2. Smith condition

We explain the Smith condition in more detail.

Let $G$ be a finite group. Let $X$ and $Y$ be semi-free $G$-CW-complexes. Let $g\colon X\to Y$ be a $G$-map and a (non-equivariant) homotopy equivalence, i.e., $g$ is a pseudo-equivalent $G$-map. Let $f\colon F=X^G\to Y$ be the restriction of $g$ to the fixed set. Let $Y$ be connected, and let $p\colon \widetilde {Y}\to Y$ be the universal cover, equipped with the free action by the fundamental group $\pi =\pi _1(Y)$.

An element $u\in G$ gives an action $u\colon Y\to Y$. The action lifts to self homeomorphisms $\widetilde {u}\colon \widetilde {Y}\to \widetilde {Y}$ of the universal cover, in the sense that $p\widetilde {u}=up$. If we fix one lifting $\widetilde {u}$ of $u$, then the other liftings of $u$ are $a\widetilde {u}$, for $a\in \pi$. Let $\Gamma '$ be the collections of all such liftings. Then $\Gamma '$ is a group fitting into an exact sequence

\[ 1\to \pi\to \Gamma'\to G/G_0\to 1. \]

We remark that $G_0$ consists of those $u\in G$ that act trivially on $Y$, because the liftings $\widetilde {u}$ can only distinguish $u\in G$ through their actions on $Y$. To further distinguish distinct elements of $G$ that may act the same way on $Y$, we introduce the group $\Gamma$ as the pullback of $\Gamma '\to G/G_0\leftarrow G$:

\[ \Gamma=\{(\widetilde{u},u)\colon \text{$\widetilde{u}\in \Gamma'$ covers $u\in G$}\}. \]

Then we get an exact sequence

(2.1)\begin{equation} 1\to \pi\to \Gamma\to G\to 1. \end{equation}

As an extreme case, if $G$ acts trivially on $Y$, then $\Gamma =\pi \times G$.

As an example, consider $G={\mathbb {Z}}_2=\langle u\rangle$ acting on the real projective space ${\mathbb {R}P}^2$ by $u([x_0,\,x_1,\,x_2])=[x_0,\,x_1,\,-x_2]=[-x_0,\,-x_1,\,x_2]$. The universal cover of ${\mathbb {R}P}^2$ is the sphere $S^2$, with the covering group $\pi$ generated by the antipode $a(x_1,\,x_2,\,x_3)=(-x_1,\,-x_2,\,-x_3)$. The action $u$ lifts to $\widetilde {u}_1(x_0,\,x_1,\,x_2)=(x_0,\,x_1,\,-x_2)$ and $\widetilde {u}_2(x_0,\,x_1,\,x_2)=(-x_0,\,-x_1,\,x_2)$. The group $\Gamma ={\mathbb {Z}}_2\times {\mathbb {Z}}_2=\langle \widetilde {u}_1\rangle \times \langle \widetilde {u}_2\rangle$, and $\pi$ is a subgroup of $\Gamma$ by $a=\widetilde {u}_1\widetilde {u}_2$.

We use $\;\widetilde {}\;$ to denote the lifting/pullback along the universal cover of $Y$. For example, we have the pullbacks (note that $\widetilde {X}$ is generally not the universal cover of $X$):

For a connected component $C$ of $Y^G$, we have the pullback

In our example, we have $({\mathbb {R}P}^2)^G=\{[x_0,\,x_1,\,0]\}\sqcup \{[0,\,0,\,1]\}={\mathbb {R}P}^1\sqcup \{[0,\,0,\,1]\}$, and $\widetilde {({\mathbb {R}P}^2)^G}=\{(x_0,\,x_1,\,0)\}\sqcup \{(0,\,0,\,1)\}\sqcup \{(0,\,0,\,-1)\}=S^1\sqcup N\sqcup S$ ($N$ and $S$ are the north and south poles).

Let $\widetilde {y}\in \widetilde {Y^G}$ and $y=p(\widetilde {y})\in Y^G$. Let $C$ and $\widehat {C}$ be the connected components of $Y^G$ and $\widetilde {Y^G}$ containing $y$ and $\widetilde {y}$. Then $\widehat {C}$ covers $C$. We may use $\widetilde {y}$ to get an isomorphism $\pi _1(Y,\,y)\cong \pi$. Then the deck transformations $\pi _{\widehat {C}}\subset \pi$ of the covering $\widehat {C}\to C$ is the image of the homomorphism $\pi _1(C,\,y)\to \pi _1(Y,\,y)$, and we get $\widetilde {C}=\pi \times _{\pi _{\widehat {C}}} \widehat {C}$.

In general, for $\widetilde {y}\in \widetilde {Y}$ and $y=p(\widetilde {y})\in Y$, the induced homomorphism $\Gamma _{\widetilde {y}}\to G_y$ of isotropy groups is an isomorphism. In particular, for $y\in Y^G$, the isomorphism gives a splitting $G=G_y\cong \Gamma _{\widetilde {y}}\subset \Gamma$ of (2.1). The splitting depends only upon the connected component $\widehat {C}$ containing $\widetilde {y}$. Therefore we may denote $\Gamma _{\widehat {C}}=\Gamma _{\widetilde {y}}$, and get $\Gamma =\pi \rtimes \Gamma _{\widehat {C}}$.

The other connected components of $\widetilde {C}$ are $a\widehat {C}$, $a\in \pi$. Therefore the connected component $C$ gives a $\pi$-conjugation class of isotropy groups (equivalently, a $\pi$-conjugation class of splittings of (2.1))

\[ \Gamma_C=\{\Gamma_{a\widehat{C}}=a\Gamma_{\widehat{C}}a^{{-}1}\colon a\in \pi\}. \]

In our example, $({\mathbb {R}P}^2)^G$ has two connected components $C_1={\mathbb {R}P}^1$ and $C_2=[0,\,0,\,1]$. Their preimages in $\widetilde {({\mathbb {R}P}^2)^G}$ are respectively $\widetilde {C}_1=S^1$ and $\widetilde {C}_2=\{N,\,S\}$. We may take $\widehat {C}_1=S^1$, $\widehat {C}_2=N$, with $a\widehat {C}_1=\widehat {C}_1$, $a\widehat {C}_2=S$. Then $\Gamma _{S^1}=\langle \widetilde {u}_1\rangle$, and $\Gamma _N=\Gamma _S=\langle \widetilde {u}_2\rangle$. The semi-direct products $\Gamma =\pi \rtimes \Gamma _{S^1}=\langle a\rangle \times \langle \widetilde {u}_1\rangle$ and $\Gamma =\pi \rtimes \Gamma _N=\pi \rtimes \Gamma _S=\langle a\rangle \times \langle \widetilde {u}_2\rangle$ are the usual products. Moreover, we have the two-fold cover $\widehat {C}_1=S^1\to C_1={\mathbb {R}P}^1$ with the covering group $\pi _{\widehat {C}_1}=\pi$, and $\widetilde {C}_1=\widehat {C}_1=\pi \times _{\pi }\widehat {C}_1$. We also have the one-fold cover $\widehat {C}_2\to C_2$ with the covering group $\pi _{\widehat {C}_2}=1$, and $\widetilde {C}_2=\widehat {C}_2\sqcup a\widehat {C}_2=\pi \times _1 \widehat {C}_2$.

