Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains

Contents
A.1.   Introduction 2  
A.2.   A  Leray  theorem  for  infinite chains 6  
A.3.   Compactly  finite and $l_{\hskip 0.70004pt1}$-homology 12  
A.4.   Extensions of  coverings and $l_{\hskip 0.70004pt1}$-homology 19  
A.5.   Removing  weakly $l_{\hskip 0.70004pt1}$-acyclic subspaces 20  
Appendix.   Double complexes 26  
References   27

The paper  is  devoted  to an adaptation of  author’s approach  [$I_{3}$]  to  Leray  theorems  in  bounded cohomology  theory  to  infinite chains.   The paper  may  be considered as a continuation of  the paper  [$I_{3}$],   but  depends on  it  mostly  for  the motivation of  proofs.   Among  the main results are a stronger and  more general  form of  Gromov’s  Vanishing-finiteness  theorem and a generalization of  the first  part  of  his  Cutting-of  theorem.   The proofs do not  depend on any  tools specific for  the bounded cohomology and $l_{\hskip 0.63004pt1}$-homology  theory,   but  use  the fact  that $l_{\hskip 0.63004pt1}$-homology depend only  on  the fundamental  group.   

1. Introduction

Locally,  compactly,  and  star  finite families.   A  family  of  subsets  $\mathcal{U}\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.99997ptU_{\hskip 0.70004pti}\hskip 1.99997pt\}_{\hskip 0.70004pti\hskip 0.70004pt\in\hskip 1.39998ptI}$  of  a set  $X$  is  a map  $i\hskip 3.99994pt\longmapsto\hskip 3.99994ptU_{\hskip 0.70004pti}\hskip 3.00003pt\subset\hskip 3.99994ptX$  from a set  $I$  to  the set  of  all  subsets of  $X$.   The  family  $\mathcal{U}$  is  said  to be  star  finite  if  for every $i\hskip 1.99997pt\in\hskip 3.00003ptI$  the intersection  $U_{\hskip 0.70004pti}\hskip 1.99997pt\cap\hskip 1.99997ptU_{\hskip 0.70004ptj}$  is  non-empty  for only  a  finite number of  $j\hskip 1.99997pt\in\hskip 3.00003ptI$.   Usually,   but  not  always,  only  the set $\{\hskip 1.99997ptU_{\hskip 0.70004pti}\hskip 1.99997pt|\hskip 1.99997pti\hskip 1.99997pt\in\hskip 1.99997ptI\hskip 3.00003pt\}$  matters,  and  we write $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$ instead of  “$U\hskip 3.99994pt=\hskip 3.99994ptU_{\hskip 0.70004pti}$  for some  $i\hskip 1.99997pt\in\hskip 3.00003ptI$”.

Let $X$ be a  topological  space and $\mathcal{U}\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.99997ptU_{\hskip 0.70004pti}\hskip 1.99997pt\}_{\hskip 0.70004pti\hskip 0.70004pt\in\hskip 1.39998ptI}$ be a family  of  subsets of  $X$.   The  family  $\mathcal{U}$ is  said  to be  locally  finite  if  for every $x\hskip 1.99997pt\in\hskip 1.99997ptX$ there exists an open neighborhood $V$ of  $x$ such  that $x\hskip 1.99997pt\in\hskip 1.99997ptU$ and $V\hskip 1.99997pt\cap\hskip 1.99997ptU_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$ for only  a  finite number of  $i\hskip 1.99997pt\in\hskip 3.00003ptI$,   and  compactly  finite  if  for every  compact  $Z\hskip 1.99997pt\subset\hskip 1.99997ptX$  the intersection  $Z\hskip 1.99997pt\cap\hskip 1.99997ptU_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$  for only  a  finite number of  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.   For  locally  compact  spaces  the notions of  locally  finite and compactly  finite families are equivalent.   Eventually  our assumptions will  imply  that  $X$  is  locally  compact,   but  we prefer  to be precise about  which  finiteness condition  is  used.   Gromov  [Gro],  Löh  and  Sauer  [$LS$],  and  Frigerio  and  Moraschini  [F M]  call  compactly  finite  families  “locally  finite”.

