at.algebraic topology – Torsion in homology or basic group of subsets of Euclidean 3-space retort

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at.algebraic topology – Torsion in homology or basic group of subsets of Euclidean 3-space

I do not arbitrator that

torsion within the homology has been dominated out

Certainly, torsion in Cech cohomology has been dominated out for a compact subset. The “habitual” common coefficient system, relating Cech cohomology to $operatorname{Hom}$ and $operatorname{Ext}$ of Steenrod homology, is just not precise for capricious compact subsets of $Bbb R^3$ (though it’s precise for ANRs, probably non-compact). The “reversed” common coefficient system, relating Steenrod homology to $operatorname{Hom}$ and $operatorname{Ext}$ of Cech cohomology is precise for compact metric areas, however it doesn’t assist, as a result of $operatorname{Ext}(Bbb Z[frac1p],Bbb Z)simeqBbb Z_p/Bbb ZsupsetBbb Z_{(p)}/Bbb Z$, which comprises $q$-torsion for all primes $qne p$. (Here $Bbb Z_{(p)}$ denotes the localization on the prime $p$, and $Bbb Z_p$ denotes the $p$-adic integers.
The two UCFs can breathe create in Bredon’s Sheaf Theory, 2nd version, equation (9) on p.292
in Section V.3 and Theorem V.12.8.)

The comment on $operatorname{Ext}$ can breathe made into an precise occasion. The $p$-adic solenoid $Sigma$ is a subset of $Bbb R^3$. The zeroth Steenrod homology $H_0(Sigma)$ is isomorphic by the Alexander duality to $H^2(Bbb R^3setminusSigma)$. This is a cohomology group of an launch $3$-manifold contained in $Bbb R^3$, but it’s isomorphic to $Bbb Zoplus(Bbb Z_p/Bbb Z)$ (utilizing the UCF, or the Milnor quick require sequence with $lim^1$), which comprises torsion. Of passage, each cocycle representing torsion is “vanishing”, i.e. its restriction to every compact submanifold is null-cohomologous inside that submanifold.

By comparable arguments, $H_i(X)$ (Steenrod homology) comprises no torsion for $i>0$ for each compact subset $X$ of $Bbb R^3$.

It is patent that “Cech homology” comprises no torsion (airplane for a noncompact subset $X$ of $Bbb R^3$), as a result of it’s the inverse limit of the homology teams of polyhedral neighborhoods of $X$ in $Bbb R^3$. But I do not arbitrator that is to breathe taken severely, as a result of “Cech homology” is just not a homology concept (it doesn’t respond the require sequence of pair). The homology concept akin to Cech cohomology is Steenrod homology (which consists of “Cech homology” plus a $lim^1$-correction time period). Some references for Steenrod homology are Steenrod’s unique paper in Ann. Math. (1940), Milnor’s 1961 preprint (revealed in http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf), Massey’s bespeak Homology and Cohomology Theory. An Approach Based on Alexander-Spanier Cochains, Bredon’s bespeak Sheaf Theory (so long as the sheaf is ceaseless and has finitely generated stalks) and this paper http://entrance.math.ucdavis.edu/math/0812.1407

As for torsion in eccentric $4$-homology of the Barratt-Milnor occasion, that is actually a query about framed floor hyperlinks in $S^4$ (graze the proof of theorem 1.1 within the linked paper).

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