Denote the homomorphism $G\cong \Gamma _{\widehat {C}}\subset \Gamma$ by $u\to \widetilde {u}$. Then the elements of $\Gamma =\pi \rtimes \Gamma _{\widehat {C}}\cong \pi \rtimes G$ are $a\widetilde {u}$, with $a\in \pi$ and $u\in G$. The multiplication in $\Gamma$ is given by $a_1\widetilde {u}_1a_2\widetilde {u}_2=a_1u_1(a_2)\widetilde {u_1u_2}$. Here $u(a)$ is obtained by regarding $a\in \pi _1(Y,\,y)$ as a loop at $y$ and applying the action of $u\in G$ to the loop. In particular, if $a\in \pi _1(C,\,y)$ lies in the deck transformation group $\pi _{{\widehat {C}}}$, then $u(a)=a$. This means $\Gamma _{\widehat {C}}$ acts trivially on $\pi _{{\widehat {C}}}$, and $\pi _{\widehat {C}}\times \Gamma _{\widehat {C}}\cong \pi _{\widehat {C}}\times G$ is a subgroup of $\Gamma$.

We have the ${\mathbb {Z}}[\Gamma ]$-chain complexes

\[ C_*(\widetilde{Y^G}) =\oplus_{C\in \pi_0Y^G} C_*(\widetilde{C}),\quad C_*(\widetilde{C}) =C_*(\pi\times_{\pi_{\widehat{C}}} {\widehat{C}}) ={\mathbb{Z}}\pi\otimes_{{\mathbb{Z}}\pi_{\widehat{C}}}C_*({\widehat{C}}). \]

Since the isotropy group $\Gamma _{\widehat {C}}$ acts trivially on ${\widehat {C}}$, we may regard $C_*({\widehat {C}})$ as a ${\mathbb {Z}}[\pi _{\widehat {C}}\times \Gamma _{\widehat {C}}]$-chain complex (with trivial action by $\Gamma _{\widehat {C}}$). By $\Gamma =\pi \rtimes \Gamma _{\widehat {C}}$, we get the following interpretation of the ${\mathbb {Z}}[\Gamma ]$-chain complex $C_*(\widetilde {C})$

\[ C_*(\widetilde{C}) ={\mathbb{Z}}[\pi\rtimes \Gamma_{\widehat{C}}]\otimes_{{\mathbb{Z}}[\pi_{\widehat{C}}\times \Gamma_{\widehat{C}}]}C_*({\widehat{C}}) =\text{Ind}_{{\mathbb{Z}}[\pi_{\widehat{C}}\times G]}^{{\mathbb{Z}}[\pi\rtimes G]}C_*({\widehat{C}}). \]

Similarly, let $\widehat {F}_C\subset \widetilde {F}_C$ correspond to $\widehat {C}\subset \widetilde {C}$. Then we have

\[ C_*(\widetilde{F}) =\oplus_{C\in \pi_0Y^G}C_*(\widetilde{F}_C),\quad C_*(\widetilde{F}_C) =\text{Ind}_{{\mathbb{Z}}[\pi_Z\times G]}^{{\mathbb{Z}}[\pi\rtimes G]}C_*(\widehat{F}_C). \]

The pseudo-equivalent $G$-map $g\colon X\to Y$ between semi-free $G$-CW-complexes lifts to a pseudo-equivalent $\Gamma$-map $\widetilde {g}\colon \widetilde {X}\to \widetilde {Y}$, and $\widetilde {g}$ has the ‘fixed part” $\widetilde {f}\colon \widetilde {F}\to \widetilde {Y^G}$. Here the fixed part is not fixed by the whole $\Gamma$, but by various isotropy subgroups $\Gamma _{\widehat {C}}\subset \Gamma$. We regard the pseudo-equivalent $\Gamma$-map $\widetilde {g}$ as a pseudo-equivalent $\Gamma _{\widehat {C}}$-map. This implies that the mapping cone $C(\widetilde {g})$ is a contractible, semi-free $\Gamma _{\widehat {C}}$-space. The fixed set $C(\widetilde {g})^{\Gamma _{\widehat {C}}}$ is the mapping cone of the map $\widetilde {X}^{\Gamma _{\widehat {C}}}\to \widetilde {Y}^{\Gamma _{\widehat {C}}}$. By the classical Smith theory [Reference Smith22], $C(\widetilde {g})^{\Gamma _{\widehat {C}}}$ has trivial ${\mathbb {F}}_p$-homology for all prime factors $p$ of $|\Gamma _{\widehat {C}}|=|G|$. This implies an isomorphism $H_*(\widetilde {X}^{\Gamma _{\widehat {C}}};{\mathbb {F}}_p)\cong H_*(\widetilde {Y}^{\Gamma _{\widehat {C}}};{\mathbb {F}}_p)$.

We note that $\Gamma _{\widehat {C}}$ may fix several connected components $\widehat {C}_0,\,\widehat {C}_1,\,\dots,\,\widehat {C}_k$ of $\widetilde {Y^G}$, in addition to $\widehat {C}_0=\widehat {C}$. Then the isomorphism $H_*(\widetilde {X}^{\Gamma _{\widehat {C}}};{\mathbb {F}}_p)\cong H_*(\widetilde {Y}^{\Gamma _{\widehat {C}}};{\mathbb {F}}_p)$ is a direct sum of isomorphisms $H_*(\widehat {F}_{C_i};{\mathbb {F}}_p) \cong H_*(\widehat {C}_i;{\mathbb {F}}_p)$. For $i=0$, this gives the local Smith condition

(2.2)\begin{equation} H_*(\widehat{F}_C;{\mathbb{F}}_p) \cong H_*(\widehat{C};{\mathbb{F}}_p),\quad \text{for all prime factors $p$ of $|G|$}. \end{equation}

For our example, we have the Smith condition on $\widehat {C}_1=S^1$ with the antipode action by $\pi =\langle a\rangle$, and the Smith condition on $\widehat {C}_2=N$ with the trivial group action. The first condition is obtained by applying the usual Smith condition to the action of $\Gamma _{S^1}=\langle \widetilde {u}_1\rangle$ on $S^2$. The second condition is obtained by applying the usual Smith condition to the action of $\Gamma _N=\Gamma _S=\langle \widetilde {u}_2\rangle$ on $S^2$. Although the second condition consists of conditions on $N$ and $S$, the two conditions are equivalent by the action of $\pi =\langle a\rangle$.

Finally, we combine the local Smith conditions into the global Smith condition in theorems 1.4 and 1.5. We pick one connected component $\widehat {C}$ of $\widetilde {C}$ for each connected component $C$ of $Y^G$. Then

\[ \widetilde{Y^G} =\sqcup_{C\in \pi_0Y^G}\pi\times_{\pi_{\widehat{C}}}\widehat{C}, \]

and

\begin{align*} H_*(Y^G;{\mathbb{F}}_p\pi) &=H_*(\widetilde{Y^G};{\mathbb{F}}_p) \\ &=\oplus_{C\in \pi_0Y^G}H_*(\pi\times_{\pi_{\widehat{C}}}\widehat{C};{\mathbb{F}}_p) \\ &=\oplus_{C\in \pi_0Y^G}{\mathbb{F}}_p\pi\otimes_{{\mathbb{F}}_p\pi_{\widehat{C}}}H_*(\widehat{C};{\mathbb{F}}_p), \end{align*}

Similarly, we have

\[ H_*(F;{\mathbb{F}}_p\pi) =H_*(\widetilde{F};{\mathbb{F}}_p) =\oplus_{C\in \pi_0Y^G}{\mathbb{F}}_p\pi\otimes_{{\mathbb{F}}_p\pi_{\widehat{C}}}H_*(\widehat{F}_C;{\mathbb{F}}_p). \]

Then the direct sum of the local Smith condition (2.2) is the global Smith condition

(2.3)\begin{equation} H_*(F;{\mathbb{F}}_p\pi) \cong H_*(Y^G;{\mathbb{F}}_p\pi),\quad \text{for all prime factors $p$ of $|G|$}. \end{equation}

For our example, the global Smith condition is the direct sum of the Smith conditions on $S^1$, $N$, and $S$.