Coverings.   The family  $\mathcal{U}$  is  said  to be a  covering  of  $X$  if  the union  $\cup_{\hskip 0.70004pti\hskip 0.70004pt\in\hskip 1.39998ptI}\hskip 1.99997ptU_{\hskip 0.70004pti}$  is  equal  to  $X$.   For a covering  $\mathcal{U}$  we will  denote by  $\mathcal{U}^{\hskip 0.70004pt\cap}$  the collection of  all  non-empty  finite intersection of  elements of  $\mathcal{U}$.   A covering  $\mathcal{U}$  is  said  to be  open if  every  $U_{\hskip 0.70004pti}$  is  open,   and  proper  if  the interiors  $\operatorname{int}\hskip 1.49994ptU_{\hskip 0.70004pti}$  form a covering  of  $X$  and  the closures of  the sets  $U_{\hskip 0.70004pti}$  are compact.   Clearly,   if  there exists a proper covering  of  $X$,   then  $X$  is  locally  compact.   It  is  easy  to see  that  a proper covering  is  compactly  finite  if  and  only  if  it  is  star finite.

Locally  and compactly  finite singular  chains and  homology.   Recall  that  a singular $n$-simplex  in  $X$  is  a continuous map  $\sigma\hskip 1.00006pt\colon\hskip 1.00006pt\Delta^{n}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptX$.   Let  $S_{\hskip 0.35002ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  be  the set  of  singular $n$-simplices in  $X$.   A subset  $I\hskip 1.99997pt\subset\hskip 1.99997ptS_{\hskip 0.35002ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  said  to be  locally  finite  if  the family  of  images  $\{\hskip 1.99997pt\sigma\hskip 1.00006pt(\hskip 1.00006pt\Delta^{n}\hskip 1.49994pt)\hskip 1.99997pt\}_{\hskip 1.39998pt\sigma\hskip 0.70004pt\in\hskip 1.39998ptI}$  is  locally  finite,   and  compactly  finite  if  this family  is  compactly  finite.   An  infinite  singular $n$-chain  in  $X$  is  defined as a  formal  sum

(1.1)

$$\quad c\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998pt\sigma\hskip 1.39998pt\in\hskip 1.39998ptS_{\hskip 0.25002ptn}\hskip 0.70004pt(\hskip 1.04996ptX\hskip 1.04996pt)}\hskip 1.99997pta_{\hskip 0.70004pt\sigma}\hskip 1.00006pt\sigma$$

with coefficients  $a_{\hskip 0.70004pt\sigma}\hskip 1.99997pt\in\hskip 1.99997ptA$,   where  $A$  is  some abelian  group.   Let $C_{\hskip 0.70004ptn}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  be  the group of  such chains.   The chain $c$  is  said  to be  locally  finite  if

$$\quad\mathcal{S}_{\hskip 0.70004ptc}\hskip 3.99994pt=\hskip 3.99994pt\bigl{\{}\hskip 3.00003pt\sigma\hskip 1.00006pt(\hskip 1.49994pt\Delta^{n}\hskip 1.49994pt)\hskip 3.00003pt\bigl{|}\hskip 3.00003pt\sigma\hskip 1.99997pt\in\hskip 1.99997ptS_{\hskip 0.35002ptn}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 0.50003pt,\hskip 3.00003pta_{\hskip 0.70004pt\sigma}\hskip 3.99994pt\neq\hskip 3.99994pt0\hskip 3.00003pt\bigr{\}}$$

is  locally  finite,   and  compactly  finite  if  $\mathcal{S}_{\hskip 0.70004ptc}$  is  compactly  finite.   The groups of  locally  and  compactly  finite chains are denoted  by  $C_{\hskip 0.70004ptn}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  and  $C_{\hskip 0.70004ptn}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  respectively.   It  is  easy  to see  that  every  locally  finite chain  is  compactly  finite.   If  $c$  is  compactly  finite chain,   then  in  the usual  formula for  the boundary  $\partial\hskip 1.00006ptc$  the coefficient  in  front  of  each singular simplex  is  a  finite sum.   Therefore  the boundaries  $\partial\hskip 1.00006ptc$  of  compactly  finite chains,   and  hence of  locally  finite chains,   are  well  defined.   Since a singular simplex  has only  finite number of  faces,   the boundary  of  a  locally  finite singular chain  is  locally  finite,   and  the boundary  of  a  compactly  finite singular chain  is  compactly  finite.   This  leads  to  two  types of  singular  homology  groups based on  in  infinite chains,   namely  the homology  groups  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  based on  locally  finite chains,   and  the homology  groups  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  based on  compactly  finite chains.   From now on we will  assume  that  $A\hskip 3.99994pt=\hskip 3.99994pt\mathbf{R}$  and  omit  the coefficient  group.

