Absolute Hurewicz theorem at the first nonzero degree

The space is nonempty and path connected by [F3]. It has a vertex $v$: take a cell containing a point; if its dimension is positive, its nonempty boundary maps to the preceding skeleton, so finite descent in dimension reaches a zero-cell. Thus $(X,\{v\})$ is a CW pair. Its point subspace is simply connected; its component map is surjective; and its relative groups in positive degrees are exactly the absolute based groups by [F3], so the pair is $(n-1)$-connected. Applying [F1] yields the relative Hurewicz isomorphism in degree $n$ and lower relative homology vanishing. In positive degrees the canonical $H_i(X)\to H_i(X,\{v\})$ is an isomorphism by [F4]. Under the corresponding homotopy identification, the relative disk representative is the sphere representative precomposed with $D^n\to D^n\mathrm{/}{S}^{n-1}$; By [F8] for the standard finite CW pair $(D^n,S^{n-1})$, the image of the disk orientation is a generator of the point-relative sphere homology. Use the sphere orientation corresponding to that generator under [F4]. Evaluating the two pushforwards then gives the same Hurewicz map under that homology isomorphism. Hence $h$ is an isomorphism at $v$ and $H_i(X)=0$ for $0<i<n$.

By [F6], a nonempty path-connected space has $H_0=\mathbb{Z}$, and its augmentation is the identity on the generator represented by any point. Its kernel is therefore zero. The degree-zero homology of the reduced complex is exactly this kernel: its cycles are the augmentation-zero chains and its boundaries are the same ordinary boundaries. Thus ${\tilde{H}}_0(X)=0$. This degree-zero calculation holds for every nonempty path-connected space, without any higher connectivity.

We record the transport check for an actual homotopy equivalence $f:T\to K$, with inverse $g$ and homotopies ${\mathrm{i}\mathrm{d}}_T\simeq gf$, ${\mathrm{i}\mathrm{d}}_K\simeq fg$. At $t\in T$, let $\alpha :t\to gf(t)$ be the first track and put $L={\beta }_{\alpha }{g}_{\ast }:{\pi }_i(K,f(t))\to {\pi }_i(T,t)$ for $i\ge 1$. By [F5], $L{f}_{\ast }=1$. The radial-shell formula commutes pointwise with postcomposition, so $f_{\ast }L={\beta }_{f\alpha }(fg{)}_{\ast }\mathrm{.}$ The second inverse homotopy makes $(fg{)}_{\ast }:{\pi }_i(K,f(t))\to {\pi }_i(K,fgf(t))$ an isomorphism by [F5]: composing it with transport along that homotopy's track is the identity. Thus $f_{\ast }L$ is an isomorphism. The equation $L{f}_{\ast }=1$ gives injectivity of $f_{\ast }$, and surjectivity of $f_{\ast }L$ gives surjectivity of $f_{\ast }$. The component functions of $f,g$ are inverse because the two homotopies join each point to its composite image. This proves component and all-basepoint homotopy invariance for this actual equivalence, without assuming its inverse is based.

For any $x\in X$, take one path $\gamma :v\to x$. The transport ${\beta }_{\gamma }:{\pi }_n(X,x)\to {\pi }_n(X,v)$ is an isomorphism by [F5]. Its moving-boundary radial-shell homotopy, including removal of the initial constant shell, descends by [F7] to a homotopy of sphere maps from a representative at $x$ to its transported representative at $v$. The basepoint may move, but [F4]'s absolute prism calculation makes their images of the sphere orientation class equal. Consequently $h_v{\beta }_{\gamma }=h_x\mathrm{.}$ Since both $h_v$ and ${\beta }_{\gamma }$ are isomorphisms, so is $h_x$. No claim that $\{x\}$ is a CW subcomplex was used. For each $x$ only one path was instantiated, not a family over all points.

Now suppose $T$ is $(n-1)$-connected and is supplied with an actual homotopy equivalence $f:T\to K$ to a CW complex. Step 1.3 implies that $K$ is nonempty and path connected and has zero homotopy groups below $n$: at points in the image use the isomorphisms, and at any other point use a path from an image point and [F5]. Steps 1.1–2.1 apply to $K$. On homology the inverse maps and inverse homotopies give inverse induced maps by [F4]'s prism identity. The augmentations commute with continuous postcomposition on point simplices, so these isomorphisms also identify reduced degree-zero homology. Naturality [F4] gives $h_{K,f(t)}{f}_{\ast }=f_{\ast }{h}_{T,t},$ where both horizontal maps induced by $f$ are isomorphisms. Solving this equality with their inverses transfers the Hurewicz isomorphism to $T$ at each $t$, and the homology isomorphisms transfer all lower vanishing. This proof uses the stipulated inverse and inverse homotopies, not the weaker fact that some CW approximation is a weak equivalence.

For the separate degree-one assertion, use [F2] directly at the given point of any path-connected space. It gives surjectivity and exactly the commutator subgroup as kernel, without AC. The argument of step 1.2 gives its reduced $H_0=0$ and uses no CW structure or choice. A point has trivial positive groups in these formulas. Empty spaces are excluded by nonemptiness or a supplied basepoint. At $n=2$ the point-pair application of [F1] satisfies its simple-connectivity hypothesis, while at $n=1$ only abelianization is claimed, not an isomorphism from an arbitrary nonabelian fundamental group. Constant maps, zero classes and moving basepoints were retained in the pushforward and transport formulas. For $n\ge 2$ the sole inherited AC uses are those in [F1] stated in [A1]; the transfer through an already supplied equivalence adds none. This proves every assertion.