3. Proof of main theorems

The proof of the main theorems in the introduction involves an equivariant version of Wall's finiteness obstruction [Reference Wall28].

Proof of theorem 1.4. Proof of theorem 1.4

We assume that the first (Smith) condition is satisfied and try to construct a pseudo-equivalence extension $g$. In the process, we will encounter the obstruction in the second condition, and will see that it is well-defined.

By $H_0(F;{\mathbb {F}}_p\pi )\cong H_0(Y;{\mathbb {F}}_p\pi )={\mathbb {F}}_p$, we know $F$ is connected. We choose a base point in $F$ and use its image in $Y$ as the base point of $Y$. For any loop $\epsilon$ in $Y$ at the base point, we may attach $G$ copies of loops to $F$, and equivariantly map these loops to the loop $\epsilon$. Since $Y$ is a finite CW-complex, we may attach finitely many such loops to get a finite, semi-free $G$-CW-complex $X^1$ with fixed point set $F$, and get a $G$-map $f^1\colon X^1\to Y$ that is surjective on $\pi _1$.

Since $X^1$ is a finite $G$-CW-complex, there are finitely many loops $\epsilon _i$ generating $\pi _1(X^1)$. Since $f^1$ is surjective on $\pi _1$, the images $f^1(\epsilon _i)$ generate $\pi$, which is finitely presented because $Y$ is a finite CW-complex. Therefore $\pi$ can be presented by $f^1(\epsilon _i)$ as generators, with finitely many words $f^1(w_j)$ of these loops as relations. For each such word $f^1(w_j)$, we glue $G$-copies of $D^2$ to $X^1$ along $Gw_j$ and equivariantly map these $2$-cells to $Y$. We thus get a finite, semi-free $G$-CW-complex $X^{1.5}$ with fixed point set $F$, and extend $f^1$ to a $G$-map $f^{1.5}\colon X^{1.5}\to Y$ that induces an isomorphism on $\pi _1$.

Since $f^{1.5}$ induces an isomorphism on $\pi _1$, by the Hurewicz theorem, we have $\pi _2(f^{1.5})=H_2(f^{1.5};{\mathbb {Z}}\pi )$. This implies that $\pi _2(f^{1.5})$ is finitely generated as a ${\mathbb {Z}}\pi$-module. In fact, as $G$ is a finite group, the $G$-action also makes $\pi _2(f^{1.5})$ into a finitely generated ${\mathbb {Z}}[\pi \times G]$-module. We represent a finite set of ${\mathbb {Z}}[\pi \times G]$-generators by maps $S^1\to X^{1.5}$ and $D^2\to Y$ compatible with $f^2$. Then we glue $G$-copies of $D^2$ along $G(S^1\to X^{1.5})$ to $X^{1.5}$ and equivariantly map these $2$-cells to $Y$ by $G(D^2\to Y)$ (for the current case that the $G$-action on $Y$ is trivial, this is $D^2\to Y$). We get a finite, semi-free $G$-CW-complex $X^2$ with fixed point set $F$, and extend $f^{1.5}$ to a $G$-map $f^2\colon X^2\to Y$, such that $f^2$ satisfies $\pi _1(f^2)=\pi _2(f^2)=0$.

The construction from $f^{1.5}$ to $f^2$ can be inductively extended to higher dimensions. If we have a $G$-map $f^i\colon X^i\to Y$ satisfying $\pi _j(f^i)=0$ for $j\le i$, then we can use a finite set of ${\mathbb {Z}}[\pi \times G]$-generators of $\pi _{i+1}(f^i)=H_{i+1}(f^i;{\mathbb {Z}}\pi )$ to equivariantly attach $(i+1)$-dimensional free $G$-cells to $X^i$, and extend $f^i$ to a $G$-map $f^{i+1}\colon X^{i+1}\to Y$ satisfying $\pi _j(f^{i+1})=0$ for $j\le i+1$. Inductively, we get $f^n\colon X^n\to Y$ for some $n>\max \{\dim F,\,\dim Y\}$, such that $\pi _j(f^n)=0$ for $j\le n$.

Let us consider the effect of one more construction to get $f^{n+1}\colon X^{n+1}\to Y$. The generators used for the construction can be interpreted as a basis of a finitely generated free ${\mathbb {Z}}[\pi \times G]$-module $A$ in a surjective ${\mathbb {Z}}[\pi \times G]$-homomorphism $A\to H_{n+1}(f^n;{\mathbb {Z}}\pi )$. By $n+1>\max \{\dim X^n,\,\dim Y\}$, we have $H_{n+2}(f^n;{\mathbb {Z}}\pi )=0$ and an exact sequence

\begin{align*} & H_{n+2}(f^n;{\mathbb{Z}}\pi)=0 \to H_{n+2}(f^{n+1};{\mathbb{Z}}\pi) \to H_{n+1}(X^{n+1},X^n;{\mathbb{Z}}\pi)=A \\ & \to H_{n+1}(f^n;{\mathbb{Z}}\pi)\to H_{n+1}(f^{n+1};{\mathbb{Z}}\pi) \to H_n(X^{n+1},X^n;{\mathbb{Z}}\pi)=0 \to\cdots \end{align*}

By the Hurewicz theorem, the exact sequence gives $\pi _j(f^{n+1})=H_j(f^{n+1};{\mathbb {Z}}\pi )=0$ for $j\le n+1$, and a short exact sequence

\[ 0 \to \pi_{n+2}(f^{n+1}) =H_{n+2}(f^{n+1};{\mathbb{Z}}\pi) \to A \to H_{n+1}(f^n;{\mathbb{Z}}\pi)\to 0. \]

Note that $\pi _{n+2}(f^{n+1})$ is to be used for the further construction based on $f^{n+1}$. Therefore, the short exact sequence shows that, if $H_{n+1}(f^n;{\mathbb {Z}}\pi )$ has a finite resolution of finitely generated free ${\mathbb {Z}}[\pi \times G]$-modules

\[ 0\to A_k\to \cdots\to A_2\to A_1\to H_{n+1}(f^n;{\mathbb{Z}}\pi)\to 0, \]

then the resolution can be used as a recipe for constructing a $G$-map $f^{n+k}\colon X^{n+k}\to Y$, such that $X^{n+k}$ is a finite, semi-free $G$-CW-complex with fixed point set $F$, and $f^{n+k}$ extends $f$ and is a (non-equivariant) homotopy equivalence. This $f^{n+k}$ is the pseudo-equivalence extension in the theorem.

Next we argue that the Smith condition implies that the ${\mathbb {Z}}[\pi \times G]$-module $H_{n+1}(f^n;{\mathbb {Z}}\pi )$ has a finite resolution by finitely generated projective ${\mathbb {Z}}[\pi \times G]$-modules. This induces (by § 3 of [Reference Mislin14], for example) an element $[H_{n+1}(f^n;{\mathbb {Z}}\pi )]\in \widetilde {K}_0({\mathbb {Z}}[\pi \times G])$, such that the element vanishes if and only if all the projective modules in the resolution can be chosen to be free. Therefore the element is the obstruction for completing our construction.