The norms of  infinite singular  chains.   The $l_{\hskip 0.70004pt1}$-norm  $\|\hskip 1.99997ptc\hskip 1.99997pt\|$  of  the singular chain  (1.1)  is

$$\quad\|\hskip 1.99997ptc\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998pt\sigma\hskip 1.39998pt\in\hskip 1.39998ptS_{\hskip 0.25002ptn}\hskip 0.70004pt(\hskip 1.04996ptX\hskip 1.04996pt)}\hskip 1.99997pt|\hskip 1.99997pta_{\hskip 0.70004pt\sigma}\hskip 1.99997pt|\hskip 3.00003pt.$$

It  may  happen  that  $\|\hskip 1.99997ptc\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt\infty$.   The $l_{\hskip 0.70004pt1}$-norm  $\|\hskip 1.99997pth\hskip 1.99997pt\|$ of  a homology  class  $h\hskip 1.99997pt\in\hskip 3.00003ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  or  $h\hskip 1.99997pt\in\hskip 3.00003ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  defined as  $\|\hskip 1.99997pth\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.00006pt\inf\hskip 3.00003pt\|\hskip 1.99997ptc\hskip 1.99997pt\|$,   where  the infimum  is  taken over  all  chains $c$ representing  the homology class $h$.   Again,   it  may  happen  that  $\|\hskip 1.99997pth\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt\infty$.

Singular $l_{\hskip 0.70004pt1}$-homology.   For  an  integer  $n\hskip 1.99997pt\geqslant\hskip 1.99997pt0$  let  $L_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  be  the vector space  of  infinite singular $n$-chains with  real  coefficients having  finite $l_{\hskip 0.70004pt1}$-norm.   Such chains are called  $l_{\hskip 0.70004pt1}$-chains  of  dimension $n$.   The $l_{\hskip 0.70004pt1}$-norm  turns  $L_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  into a  Banach  space.   The vector space  $C_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  of  finite singular $n$-chains in  $X$  is  dense in  $L_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  and  the boundary  operator

$$\quad\partial\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 1.04996ptn\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$$

extends by continuity  to  a map  $\partial\hskip 1.00006pt\colon\hskip 1.00006ptL_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptL_{\hskip 1.04996ptn\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   also called  the  boundary  operator.   These boundary  operators  turn  $L_{\hskip 1.04996pt\bullet}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  into a chain complex.   The homology of  is  complex are knows as  $l_{\hskip 0.70004pt1}$-homology  of  $X$  and are denoted  by  $H_{\hskip 0.70004pt*}^{{\hskip 0.70004ptl_{\hskip 0.50003pt1}}}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   The real  vector spaces  $H_{\hskip 0.70004ptn}^{{\hskip 0.70004ptl_{\hskip 0.50003pt1}}}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$  inherit $l_{\hskip 0.70004pt1}$-norms from  $L_{\hskip 1.04996ptn}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   but  in  general  are not  Banach  spaces,   because non-zero $l_{\hskip 0.70004pt1}$-homology  classes may  have $l_{\hskip 0.70004pt1}$-norm equal  to $0$.

Acyclicity of  subsets.   As  in  [$I_{3}$],   let  us  call  a  topological  space  $X$  boundedly  acyclic  if  its bounded cohomology  are isomorphic  to  the bounded cohomology  of  a point.   This property  is  equivalent  to $X$ being  path connected and  its fundamental  group being boundedly  acyclic in an obvious sense.   By  a  theorem of  Sh.  Matsumoto  and  Sh.  Morita  [$MM$]  the space  $X$  is  boundedly  acyclic  if  and  only  if  it  is  path connected and  $H_{\hskip 0.70004pt*}^{{\hskip 0.70004ptl_{\hskip 0.50003pt1}}}\hskip 0.50003pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt0$  for  $n\hskip 1.99997pt\geqslant\hskip 1.99997pt1$.   See also  [$I_{3}$],   Theorem  5.1  for a proof.   In  this paper  we are dealing only  with  homology  and  will  call  boundedly  acyclic spaces and  groups  $l_{\hskip 0.70004pt1}$-acyclic.

In  [$I_{3}$]  a path connected  subset  $Z$ of  $X$  was called  weakly  boundedly  acyclic  if  the image of  the inclusion  homomorphism  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptZ\hskip 0.50003pt,\hskip 1.99997ptz\hskip 1.49994pt)\hskip 1.49994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 1.99997ptz\hskip 1.49994pt)$  is  boundedly  acyclic,   i.e.  is  $l_{\hskip 0.70004pt1}$-acyclic in our current  terminology.   The ambient  space $X$ was fixed.   Now  we need a more flexible  version of  this notion.   Suppose  that  $Z\hskip 1.99997pt\subset\hskip 1.99997ptY\hskip 1.99997pt\subset\hskip 1.99997ptX$.   The subset  $Z$  is  said  to be  weakly $l_{\hskip 0.70004pt1}$-acyclic  in  $Y$  if  the image of  the homomorphism  $\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptZ\hskip 0.50003pt,\hskip 1.99997ptz\hskip 1.49994pt)\hskip 1.49994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.49994pt\pi_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 0.50003pt,\hskip 1.99997ptz\hskip 1.49994pt)$  is  $l_{\hskip 0.70004pt1}$-acyclic.

