Holmstrom Singular cohomology

Singular cohomology

Hatcher: http://www.math.cornell.edu/~hatcher/#ATI

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Singular cohomology

Let $X$ be a topological space. Define the singular chain complex $S(X)$. Now $H_*(X; A)$ and $H^*(X; A)$ is the homology and the cohomology of $S(X) \otimes A$ and $Hom(S(X),A)$, respectively.

This is a bifunctor, contravariant in $X$ and covariant in the coefficient group $A$.

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Singular cohomology

Is $HZ$ initial among (ring) spectra in the stable homotopy category?

Are there interesting examples of coefficient groups which are not rings?

Notation

The $n$-the cohomology group of a space $X$, with coefficients in a group $G$:

$H^n(X; G)$

Taking the coefficients to be a ring $R$, we have the cohomology ring:

$H^*(X; R)$

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Singular cohomology

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Singular cohomology

AT (Algebraic topology)

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Singular cohomology

Completely unverified, from MathOverflow: “Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally contractible and $F$ is the sheaf of locally constant functions.”

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Singular cohomology

Spectrum: HR, sometimes H in place of $H\mathbb{Z}$

The representing space of $H^n(-, \pi)$ is denoted by $K(\pi,n)$. The decomposition of the coefficient group into p-primary parts and a free part corresponds to a product decomposition of $K(\pi,n)$.

For homotopy groups, we have $\pi_k(X \times Y) = \pi_k(X) \oplus \pi_k(Y)$. For cohomology groups, one has instead (neglecting torsion) $H^k(X \times Y) = \Sigma_{p+q=k} H^p(X) \otimes H^q(Y)$.

It is claimed here that singular cohomology is not Brown representable; there are also other interesting remarks: http://mathoverflow.net/questions/82956/cohomology-of-a-space-with-local-coefficients-and-singular-cohomological-dimensio

nLab page on Singular cohomology