We identify this obstruction with the $K$-theory element represented by the ${\mathbb {Z}}\pi$-chain complex $C_*(\widetilde {f})$, regarded as a ${\mathbb {Z}}[\pi \times G]$-chain complex with the trivial $G$-action. This is crucial for detailed calculations. This fact is essentially Lemma 1.7 of [Reference Assadi and Vogel2]; but we give a detailed explanation anyway.

There is a (possibly infinite) free ${\mathbb {Z}}G$-resolution $P$ of the trivial ${\mathbb {Z}}G$-module ${\mathbb {Z}}$, such that $P$ is finitely generated in each dimension. Then we have

\[ C_*(\widetilde{f})=C_*(\widetilde{f})\otimes {\mathbb{Z}} \simeq C_*(\widetilde{f})\otimes P, \]

where each term in the ${\mathbb {Z}}[\pi \times G]$-chain complex $C_*(\widetilde {f})\otimes P$ is finitely generated and free.

There is also a (infinitely generated) projective ${\mathbb {Z}}[{1}/{|G|}][G]$-resolution $P'$ of the trivial ${\mathbb {Z}}[{1}/{|G|}][G]$-module ${\mathbb {Z}}[{1}/{|G|}]$, such that $P'$ is nonzero only in dimensions $0$ and $1$. This implies the second chain homotopy equivalence below

\[ C_*(\widetilde{f}) \simeq C_*(\widetilde{f})\otimes {\mathbb{Z}}[\tfrac{1}{|G|}] \simeq C_*(\widetilde{f})\otimes P'. \]

The first chain homotopy equivalence is due to the Smith condition, $H_*(C_*(\widetilde {f});{\mathbb {F}}_p)=H_*(f;{\mathbb {F}}_p\pi )=0$, for all prime factors $p$ of $|G|$. Since $C_*(\widetilde {f})$ vanishes above dimension $n$, and $P'$ vanishes away from dimensions $0$ and $1$, the chain complex $C_*(\widetilde {f})\otimes P'$ has cohomological dimension $\le n+1$.

By Theorem 3.5 of [Reference Mislin14], and the special properties of the two ${\mathbb {Z}}[\pi \times G]$-chain complexes that are chain homotopy equivalent to $C_*(\widetilde {f})$, we know $C_*(\widetilde {f})$ is chain homotopy equivalent to a finite chain complex of finitely generated projective ${\mathbb {Z}}[\pi \times G]$-modules. This gives a well-defined element $[C_*(\widetilde {f})]\in \widetilde {K}_0({\mathbb {Z}}[\pi \times G])$.

The chain complex $C_*(\widetilde {f})$ fits into an exact sequence of ${\mathbb {Z}}[\pi \times G]$-chain complexes

\[ 0\to C_*(\widetilde{f}) \to C_*(\widetilde{f}^n) \to C_{*-1}(\widetilde{X}^n,\widetilde{F}) \to 0. \]

Since $C_{*-1}(\widetilde {X}^n,\,\widetilde {F})$ is a finite chain complex of finitely generated free ${\mathbb {Z}}[\pi \times G]$-modules, $C_*(\widetilde {f}^n)$ is also chain homotopy equivalent to a finite chain complex of finitely generated projective ${\mathbb {Z}}[\pi \times G]$-modules, and $[C_*(\widetilde {f}^n)]=[C_*(\widetilde {f})]\in \widetilde {K}_0(\pi \times G)$. On the other hand, we know the homology of $C_*(\widetilde {f}^n)$ vanishes in all dimensions except for $H_{n+1}(f^n;{\mathbb {Z}}\pi )$. Therefore the ${\mathbb {Z}}[\pi \times G]$-chain complex $\cdots \to 0\to H_{n+1}(f^n;{\mathbb {Z}}\pi )\to 0\to \cdots$ is chain homotopy equivalent to $C_*(\widetilde {f}^n)$. This implies that $H_{n+1}(f^n;{\mathbb {Z}}\pi )$ has a finite resolution by finitely generated projective ${\mathbb {Z}}[\pi \times G]$-modules, and

\[ ({-}1)^{n+1}[H_{n+1}(f^n;{\mathbb{Z}}\pi)] =[C_*(\widetilde{f}^n)] =[C_*(\widetilde{f})] \in \widetilde{K}_0({\mathbb{Z}}[\pi\times G]). \]

The equality to $[C_*(\widetilde {f})]$ shows that the obstruction $(-1)^{n+1}[H_{n+1}(f^n;{\mathbb {Z}}\pi )]$ is independent of our choice of construction.

Proof of Theorem 1.5. Proof of theorem 1.5

The proof is similar to theorem 1.4. The inductive construction of $f^n\colon X^n\to Y$ is the same, except that the new cells can be mapped to $Y$ instead of just the fixed set, and all the homotopy groups and homology groups (at the universal cover level) are ${\mathbb {Z}}[\Gamma ]$-modules. For sufficiently large $n$, the obstruction for constructing the pseudo-equivalence is $[H_{n+1}(f^n;{\mathbb {Z}}\pi )]\in \widetilde {K}_0({\mathbb {Z}}[\Gamma ])$. We need to argue that $C_*(\widetilde {f})$ represents an element in $\widetilde {K}_0({\mathbb {Z}}[\Gamma ])$ that is the same as $(-1)^{n+1}[H_{n+1}(f^n;{\mathbb {Z}}\pi )]$.

In § 2, we saw that the global Smith condition (2.3) in theorem 1.5 is equivalent to the local Smith condition (2.2) for each connected component $C$ of $Y^G$. Following the same argument for theorem 1.4, we know the ${\mathbb {Z}}[\pi _{\widehat {C}}\times \Gamma _{\widehat {C}}]$-chain complex $C_*(\widehat {F}_C\to \widehat {C})$ is chain homotopy equivalent to a finite chain complex of finitely generated projective ${\mathbb {Z}}[\pi _{\widehat {C}}\times \Gamma _{\widehat {C}}]$-modules. This gives a well-defined $K$-theory element

\[ [C_*(\widehat{F}_C\to \widehat{C})] \in \widetilde{K}_0({\mathbb{Z}}[\pi_{\widehat{C}}\times \Gamma_{\widehat{C}}]). \]

By $\pi _{\widehat {C}}\times \Gamma _{\widehat {C}}\subset \pi \rtimes \Gamma _{\widehat {C}}=\Gamma$, this inducts to

\begin{align*} [C_*(\widetilde{F}_C\to \widetilde{C})] &=[{\mathbb{Z}}[\Gamma]\otimes_{{\mathbb{Z}}[\pi_{\widehat{C}}\times \Gamma_{\widehat{C}}]}C_*(\widehat{F}_C\to \widehat{C})] \\ &=\text{Ind}_{{\mathbb{Z}}[\pi_{\widehat{C}}\times \Gamma_{\widehat{C}}]}^{{\mathbb{Z}}[\Gamma]}[C_*(\widehat{F}_C\to \widehat{C})] \in \widetilde{K}_0({\mathbb{Z}}[\Gamma]), \end{align*}

which further adds up to

\[ [C_*(\widetilde{f})] =\sum_{C\in \pi_0Y^G}[C_*(\widetilde{F}_C\to \widetilde{C})] \in \widetilde{K}_0({\mathbb{Z}}[\Gamma]). \]