Acyclicity of  families and coverings.   Let $\mathcal{U}$ be a family  of  subsets  of  $X$.   It  is  said  to be  $l_{\hskip 0.70004pt1}$-acyclic  if  every  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  is  $l_{\hskip 0.70004pt1}$-acyclic.   A covering $\mathcal{U}$ of  $X$  is  said  to be  $l_{\hskip 0.70004pt1}$-acyclic  (as a covering )  if  the family  $\mathcal{U}^{\hskip 0.70004pt\cap}$  is  $l_{\hskip 0.70004pt1}$-acyclic.

A  family  $\mathcal{U}$  is  said  to be  almost  $l_{\hskip 0.70004pt1}$-acyclic  if  every  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  is  $l_{\hskip 0.70004pt1}$-acyclic,   except ,  perhaps,   of  a single exceptional  element  $U_{\hskip 0.35002pte}\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   A covering $\mathcal{U}$ of  $X$  is  said  to be  almost  $l_{\hskip 0.70004pt1}$-acyclic  if  the family  $\mathcal{U}^{\hskip 0.70004pt\cap}$  is  almost  $l_{\hskip 0.70004pt1}$-acyclic and  the exceptional  set  $U_{\hskip 0.35002pte}$ belongs  to  $\mathcal{U}$.

We need an analogue of  weakly  boundedly  acyclic coverings from  [$I_{3}$].   Requiring  sets  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  to be weakly $l_{\hskip 0.70004pt1}$-acyclic in  $X$  is  not  sufficient  for working  with compactly  finite chains.

A family $\mathcal{U}$ of  subsets  of  $X$  is  said  to be  weakly $l_{\hskip 0.70004pt1}$-acyclic  if  for every  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  a subset  $U_{\hskip 0.70004pt+}\hskip 1.99997pt\subset\hskip 1.99997ptX$  is  given,   such  that  $U\hskip 1.99997pt\subset\hskip 1.99997ptU_{\hskip 0.70004pt+}$,   the set  $U$  is  weakly $l_{\hskip 0.70004pt1}$-acyclic  in  $U_{\hskip 0.70004pt+}$,   and  the family  of  subsets  $U_{\hskip 0.70004pt+}$  is  compactly  finite.   A  finite number of  subsets  $U_{\hskip 0.70004pt+}$  can  be equal  to  $X$.   A covering $\mathcal{U}$ of  $X$  is  said  to be  weakly $l_{\hskip 0.70004pt1}$-acyclic  if  the family  $\mathcal{U}^{\hskip 0.70004pt\cap}$  is  weakly $l_{\hskip 0.70004pt1}$-acyclic.

Similarly,   a  family  $\mathcal{U}$  is  said  to be  almost  weakly $l_{\hskip 0.70004pt1}$-acyclic  if  subsets  $U_{\hskip 0.70004pt+}\hskip 1.99997pt\subset\hskip 1.99997ptX$  with  the properties  listed  above are given  for  every subset  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$,   except ,  perhaps,   of  a single exceptional  set  $U_{\hskip 0.35002pte}\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   A covering $\mathcal{U}$ of  $X$  is  said  to be  almost  weakly  $l_{\hskip 0.70004pt1}$-acyclic  if  the family  $\mathcal{U}^{\hskip 0.70004pt\cap}$  is  almost  weakly  $l_{\hskip 0.70004pt1}$-acyclic and  the exceptional  set  $U_{\hskip 0.35002pte}$ belongs  to  $\mathcal{U}$.

Infinite chains  in  simplicial  complexes.   A simplicial  complex $S$  is  said  to be  star  finite  if  each  its simplex  is  contained  in  only  a finite number of  simplices.   Equivalently,  $S$  is  star  finite  if  the family  of  its  simplices  is  star  finite.   If  $\mathcal{U}$  is  a family  of  subsets of  $X$,   then  $\mathcal{U}$  is  star  finite  if  and  only  if  the nerve  $N_{\hskip 1.04996pt\mathcal{U}}$  of  $\mathcal{U}$  is  star  finite.

An  infinite $n$-chain  of  a simplicial  complex $S$  is  a potentially  infinte formal  sum of  $n$-simplices of  $S$  with coefficients in some abelian  group  $A$.   If  $S$  is  star  finite,   then  the usual  formula defines  the boundaries  $\partial\hskip 1.00006ptc$  of  infinite $n$-chains of  $S$.   This  leads  to homology  groups  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptS\hskip 0.50003pt,\hskip 1.99997ptA\hskip 1.49994pt)$  based on  infinite chains.   We will  always assume  that  $A\hskip 3.99994pt=\hskip 3.99994pt\mathbf{R}$.

