Intereting Posts

Number of solutions to the congruence $x^q \equiv 1 \mod p$.
Does weak compactness imply boundedness in a normed vector space (not necessarily complete)?
Defining a Perplexing Two-Dimensional Sequence Explicitly
A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
I need a better explanation of $(\epsilon,\delta)$-definition of limit
How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$
Sum of two periodic functions
Geometric basis for the real numbers
Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.
How generalize the bicommutant theorem?
Is there a multiple function composition operator?
How many arrangements of a bookshelf exist where certain books must be to the left/right of other books?
(Fast way to) Get a combination given its position in (reverse-)lexicographic order
limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$

Is it true that the homology of a manifold with field coefficients determines the homology

over the integers? I know that by the universal coefficient theorem that

$H_k(X; \mathbb{F}) \cong H_k(X; \mathbb{Z}) \bigotimes \mathbb{F}$ since the Tor part vanishes for fields. Then I think we can use the structure theorem of abelian groups (since $H_k(X; \mathbb{Z})$ is an abelian group). Is this true? Is there a reference for this claim?

- The First Homology Group is the Abelianization of the Fundamental Group.
- If two chain maps over a PID induce the same homomorphism, then they are homotopic
- Original Papers on Singular Homology/Cohomology.
- Manifold with 3 nondegenerate critical points
- Finding a space with given homology groups and fundamental group
- Homology of the loop space
- Why do universal $\delta$-functors annihilate injectives?
- Applications for Homology
- Group structure on pointed homotopy classes
- Does taking the direct limit of chain complexes commute with taking homology?

No. More generally, let $X$ be a space whose integral homology is levelwise finitely generated. This is true for any compact manifold but not true for general noncompact manifolds, e.g. a surface of infinite genus. Then the integral homology groups $H_k(X, \mathbb{Z})$ are determined by their rank and by the number of times the summand $\mathbb{Z}_{p^k}$ shows up in their torsion subgroups.

The rank, and only the rank, can be detected by taking homology with coefficients in $\mathbb{Q}$, or more generally any field of characteristic $0$. Some information about the $p$-torsion can be detected by taking homology with coefficients in $\mathbb{F}_p$, or more generally any field of characteristic $p$, although less directly than you suggest: universal coefficients gives a short exact sequence

$$0 \to H_i(X, \mathbb{Z}) \otimes \mathbb{F}_p \to H_i(X, \mathbb{F}_p) \to \text{Tor}(H_{i-1}(X, \mathbb{Z}), \mathbb{F}_p) \to 0.$$

The Tor term here does not vanish in general; its rank is the rank of the $p$-torsion in $H_{i-1}(X, \mathbb{F}_p)$. The rank of the leftmost term is the rank of $H_i(X, \mathbb{Z})$ plus the sum of the number of times $\mathbb{Z}_{p^k}$ shows up over all $k$.

So the problem is that taking homology with coefficients in a field cannot tell you all of the information in the torsion: in particular it cannot distinguish between an integral homology group with torsion of the form $\mathbb{Z}_{p^2}$ and torsion of the form $\mathbb{Z}_{p^3}$, for example (and manifolds with torsion of this form are not hard to write down, e.g. using lens spaces). You can detect this by taking homology with coefficients in $\mathbb{Z}_{p^k}$ for all $k$, or more cleanly by taking homology with coefficients in the $p$-adic integers, which to avoid collision with my previous notation I’ll write $\widehat{\mathbb{Z}}_p$. A keyword to look up here is fracture theorem.

- Is $V$ a simple $\text{End}_kV$-module?
- Is this fraction undefined? Infinite Probability Question.
- Parametric equations for hypocycloid and epicycloid
- Is $\nabla$ a vector?
- Functions with the property $\frac{d}{dx}(f(x)\cdot f(x))=f(2x)$
- Derivative of sinc function
- Probability that one part of a randomly cut equilateral triangle covers the other without flipping
- Do you prove all theorems whilst studying?
- Finding Divisibility of Sequence of Numbers Generated Recursively
- deformation retract of $GL_n^{+}(\mathbb{R})$
- What's the solution of the functional equation
- Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
- f is monotone and the integral is bounded. Prove that $\lim_{x→∞}xf(x)=0$
- How to formalize $\text{span}(S)=\{c_1v_1+\cdots+c_kv_k\mid v_1,~\cdots,~v_k\in S,~c_1,~\cdots,~c_k\in F\}$ rigorously in first order language?
- Uncountable closed set of irrational numbers