Now we know the ${\mathbb {Z}}[\Gamma ]$–chain complex $C_*(\widetilde {f})$ is chain homotopy equivalent to a finite chain complex of finitely generated projective ${\mathbb {Z}}[\Gamma ]$-modules and gives a $K$-theory element. Then we have short exact sequences of ${\mathbb {Z}}[\Gamma ]$-chain complexes

\begin{align*} 0&\to C_*(\widetilde{f}\colon \widetilde{F}\to \widetilde{Y^G}) \to C_*(\widetilde{F}\to \widetilde{Y}) \to C_{*-1}(\widetilde{Y},\widetilde{Y^G})\to 0, \\ 0&\to C_*(\widetilde{F}\to \widetilde{Y}) \to C_*(\widetilde{f}^n\colon \widetilde{X}^n\to \widetilde{Y}) \to C_{*-1}(\widetilde{X}^n,\widetilde{F}) \to 0. \end{align*}

Since $Y$ is a semi-free $G$-CW-complex, and $X^n$ is obtained by glueing free $G$-cells to $F$, both $C_{*-1}(\widetilde {Y},\,\widetilde {Y^G})$ and $C_{*-1}(\widetilde {X}^n,\,\widetilde {F})$ are finite chain complexes of finitely generated free ${\mathbb {Z}}[\Gamma ]$-modules. Therefore

\begin{align*} [C_*(\widetilde{f})] &=[C_*(\widetilde{F}\to \widetilde{Y})] \\ &=[C_*(\widetilde{f}^n\colon \widetilde{X}^n\to \widetilde{Y})] \\ &=({-}1)^{n+1}[H_{n+1}(f^n;{\mathbb{Z}}\pi)] \in \widetilde{K}_0({\mathbb{Z}}[\Gamma]). \end{align*}

This completes the identification of the $K$-theory obstruction.

The main theorems can be modified to get the pseudo-equivalence extension to be a simple homotopy equivalence. The only change is the $K$-theory in which the obstruction lives. The following is the analogue of theorem 1.4 making use of an algebraic $K$-theory introduced in [Reference Assadi and Vogel2].

Assadi and Vogel [Reference Assadi and Vogel2] introduced the Grothendick group $Wh_1^T(\pi \subset \pi \times G)$ of the additive category of finitely generated ${\mathbb {Z}}\pi$-based projective ${\mathbb {Z}}[\pi \times G]$-modules, such that the ${\mathbb {Z}}[\pi \times G]$-module ${\mathbb {Z}}[\pi \times G]$ with the choice of $G$ as ${\mathbb {Z}}\pi$-basis is trivial in $Wh_1^T$. They showed that there is an exact sequence ($\widetilde {K}_0(\pi )$ is denoted $Wh_0(\pi )$ in [Reference Assadi and Vogel2])

\[ Wh_1(\pi\times G)\overset{T}{\to} Wh_1(\pi)\overset{\beta}{\to} Wh_1^T(\pi\subset\pi\times G)\overset{\alpha}{\to} \widetilde{K}_0({\mathbb{Z}}[\pi\times G])\overset{T}{\to} \widetilde{K}_0({\mathbb{Z}}\pi). \]

The cells of $F$ and $Y$ give natural ${\mathbb {Z}}\pi$-bases for the modules in $C_*(\widetilde {f})$. By Lemmas 1.6 and 1.7 of [Reference Assadi and Vogel2], under the Smith condition, the chain complex $C_*(\widetilde {f})$ with the natural ${\mathbb {Z}}\pi$-bases gives a well-defined element $[C_*(\widetilde {f})]\in Wh_1^T(\pi \subset \pi \times G)$. The image of this element in $\widetilde {K}_0({\mathbb {Z}}[\pi \times G])$ is the obstruction in the main theorem.

The proof of theorem 3.1 is the same as the proof of theorem 1.4, with additional tracking of the basis in the construction. The key point is that the free $G$-cells used in the construction give the chain complex $C_{*-1}(\widetilde {X}^n,\,\widetilde {F})$, where each term is the direct sum of finitely many copies of ${\mathbb {Z}}[\pi \times G]$ with the ${\mathbb {Z}}\pi$-basis $G$. Therefore $[C_{*-1}(\widetilde {X}^n,\,\widetilde {F})]=0\in Wh_1^T(\pi \subset \pi \times G)$.

In the context of theorem 1.5, we may also extend the $K$-theory to $Wh_1^T(\pi \subset \Gamma )$, by considering the additive category of finitely generated ${\mathbb {Z}}\pi$-based projective ${\mathbb {Z}}[\Gamma ]$-modules. For the ${\mathbb {Z}}[\Gamma ]$-module ${\mathbb {Z}}[\Gamma ]$, we find a subset $\Gamma _0\subset \Gamma$, such that the multiplication map $\pi \times \Gamma _0\to \Gamma$ is a one-to-one correspondence. Then $\Gamma _0$ is a ${\mathbb {Z}}\pi$-basis of ${\mathbb {Z}}[\Gamma ]$, and this represents the trivial element in $Wh_1^T(\pi \subset \Gamma )$. Using this $K$-theory, we also have the simple homotopy version of theorem 1.5.

The main theorems can also be modified to treat compact ANR-spaces in place of finite CW-complexes. We only present the analogue of theorem 1.4.

The topological $K$-theory $\widetilde {K}_0^{top}$ was introduced by M. Steinberger and J. West [Reference Steinberger23, Reference Steinberger and West24], and by F. Quinn [Reference Quinn19, Reference Quinn20] as the $K$-theoretical obstruction for the topological version of the finiteness theorems for $G$-ANR-spaces. It fits into an exact sequence

(3.1)\begin{align} & H_1(\pi;\widetilde{\bf K}({\mathbb{Z}}[G]))\to Wh(\pi\times G)\to Wh^{top}(\pi\subset\pi\times G) \nonumber\\ &\to H_0(\pi;\widetilde{\bf K}({\mathbb{Z}}[G]))\to \widetilde{K}_0({\mathbb{Z}}[\pi\times G])\to \widetilde{K}_0^{top}({\mathbb{Z}}\pi\subset {\mathbb{Z}}[\pi\times G]) \to\cdots \end{align}

By a theorem of West [Reference West30], compact ANRs are homotopy equivalent to finite CW-complexes. Therefore we have an obstruction $[C_*(\widetilde {f})]\in \widetilde {K}_0({\mathbb {Z}}[\pi \times G])$ in our main pseudo-equivalence extension theorem. Quinn [Reference Quinn19] showed that controlled finitely dominated CW-complexes over $F$ with free $G$-actions have controlled Wall finiteness obstructions in $H_0(F;\widetilde {\bf K}({\mathbb {Z}}[G]))$, and all elements of this group can be realised. By glueing on such an element, we can change the obstruction $[C_*(\widetilde {f})]\in \widetilde {K}_0({\mathbb {Z}}[\pi \times G])$ by any element in the image of $H_0(F;\widetilde {\bf K}({\mathbb {Z}}[G]))$. We will explain in the next paragraph that the natural map $H_0(F;\widetilde {\bf K}({\mathbb {Z}}[G]))\to H_0(Y;\widetilde {\bf K}({\mathbb {Z}}[G]))\to H_0(\pi; \widetilde {\bf K}({\mathbb {Z}}[G]))$ is surjective. Therefore $H_0(F;\widetilde {\bf K}({\mathbb {Z}}[G]))$ and $H_0(\pi; \widetilde {\bf K}({\mathbb {Z}}[G]))$ have the same image in $\widetilde {K}_0({\mathbb {Z}}[\pi \times G])$. Thus we conclude that the obstruction for $G$-ANR pseudo-equivalence extension problem actually lies in the image of $\widetilde {K}_0({\mathbb {Z}}[\pi \times G])$ in $\widetilde {K}_0^{top}({\mathbb {Z}}\pi \subset {\mathbb {Z}}[\pi \times G])$.