Leray  homomorphisms.   Suppose  that  $\mathcal{U}$  is  a star  finite proper covering of  $X$.   Let  $N_{\hskip 1.04996pt\mathcal{U}}$  be  the nerve of  $\mathcal{U}$.   Then  there  is  a canonical  Leray  homomorphism

$$\quad l_{\hskip 0.70004pt\mathcal{U}}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN_{\hskip 1.04996pt\mathcal{U}}\hskip 1.49994pt)\hskip 3.00003pt.$$

See  Section  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.   The  first  Leray  theorem  for $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  the following  theorem.

Theorem  A.   Suppose  that  $\mathcal{U}$ is  a star  finite proper covering,   and  that  $\mathcal{U}$ is  countable and  weakly $l_{\hskip 0.70004pt1}$-acyclic.   If  a  compactly  finite  homology  class  $h\hskip 1.99997pt\in\hskip 3.00003ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  belongs  to  the kernel  of  the  Leray  homomorphism  $l_{\hskip 0.70004pt\mathcal{U}}$,   then  $\|\hskip 1.99997pth\hskip 1.99997pt\|\hskip 3.99994pt=\hskip 3.99994pt0$.   

The second  Leray  theorem  is  concerned  with almost  weakly $l_{\hskip 0.70004pt1}$-acyclic coverings.

Theorem  B.   Suppose  that  $\mathcal{U}$ is  a star  finite proper covering,   and  that  $\mathcal{U}$ is  countable and  almost  weakly $l_{\hskip 0.70004pt1}$-acyclic.   If  a  compactly  finite  homology  class  $h\hskip 1.99997pt\in\hskip 3.00003ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  belongs  to  the kernel  of  the  Leray  homomorphism  $l_{\hskip 0.70004pt\mathcal{U}}$,   then  $\|\hskip 1.99997pth\hskip 1.99997pt\|\hskip 3.99994pt<\hskip 3.99994pt\infty$.   

See  Theorems  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  and  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  respectively.   These  results are motivated  by  Gromov’s  Vanishing-Finiteness  theorem.   See  [Gro],   Section  4.2.   Like  Leray  theorems of  [$I_{3}$],  Theorems  A  and  B  are deduced  from an abstract  Leray  theorem,   Theorem  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.   The proofs are elementary  and are based on an adaptation of  the methods of  [$I_{1}$],  [$I_{2}$],  and  [$I_{3}$]  to compactly  finite chains.   The same methods  lead  to a proof  of  the  Vanishing-Finiteness  theorem.   See  Section  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.

The assumptions of  Theorems  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  and  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  are much  weaker  than  Gromov’s.   In  the  Vanishing-Finiteness  theorem  the space  $X$  is  assumed  to be a manifold,   instead of  $l_{\hskip 0.70004pt1}$-acyclicity  the stronger  amenability  property  is  used,   and  it  is  assumed  that  $h\hskip 1.99997pt\in\hskip 3.00003ptH_{\hskip 0.70004ptn}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  for some $n$  strictly  larger  than  the dimension of  $N_{\hskip 1.04996pt\mathcal{U}}$.

Gromov’s  proof  of  the  Vanishing-Finiteness  theorem  was recently  reconstructed  by  R.  Frigerio  and  M.  Moraschini  [F M].   Their  proof  is  based on  Gromov’s  theories of  multicomplexes and of  diffusion of  chains,   and  is  far from  being elementary.   Technical  difficulties forced  Frigerio  and  Moraschini  to  consider only  triangulable spaces,   although  they  conjectured  that  this assumption  is  superfluous.   Theorems  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  and  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  together  imply  this conjecture.

Removing  subspaces.   Let  $X$  be a  topological  space,   and  let  $Y\hskip 1.99997pt\subset\hskip 1.99997ptX$  be a closed subset.   There exists natural  (in an  informal  sense)  chain  maps

$$\quad r_{\hskip 0.70004pt\smallsetminus\hskip 0.70004ptY}\hskip 1.99997pt\colon\hskip 1.00006ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.99997pt\smallsetminus\hskip 1.99997ptY\hskip 1.49994pt)\hskip 3.00003pt.$$

