_2)={\mathbb Z}_2[x,y]/(xy,x^3+y^3,x^4,y^4) $$ but the ring structures differ by whether there are nonzero …

If it's commutativity of the diagram at the bottom of page 26 that you are worried about, this is just a formality. For convenience, insert another arrow into the diagram going from the middle group …

Taking the union over all $n$, the infinite-dimensional group $\mathrm{SO}$ has mod $2$ cohomology a polynomial ring with one generator in each odd degree and mod $2$ homology an exterior algebra with … Restricting to a finite dimensional $\mathrm{SO}_n$ has the effect of restricting the homology and cohomology algebras to a finite number of generators and truncating the polynomial algebra by relations …