Carter's vanishing theorem [Reference Carter9] says that $K_{-i}({\mathbb {Z}}[G])=0$ for all finite groups $G$ and $i>1$. Therefore the spectral sequence that computes $H_0(F;\widetilde {\bf K}({\mathbb {Z}}[G]))$ consists of only $H_0(F;\widetilde {K}_0({\mathbb {Z}}[G]))$ and $H_1(F;\widetilde {K}_{-1}({\mathbb {Z}}[G]))$. The same is true for $H_0(Y;\widetilde {\bf K}({\mathbb {Z}}[G]))$ and $H_0(\pi; \widetilde {\bf K}({\mathbb {Z}}[G]))$. To show the surjection, therefore, we only need to show that $H_i(F;\widetilde {K}_{-i}({\mathbb {Z}}[G]))\to H_i(Y;\widetilde {K}_{-i}({\mathbb {Z}}[G]))\to H_i(\pi; \widetilde {K}_{-i}({\mathbb {Z}}[G]))$ is surjective for $i=0,\,1$. The Smith condition implies that $\pi _iF\to \pi _iY$ is surjective for $i=0,\,1$. This implies the surjections on $H_i(*;\widetilde {K}_{-i}({\mathbb {Z}}[G]))$ for $i=0,\,1$.

4. Calculations and examples

Let $T(r)$ be the mapping torus of a map $S^d\to S^d$ of degree $r$. Let

\[ f\colon F=T(r)\to Y=S^1. \]

be the projection map. For a finite group $G$ of order $n$, we try to extend $F$ to be the fixed set of a finite, semi-free $G$-CW-complex $X$, and extend $f$ to a pseudo-equivalence $g\colon X\to S^1$.

We have $\pi _1(Y)=\langle t\rangle =\{t^i\colon i\in {\mathbb {Z}}\}\cong {\mathbb {Z}}$, and the only non-trivial ${\mathbb {Z}}\langle t\rangle$-homologyFootnote ² of $f$ is

\[ H_d(f;{\mathbb{Z}}\langle t\rangle) ={\mathbb{Z}}\langle t\rangle/(rt-1). \]

For a prime $p$, we have $H_d(f;{\mathbb {Z}}_p\langle t\rangle )=0$ if and only if $p|r$. Therefore, the Smith condition is satisfied for $G$ if and only if

\[ p|n\implies p|r. \]

This is equivalent to $n$ dividing some power of $r$. Under this assumption, the condition for the semi-free pseudo-equivalence extension is the vanishing of

\[ [{\mathbb{Z}}\langle t\rangle/(rt-1)] \in \widetilde{K}_0({\mathbb{Z}}[G]\langle t\rangle). \]

Example 1.1 in the introduction is a direct consequence of proposition 4.1. In fact, $T(p)$ is not only not the fixed set of a semi-free ${\mathbb {Z}}_{p^2}$-action on homotopy circle, it is also not the fixed set of a semi-free ${\mathbb {Z}}_p\times {\mathbb {Z}}_p$-action. The proof given below also shows that $T(p^2)$ is not fixed under a semi-free ${\mathbb {Z}}_p\times {\mathbb {Z}}_{p^2}$-action or ${\mathbb {Z}}_p\times {\mathbb {Z}}_p\times {\mathbb {Z}}_p$-action, etc.

Proof. We have the Bass–Heller–Swan decompositions [Reference Bass, Heller and Swan5]

(4.1)\begin{align} \widetilde{K}_0({\mathbb{Z}}[G]\langle t\rangle) &= \widetilde{K}_0({\mathbb{Z}}[G])\oplus K_{{-}1}({\mathbb{Z}}[G])\oplus NK_0({\mathbb{Z}}[G])\oplus NK_0({\mathbb{Z}}[G]), \end{align}

(4.2)\begin{align} K_1({\mathbb{Z}}_n\langle t\rangle) &= K_1({\mathbb{Z}}_n)\oplus K_0({\mathbb{Z}}_n)\oplus NK_1({\mathbb{Z}}_n)\oplus NK_1({\mathbb{Z}}_n). \end{align}

We also have the pullbacks of rings [Reference Milnor13] ($\Sigma _G=\sum _{g\in G}g$)

that induce the Swan homomorphismsFootnote ³ that are compatible with the Bass–Heller–Swan decompositions

\begin{align*} \partial &\colon K_1({\mathbb{Z}}_n\langle t\rangle)\to \widetilde{K}_0({\mathbb{Z}}[G]\langle t\rangle), \\ \partial &\colon K_1({\mathbb{Z}}_n)\to \widetilde{K}_0({\mathbb{Z}}[G]), \\ \partial &\colon K_0({\mathbb{Z}}_n)\to K_{{-}1}({\mathbb{Z}}[G]), \\ \partial &\colon NK_1({\mathbb{Z}}_n)\to NK_0({\mathbb{Z}}[G]). \end{align*}

We note that our obstruction is an image of the Swan homomorphism

\[ [{\mathbb{Z}}\langle t\rangle/(rt-1)]=\partial[rt-1],\quad [rt-1]\in K_1({\mathbb{Z}}_n\langle t\rangle). \]

Since $n$ divides $r$, we get $[rt-1]=[-1]$, which comes from $K_1({\mathbb {Z}}\langle t\rangle )$. Therefore, $\partial [rt-1]=0$, and the sufficient part of the proposition.

For the necessary part, we note that the Smith condition requires $n$ dividing some power of $r$. Now we identify the obstruction in the Bass–Heller–Swan decomposition. By the proof of the decomposition in [Reference Rosenberg21, Theorem 3.2.22], we write the automorphism of $R\langle t\rangle$ as an automorphism of $R$ with a nilpotent correction

\[ rt-1 =r(t-1)+(r-1) =(r-1)[1+(r-1)^{{-}1}r(t-1)] \in {\mathbb{Z}}_n\langle t\rangle, \]

where $r-1$ is invertible and $(r-1)^{-1}r$ is nilpotent by the Smith condition. This shows that the element $[rt-1]$ becomes $([r-1],\,0,\,0,\,[(r-1)^{-1}r])$ in the decomposition, and our obstruction is $\partial [r-1]\in \widetilde {K}_0({\mathbb {Z}}[G])$ and $\partial [(r-1)^{-1}r]\in NK_0({\mathbb {Z}}[G])$. We will concentrate on the vanishing of $\partial [(r-1)^{-1}r]$, which is the image of $(r-1)^{-1}(rt-1)\in K_1({\mathbb {Z}}_n\langle t\rangle )$ under the Swan homomorphism.

Now we assume $G$ is abelian. Then we have the determinant maps from $K_1$ to the groups of invertible elements. The Swan homomorphism is part of an exact sequence compatible with the determinants

The vanishing of $\partial [(r-1)^{-1}r]$ implies that $[(r-1)^{-1}(rt-1)]$ is in the image of $\alpha$. This further implies that $\det [(r-1)^{-1}(rt-1)]=(r-1)^{-1}(rt-1)$ is in the image of $\beta$. In the appendix of this paper, we prove that $({\mathbb {Z}}[G]/\Sigma _G)\langle t\rangle ^*=({\mathbb {Z}}[G]/\Sigma _G)^*\langle t\rangle$. In other words, the units (i.e., multiplicatively invertible elements) in $({\mathbb {Z}}[G]/\Sigma _G)\langle t\rangle$ are monomials. If $n$ does not divide $r$, then $(r-1)^{-1}(rt-1)$ is not a monomial, and we get a contradiction. This proves the necessary part.