See  Section  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.   The maps  $r_{\hskip 0.70004pt\smallsetminus\hskip 0.70004ptY}$  depend on many choices,   but  the maps

$$\quad r_{\hskip 0.70004pt\smallsetminus\hskip 0.70004ptY\hskip 1.04996pt*}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.99997pt\smallsetminus\hskip 1.99997ptY\hskip 1.49994pt)\hskip 3.00003pt.$$

induced  by  $r_{\hskip 0.70004pt\smallsetminus\hskip 0.70004ptY}$  does not  depend on  these choices.   Suppose now  that  $Y$  is  presented as  the union of  a  family  $\mathcal{Z}$  of  pair-wise disjoint  compact  subspaces of  $X$.   Suppose  further  that  for  every  $Z\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{Z}$  a compact  neighborhood  $C_{\hskip 1.04996ptZ}$  of  $Z$  is  given,   and  that  the neighborhoods  $C_{\hskip 1.04996ptZ}$  are pair-wise disjoint.   Suppose  that  every  $C_{\hskip 1.04996ptZ}$  is  Hausdorff  and  path connected.

Theorem  C.   Suppose  that  $\mathcal{Z}$ is  countable and  the  family  of  sets  $C_{\hskip 1.04996ptZ}$  is  weakly  $l_{\hskip 0.70004pt1}$-acyclic.   Then  $\|\hskip 1.99997ptr_{\hskip 0.70004pt\smallsetminus\hskip 0.70004ptY}\hskip 1.49994pt(\hskip 1.00006pth\hskip 1.49994pt)\hskip 1.99997pt\|\hskip 3.99994pt\geqslant\hskip 3.99994pt\|\hskip 1.99997pth\hskip 1.99997pt\|$  for every  homology  class  $h\hskip 1.99997pt\in\hskip 1.99997ptH_{\hskip 0.70004ptn}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   

See  Theorem  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.   Implicitly  this  theorem  is  concerned  with  the covering of  $X$  by  $X\hskip 1.99997pt\smallsetminus\hskip 1.99997ptY$  and  the sets  $C_{\hskip 1.04996ptZ}$.   Since  this covering  is  very simple,   there  is  no need  to involve  it  or  related  double complexes explicitly.   Theorem  C  was motivated  by  Gromov’s  Cutting-of  theorem  from  [Gro],   Section  4.2,   and easily  implies  its  first  claim.   See  Section  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.

2. A  Leray  theorem  for  infinite  chains

Generalized  chains.   Let $X$ be a  topological  space.   Let $\operatorname{\textit{sub}}\hskip 1.99997ptX$ be  the category  having  subspaces of  $X$  as objects and  inclusions  $Y\hskip 1.99997pt\subset\hskip 3.00003ptZ$  as morphisms.   Let  $e_{\hskip 0.70004pt\bullet}$  be  a covariant  functor  from  $\operatorname{\textit{sub}}\hskip 1.99997ptX$  to augmented chain complexes of  modules over a ring  $R$.   The  functor  $e_{\hskip 0.70004pt\bullet}$  assigns  to a subspace $Z\hskip 1.99997pt\subset\hskip 3.00003ptX$ a  complex

(2.1)

For every  $Y\hskip 1.99997pt\subset\hskip 1.99997ptZ$  there  is  a  inclusion  morphism  $e_{\hskip 1.04996pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptZ\hskip 1.49994pt)\hskip 1.00006pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.00006pte_{\hskip 1.04996pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   Elements  of  $e_{\hskip 0.35002ptq}\hskip 1.00006pt(\hskip 1.49994ptZ\hskip 1.49994pt)$ are  thought  as  generalized $q$-chains  of  $Z$.

The double complex of  a covering.   Let $\mathcal{U}$ be a star  finite covering  of  $X$  and  let  $N$  be  its  nerve.   For  $p\hskip 1.99997pt\geqslant\hskip 1.99997pt0$  let  $N_{\hskip 0.70004ptp}$  be  the set  of  $p$-dimensional  simplices of  $N$.   For  $p\hskip 0.50003pt,\hskip 3.00003ptq\hskip 1.99997pt\geqslant\hskip 1.99997pt0$  let

$$\quad c_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004ptq}\hskip 1.49994pt)\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt\prod\nolimits_{\hskip 1.39998pt\sigma\hskip 1.39998pt\in\hskip 1.39998ptN_{\hskip 0.50003ptp}}\hskip 1.99997pte_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 3.00003pt.$$

So,   an element $c\hskip 1.99997pt\in\hskip 1.99997ptc_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004ptq}\hskip 1.49994pt)$ is  a  family of  generalized $q$-chains

$$\quad c\hskip 1.00006pt\colon\hskip 1.00006pt\sigma\hskip 3.99994pt\longmapsto\hskip 3.99994ptc_{\hskip 0.70004pt\sigma}\hskip 3.99994pt\in\hskip 3.99994pte_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 3.00003pt,$$

where  $\sigma\hskip 1.99997pt\in\hskip 1.99997ptN_{\hskip 0.70004ptp}$,   thought  as an  infinite formal  sum

$$\quad c\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998pt\sigma\hskip 1.39998pt\in\hskip 1.39998ptN_{\hskip 0.50003ptp}}\hskip 1.99997ptc_{\hskip 0.70004pt\sigma}\hskip 3.00003pt.$$

