Intereting Posts

find the fourier cosine transform of the function defined by $\displaystyle f(x)= \frac1{1+x^2}$
$\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
Significance of the Riemann hypothesis to algebraic number theory?
Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1
Is there a slowest rate of divergence of a series?
The Laplace transform of the first hitting time of Brownian motion
An extension of a game with two dice
Find $\lim_{x\to 1}\frac{p}{1-x^p}-\frac{q}{1-x^q}$
Uncountable closed set of irrational numbers
Is there a simpler way to falsify this?
Prove that formula is or is not a tautology
Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?
Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)
Evaluate $\displaystyle\int_2^3 {\text{d}x\over x \log(x + 5)}$

is it true that when we compute homologies and cohomologies with coefficients in a field then homology and cohomology groups are isomorphic to each other?

That is valid when homology groups are free with integer coefficients.

- Are these two spaces homotopy equivalent?
- Products of homology groups
- Is such an infinite dimensional metric space, weakly contractible?
- Homotopy equivalence of universal cover
- $K(\mathbb R P^n)$ from $K(\mathbb C P^k)$
- Geometric interpretation of reduction of structure group to $SU(n)$.

- Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem
- Are there nontrivial continuous maps between complex projective spaces?
- Why does «Massey cube» of an odd element lie in 3-torsion?
- Fundamental Group of Punctured Plane
- De Rham Cohomology of $M \times \mathbb{S}^1$
- The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian.
- Relation between Stiefel-Whitney class and Chern class
- Normal subgroups of free groups: finitely generated $\implies$ finite index.
- Deck transformation acting properly discontinuously assumed covering space is path-connected
- The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

By the universal coefficients theorem, since a field is a PID, one has:

$$H^n(X;\Bbbk) \cong \hom_\Bbbk(H_n(X;\Bbbk), \Bbbk) \oplus \operatorname{Ext}_\Bbbk^1(H_{n-1}(X;\Bbbk), \Bbbk).$$

But over a field all $\operatorname{Ext}$’s vanish, and thus:

$$H^n(X;\Bbbk) \cong \hom_\Bbbk(H_n(X;\Bbbk),\Bbbk).$$

Now, **if the homology groups with $\Bbbk$-coefficients are all finitely generated**, then this means that $H^n(X;\Bbbk) \cong H_n(X;\Bbbk)$, because two vector spaces of the same dimension are isomorphic, and the dimension of the dual of a finite-dimensional vector space is the same. But in general, if $H_n(X;\Bbbk)$ is infinite dimensional, then depending on its (infinite) dimension, it may or may not be isomorphic to its dual.

For an explicit example, let $\Bbbk = \mathbb{Q}$, and let $X = \mathbb{N}$ be a countable discrete space. Then $H_0(X;\mathbb{Q}) = \bigoplus_{n \in \mathbb{N}} \mathbb{Q}$, while $H^0(X;\mathbb{Q}) \cong \prod_{n \in \mathbb{N}} \mathbb{Q}$, and the two are not isomorphic – the first has dimension $\aleph_0$, the second has dimension $2^{\aleph_0}$.

So to conclude, in general, you cannot say that $H_n(X;\Bbbk)$ is isomorphic to $H^n(X;\Bbbk)$; but if all the $H_n(X;\Bbbk)$ are finite dimensional, then it is true.

PS:

That is valid when homology groups are free with integer coefficients.

This is only valid if the homology groups are free *and finitely generated*. Again, same counterexample $X = \mathbb{N}$: $H_0(X;\mathbb{Z}) = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$ is free, but $H^0(X;\mathbb{Z}) = \prod_{n \in \mathbb{N}} \mathbb{Z} \not\cong H_0(X;\mathbb{Z})$.

(The other answers are not really explicit to answer the question in the title, so here you go.)

The universal coefficient theorem is not needed in full generality. If $k$ is a field, then the vector space duality functor $\text{Hom}_k(.,k)$ is exact. This gives a canonical isomorphism between cohomology and the vector space dual of homology.

Given a space $X$ and an abelian group $A$, the Universal Coefficient Theorem for cohomology states that there is a natural short exact sequence $0\to \text{Ext}(H_{i-1}(X;\mathbb{Z}),A) \to H^i(X;A) \to \text{Hom}(H_i(X;\mathbb{Z}),A)\to 0$ and this sequence splits (but not naturally).

If $A$ is a field, then $\text{Ext}(H_{i-1}(X;\mathbb{Z}),A)=0$ and so $H^i(X;A)\cong \text{Hom}(H_i(X;A),A)$.

- Area in axiomatic geometry
- Number of inversions
- If $R=\{(x,y): x\text{ is wife of } y\}$, then is $R$ transitive?
- Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
- Examples when vector $(X,Y)$ is not normal 2D distribution, but X and Y are.
- If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges
- $(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$
- Three pythagorean triples
- Are all metric spaces topological spaces?
- When the trig functions moved from the right triangle to the unit circle?
- simple/dumb logarithmic conversion question
- $\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
- Understanding Dot and Cross Product
- proof that $\frac{x^p – 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible
- $n\mid \phi(a^{n}-1)$ for any $a>n$?