The necessary part of the proof shows that, if $G$ is abelian and $r$ is not a multiple of $n$ (and $n$ divides a power of $r$), then the $K$-theory obstruction $\partial [rt-1]\in \widetilde {K}_0({\mathbb {Z}}[G]\langle t\rangle )$ for $T(r)\to S^1$ has nonzero component $\partial [(r-1)^{-1}r]$ in the Bass Nil group $NK_0({\mathbb {Z}}[G])$. The Nil group is the $K$-theory of the additive category of finitely generated $R$-modules equipped with nilpotent endomorphisms. It is introduced in [Reference Bass4] to account for the difference between the $K$-theory of a ring $R$ and its polynomial extension $R[t]$ or the Laurent extension $R\langle t\rangle$. In case $\pi ={\mathbb {Z}}=\langle t\rangle$ in the exact sequence (3.1), the map $H_0({\mathbb {Z}};\widetilde {\bf K}({\mathbb {Z}}[G]))\to \widetilde {K}_0({\mathbb {Z}}[{\mathbb {Z}}\times G])$ is exactly the comparison (i.e., the inclusion in (4.1)) between $\widetilde {K}_0({\mathbb {Z}}[G])\oplus K_{-1}({\mathbb {Z}}[G])$ and $\widetilde {K}_0({\mathbb {Z}}[G]\langle t\rangle )$. Therefore, the component $\partial [(r-1)^{-1}r]$ in the $NK_0({\mathbb {Z}}[G])$ part of $\widetilde {K}_0({\mathbb {Z}}[G]\langle t\rangle )$, being nonzero, implies that the image of the obstruction $\partial [rt-1]\in \widetilde {K}_0({\mathbb {Z}}[G]\langle t\rangle )$ in $\widetilde {K}_0^{top}({\mathbb {Z}}[{\mathbb {Z}}]\subset {\mathbb {Z}}[{\mathbb {Z}}\times G])$ is nonzero. This image is the $K$-theory obstruction in theorem 3.2. Therefore, the map $T(r)\to S^1$ does not even have semi-free pseudo-equivalence extension in the $G$-ANR category.

In short, there are examples of non-existence of semi-free extension in the $G$-ANR category if and only if the order of $G$ has a square factor.

For the sufficiency part of proposition 4.1, we also give an explicit construction for the special case that $G$ acts freely on a sphere. There is, for example, the case $G$ is a cyclic group acting on the circle $S^1$ by the standard rotations.

Now we turn to another application showing other phenomena. Suppose $n$ is not a prime power. Then $n=n_1n_2$, with $n_1,\,n_2>1$ and coprime. Let $a$ satisfy $a=1$ mod $n_1$ and $a=0$ mod $n_2$. Then $b=1-a$ satisfies $b=0$ mod $n_1$ and $b=1$ mod $n_2$. Let $T(a,\,b)$ be the double torus of two maps of $S^d$ to itself of respective degrees $a,\,b$. Let $f\colon T(a,\,b)\to S^1$ be the projection map. We consider the pseudo-equivalence extension of $f$ for the action by the cyclic group $G={\mathbb {Z}}_n$.

Similar to the mapping torus in the earlier example, the only non-trivial ${\mathbb {Z}}\langle t\rangle$-homology of $f$ is

\[ H_d(f;{\mathbb{Z}}\langle t\rangle) ={\mathbb{Z}}\langle t\rangle/(at-b). \]

By $(at-b)(at^{-1}-b)=a^2=1$ mod $n_1$, and $(at-b)(at^{-1}-b)=b^2=1$ mod $n_2$, and $n_1,\,n_2$ coprime, we know $at-b$ is invertible in ${\mathbb {Z}}_n\langle t\rangle$. This verifies the Smith condition for $f$.

Example 1.2 in the introduction is a direct consequence of proposition 4.3. The proof shows that the obstruction effectively lies in the direct summand $K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n])\subset \widetilde {K}_0({\mathbb {Z}}[{\mathbb {Z}}_n]\langle t\rangle )$ according to (4.1). By theorem 3.2 and the explanation after the proof of proposition 4.1, this implies that, although $T(a,\,b)\to S^1$ has no semi-free pseudo-equivalence extension in the $G$-CW-complex category, the map does have semi-free pseudo-equivalence extension in the $G$-ANR category.

Proof. The obstruction for pseudo-equivalence extension is

\[ [{\mathbb{Z}}\langle t\rangle/(at-b)]\in \widetilde{K}_0({\mathbb{Z}}[{\mathbb{Z}}_n]\langle t\rangle). \]

This is the image of

\[ [at-b]\in {\mathbb{Z}}_n\langle t\rangle^* \subset K_1({\mathbb{Z}}_n\langle t\rangle) \]

under the Swan homomorphism. We carry out the argument similarly to proposition 4.1.

We have $a(1-a)=ab=0$ mod $n$. This means $a^2=a$ mod $n$, or $a$ is an idempotent mod $n$. In particular, $a{\mathbb {Z}}_n$ is a projective ${\mathbb {Z}}_n$-module. By the proof of [Reference Rosenberg21, Theorem 3.2.22], the obstruction $[at-b]$ on the left of (4.2) corresponds to $(0,\,[a{\mathbb {Z}}_n],\,0,\,0)$ on the right, with $[a{\mathbb {Z}}_n]\in K_0({\mathbb {Z}}_n)$. Therefore our obstruction is the image $\partial [a{\mathbb {Z}}_n]\in K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n])$ under the Swan homomorphism. By the calculation of [Reference Bass and Murthy6], this element is a non-divisible element of $K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n])$.

The Swan homomorphism fits into an exact sequence

\begin{align*} &\widetilde{K}_0({\mathbb{Z}})\oplus \widetilde{K}_0({\mathbb{Z}}[{\mathbb{Z}}_n]/\Sigma_{{\mathbb{Z}}_n}) \to \widetilde{K}_0({\mathbb{Z}}_n) \xrightarrow{\partial} K_{{-}1}({\mathbb{Z}}[{\mathbb{Z}}_n]) \\ &\quad\to K_{{-}1}({\mathbb{Z}})\oplus K_{{-}1}({\mathbb{Z}}[{\mathbb{Z}}_n]/\Sigma_{{\mathbb{Z}}_n}) \to K_{{-}1}({\mathbb{Z}}_n). \end{align*}

Here the image of $\widetilde {K}_0({\mathbb {Z}}_n)$ is the same as the image of $K_0({\mathbb {Z}}_n)$. By [Reference Bass and Murthy6, Reference Swan and Evans26], we have $\widetilde {K}_0({\mathbb {Z}})=K_{-1}({\mathbb {Z}})=0$, $\widetilde {K}_0({\mathbb {Z}}[{\mathbb {Z}}_n]/\Sigma _{{\mathbb {Z}}_n})$ is finite, and $\widetilde {K}_0({\mathbb {Z}}_n)$ and $K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n]/\Sigma _{{\mathbb {Z}}_n})$ are free abelian. Therefore $\widetilde {K}_0({\mathbb {Z}}_n)$ embeds as a direct summand of $K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n])$.