For every  $p\hskip 1.99997pt>\hskip 1.99997pt0$  (and  sometimes for $p\hskip 3.99994pt=\hskip 3.99994pt0$ also)  there  is  a canonical  morphism

$$\quad\delta_{\hskip 0.35002ptp}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 0.70004ptp\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt)\hskip 3.00003pt,$$

defined as follows.   Let  $\sigma\hskip 1.99997pt\in\hskip 1.99997ptN_{\hskip 0.70004ptp}$.   For each  face  $\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma$  there  is  an  inclusion  morphism

$$\quad\Delta_{\hskip 1.39998pt\sigma\hskip 0.35002pt,\hskip 0.70004pti}\hskip 3.00003pt\colon\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994pt|\hskip 1.99997pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma\hskip 1.99997pt|\hskip 1.49994pt)\hskip 3.00003pt.$$

For  $c_{\hskip 0.70004pt\sigma}\hskip 1.99997pt\in\hskip 1.99997pte_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)$  we set

$$\quad\delta_{\hskip 0.35002ptp}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)\hskip 1.99997pt\colon\hskip 1.99997pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma\hskip 3.99994pt\longmapsto\hskip 3.99994pt(\hskip 1.00006pt-\hskip 1.99997pt1\hskip 1.00006pt)^{\hskip 0.70004pti}\hskip 1.99997pt\Delta_{\hskip 1.39998pt\sigma\hskip 0.35002pt,\hskip 0.70004pti}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)\hskip 3.99994pt\in\hskip 3.99994pte_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\quad\ \mbox{and}$$

$$\quad\delta_{\hskip 0.35002ptp}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)\hskip 1.99997pt\colon\hskip 1.99997pt\tau\hskip 3.99994pt\longmapsto\hskip 3.99994pt0$$

if  $\tau\hskip 3.99994pt\neq\hskip 3.99994pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma$  for every  $i$.   The map  $\delta_{\hskip 0.35002ptp}$  extends  to  the direct  product  $c_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004ptq}\hskip 1.49994pt)$  by  linearity .   In order  to see  that  such an extension  to be well  defined  we need  to know  that  for every  $\rho\hskip 1.99997pt\in\hskip 3.00003ptN_{\hskip 0.70004ptp\hskip 0.70004pt-\hskip 0.70004pt1}$  only  a  finite number of  expressions  $(\hskip 1.00006pt-\hskip 1.99997pt1\hskip 1.00006pt)^{\hskip 0.70004pti}\hskip 1.99997pt\Delta_{\hskip 1.39998pt\sigma\hskip 0.35002pt,\hskip 0.70004pti}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)$  need  to be summed  to get  the value of  $\delta_{\hskip 0.35002ptp}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)$  on  $\rho$.   But $(\hskip 1.00006pt-\hskip 1.99997pt1\hskip 1.00006pt)^{\hskip 0.70004pti}\hskip 1.99997pt\Delta_{\hskip 1.39998pt\sigma\hskip 0.35002pt,\hskip 0.70004pti}\hskip 1.00006pt\left(\hskip 1.49994ptc_{\hskip 0.70004pt\sigma}\hskip 1.49994pt\right)$ enters  this sum only  if  $\rho\hskip 3.99994pt=\hskip 3.99994pt\partial_{\hskip 0.70004pti}\hskip 1.00006pt\sigma$  for  some  $\sigma\hskip 1.99997pt\in\hskip 3.00003ptN_{\hskip 0.70004ptp}$  and  some $i$.   Since  the covering  $\mathcal{U}$  is  star  finite,   its  nerve  $N$  is  also star  finite,   and  hence  there  is  indeed only  a finite number of  such  $\rho\hskip 0.50003pt,\hskip 1.99997pti$.   Therefore  $\delta_{\hskip 0.35002ptp}$  is  indeed  well  defined.   As usual,   we agree  that  $|\hskip 1.99997pt\varnothing\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994ptX$,   but  this argument  does not  work for  $p\hskip 3.99994pt=\hskip 3.99994pt0$  and  $\rho\hskip 3.99994pt=\hskip 3.99994pt\varnothing$.   In  order  to define  $\delta_{\hskip 0.70004pt0}$  one needs  to be able  to speak about  infinite generalized chains.