If $n=p_1^{m_1}\dots p_l^{m_l}$ is the decomposition into distinct primes, then

\[ K_0({\mathbb{Z}}_n) =\oplus_{i=1}^l K_0({\mathbb{Z}}_{p_i^{m_i}}) =\oplus_{i=1}^l {\mathbb{Z}}. \]

We note that the projective ${\mathbb {Z}}_n$-module $a{\mathbb {Z}}_n$ is isomorphic to ${\mathbb {Z}}_{n_1}$. Under the isomorphism $K_0({\mathbb {Z}}_n)=K_0({\mathbb {Z}}_{n_1})\oplus K_0({\mathbb {Z}}_{n_2})$, $[a{\mathbb {Z}}_n]\in K_0({\mathbb {Z}}_n)$ corresponds to $([{\mathbb {Z}}_{n_1}],\,0)\in K_0({\mathbb {Z}}_{n_1})\oplus K_0({\mathbb {Z}}_{n_2})$. If we start by choosing $n_1=p_1^{m_1}$ and $n_2={n}/{n_1}$, then the obstruction for $f\colon T(a,\,b)\to S^1$ is the image of $(1,\,0,\,\dots,\,0)\in K_0({\mathbb {Z}}_n)$ under the injective Swan homomorphism. Similarly, we can make other choices of $n_1,\,n_2$, such that the obstructions for the corresponding $f\colon T(a,\,b)\to S^1$ are the images of other unit vectors, in $K_0({\mathbb {Z}}_n)$. The upshot is that, if $n$ is not a prime power, then we can construct $f\colon F\to S^1$ satisfying the Smith condition, and the obstruction is a nonzero element in $K_{-1}({\mathbb {Z}}[{\mathbb {Z}}_n])$.

The examples in propositions 4.1 and 4.3 can fit into other spaces.

The condition on the order of $G$ is that $|G|$ is a product of more than one distinct primes.

Proof. The hypothesis that $|G|$ has no square factor implies that $NK_0({\mathbb {Z}}[G])=0$. Then for torsion free $\pi$, the Farrell–Jones conjecture asserts that

\[ K_0({\mathbb{Z}}[\pi\times G]) =K_0({\mathbb{Z}}[G])\oplus H_1(\pi;K_{{-}1}({\mathbb{Z}}[G])). \]

If $H_1\pi =0$, then $H_1(\pi; K_{-1}({\mathbb {Z}}[G]))=0$, and the Swan homomorphism relevant to constructing the $G$-action lies in the $K_0({\mathbb {Z}}[G])$ part. Thus we are reduced to the classical Swan homomorphism. By [Reference Swan25, Corollary 6.1], the Swan homomorphism vanishes for cyclic $G$.

If $H_1\pi \ne 0$, then a generator of $H_1\pi =H_1Y$ can be represented by a loop $S^1\to Y$. By proposition 4.3, there is a map $f\colon F\to S^1$ satisfying the Smith condition but has non-vanishing pseudo-equivalence extension obstruction $[C(\widetilde {f})]\in \widetilde {K}_{-1}({\mathbb {Z}}[G])=H_1(S^1,\,\widetilde {K}_{-1}({\mathbb {Z}}[G]))\subset \widetilde {K}_0({\mathbb {Z}}[G]\langle t\rangle )$, where $t$ is the generator of $\pi _1(S^1)$. We use the loop $S^1\to Y$ to extend $f$ to $f'\colon F\cup _{S_1}Y\to Y$. Then $f'$ also satisfies the Smith condition, and the pseudo-equivalence extension obstruction $[C(\widetilde {f'})]$ for $f'$ is the image of $[C(\widetilde {f})]$ under the homomorphism

\[ \widetilde{K}_{{-}1}({\mathbb{Z}}[G])=H_1(S^1,\widetilde{K}_{{-}1}({\mathbb{Z}}[G])) \to H_1(\pi,\widetilde{K}_{{-}1}({\mathbb{Z}}[G])). \]

Since the circle represents a generator of $H_1\pi$, the image obstruction is still nonzero.

The theorem implies that, if every map $F\to Y$ satisfying the Smith condition has semi-free pseudo-equivalence extension for ${\mathbb {Z}}_n$-action, then $n$ is a product of at least two distinct primes.

Finally, we study the pseudo-equivalence extension of a map between $3$-dimensional lens spaces.

Proof. The obstruction lies in $\widetilde {K}_0({\mathbb {Z}}[{\mathbb {Z}}_p\times {\mathbb {Z}}_p])$, where the first ${\mathbb {Z}}_p=\pi _1(L(p;1))$, and the second ${\mathbb {Z}}_p$ is the action group. The obstruction is given by the chain complex of the map $\widetilde {f}\colon \widetilde {L}(kp;1)\to \widetilde {L}(p;1)$ obtained by pulling back along the universal cover $\widetilde {L}(p;1)=S^3\to L(p;1)$. We have the long exact sequence

\begin{align*} & H_3(L(kp;1);{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}\to H_3(L(p;1);{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}\to H_3(f;{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}_d \\ &\quad\to H_2(L(kp;1);{\mathbb{Z}}[{\mathbb{Z}}_p])=0\to H_2(L(p;1);{\mathbb{Z}}[{\mathbb{Z}}_p])=0\to H_2(f;{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}_k \\ &\quad\to H_1(L(kp;1);{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}_{kp}\to H_1(L(p;1);{\mathbb{Z}}[{\mathbb{Z}}_p])={\mathbb{Z}}_p\to H_1(f;{\mathbb{Z}}[{\mathbb{Z}}_p])=0. \end{align*}

We note that $H_*(f;{\mathbb {Z}}[{\mathbb {Z}}_p])=0,\,0,\,{\mathbb {Z}}_k,\,{\mathbb {Z}}_d,\,0,\,\dots$ are trivial ${\mathbb {Z}}[{\mathbb {Z}}_p\times {\mathbb {Z}}_p]$-modules, and have ${\mathbb {Z}}[{\mathbb {Z}}_p\times {\mathbb {Z}}_p]$-projective resolutions. Therefore, the Euler characteristic of $H_*(f;{\mathbb {Z}}[{\mathbb {Z}}_p])$ gives an element of $K_1({\mathbb {Z}}_{p^2})$ ($p^2$ is the order of the group ${\mathbb {Z}}_p\times {\mathbb {Z}}_p$), and the obstruction is the image of this element under the Swan homomorphism for the group ${\mathbb {Z}}_p\times {\mathbb {Z}}_p$

\[ K_1({\mathbb{Z}}_{p^2})=({\mathbb{Z}}_{p^2})^*\to \widetilde{K}_0({\mathbb{Z}}[{\mathbb{Z}}_p\times {\mathbb{Z}}_p]). \]

In the multiplicative group $({\mathbb {Z}}_{p^2})^*$, the Euler characteristic of $H_*(f;{\mathbb {Z}}[{\mathbb {Z}}_p])$ is ${k}/{d}$. The group $({\mathbb {Z}}_{p^2})^*$ is additively isomorphic to ${\mathbb {Z}}_p\oplus {\mathbb {Z}}_{p-1}$. By [Reference Ullom27, Proposition 3], the image of the Swan homomorphism is an additive group ${\mathbb {Z}}_p\subset \widetilde {K}_0({\mathbb {Z}}[{\mathbb {Z}}_p\times {\mathbb {Z}}_p])$. Therefore an element $r$ is in the kernel of the Swan homomorphism $({\mathbb {Z}}_{p^2})^*\cong {\mathbb {Z}}_p\oplus {\mathbb {Z}}_{p-1}\to {\mathbb {Z}}_p$ if and only if $r^{p-1}=1$. In particular, the pseudo-equivalence extension obstruction $\frac {k}{d}\in ({\mathbb {Z}}_{p^2})^*$ for $f$ vanishes if and only if $({k}/{d})^{p-1}=1$ in $({\mathbb {Z}}_{p^2})^*$. This gives the condition $d^{p-1}=k^{p-1}$ mod $p^2$ in the proposition.