The fact  that  each  $\Delta_{\hskip 1.39998pt\sigma\hskip 0.35002pt,\hskip 0.70004pti}$  are  morphisms of  complexes implies  that  $\delta_{\hskip 0.35002ptp}$  is  a  morphism also of  complexes.   The  double complex  $c_{\hskip 1.04996pt\bullet}\hskip 0.50003pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 1.04996pt\bullet}\hskip 1.49994pt)$  of  the covering  $\mathcal{U}$  is  the double complex

(2.2)

where  the horizontal  arrows are  the products of  the maps  $d_{\hskip 0.70004pti}$  and  the vertical  arrows are  the maps  $\delta_{\hskip 0.70004pti}$.   Let  $t_{\hskip 1.04996pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)$  be  the  total  complex of  $c_{\hskip 1.04996pt\bullet}\hskip 0.50003pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 1.04996pt\bullet}\hskip 1.49994pt)$.   Let  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 0.50003pt(\hskip 1.49994ptN\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptS\hskip 0.50003pt,\hskip 1.99997ptR\hskip 1.49994pt)$  be  the complex of  infinite  simplicial  chains of  $N$  with coefficients  in  $R$.   It  is  well  defined  because  $N$  is  star  finite.   Let  $H_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptH_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 1.99997ptR\hskip 1.49994pt)$  be  the homology  of  this complex.   Since  $\delta_{\hskip 0.70004pt0}$  is  not  defined,   we will  replace  $e_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  by  the cokernel  $e_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  of  the homomorphism

$$\quad\delta_{\hskip 0.70004pt1}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt)\hskip 3.00003pt.$$

Then  we can define  $\delta_{\hskip 0.70004pt0}$  as  the canonical  map  $c_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.00006pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$.   Since  (2.2)  is  commutative,   the maps  $d_{\hskip 0.70004pti}$  induce canonical  maps

$$\quad d_{\hskip 0.70004pti}\hskip 1.00006pt\colon\hskip 1.00006pte_{\hskip 0.70004pti}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pti\hskip 0.70004pt-\hskip 0.70004pt1}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$$

turning  $e_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  into a complex.   Let  $\widetilde{H}_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  be  the homology  of  this complex.   The boundary  maps  $d_{\hskip 1.04996pt0}$  and  $\delta_{\hskip 0.70004pt1}$  lead  to morphisms

$$\quad\lambda_{\hskip 0.70004pte}\hskip 1.99997pt\colon\hskip 1.99997ptt_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 1.49994pt)\quad\ \mbox{and}\quad\ \tau_{\hskip 0.70004pte}\hskip 1.99997pt\colon\hskip 1.99997ptt_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 3.00003pt,$$

where  it  is  understood  that  the augmentation  term  is  removed  from  $e_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$.

Acyclic coverings.   Clearly,  $\mathcal{U}^{\hskip 0.70004pt\cap}$  is  the collection of  all  sets of  the form  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|$  with  $\sigma\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$.   The covering  $\mathcal{U}$  is  said  to be  $e_{\hskip 0.35002pt\bullet}$-acyclic   if  $e_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptZ\hskip 1.49994pt)$  is  exact  for every  $Z\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}^{\hskip 0.70004pt\cap}$.

2.1. Lemma.   If  $\mathcal{U}$  is  star  finite and  $e_{\hskip 0.35002pt\bullet}$-acyclic,   then  $\lambda_{\hskip 0.70004pte}\hskip 1.00006pt\colon\hskip 1.00006ptt_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 1.49994pt)$  induces an  isomorphism of  homology  groups.   

Proof.   If  $\mathcal{U}$  is  $e_{\hskip 0.35002pt\bullet}$-acyclic,   then  for every  simplex  $\sigma\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$  the complex  (2.1)  is  exact .   Since  the  term-wise products of  exact  sequences are exact,   this implies  that  every  row  of  the double complex  (2.2)  is  exact  and  $d_{\hskip 1.04996pt0}$  induces an  isomorphism of  the complex  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 1.49994pt)$  with  the kernel  of  the morphism of  complexes  $d_{\hskip 0.70004pt1}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt1}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 1.04996pt0}\hskip 1.49994pt)$.   It  remains  to apply  a well  known  theorem about  double complexes.   See  Theorem  A.2  in  [$I_{3}$].    $\blacksquare$

Infinite singular  chains.   Suppose  that  a  space $\Delta$ is  fixed and  maps  $s\hskip 1.00006pt\colon\hskip 1.00006pt\Delta\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptY$  are  treated as singular simplices.   A  finite singular  chain  is  a finite formal  sum of  singular  simplices with coefficients in  $R$.   The $R$-module of  finite singular chains  in  $Y$  is  denoted  by  $c\hskip 1.49994pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   An  infinite singular  chain  is  a finite or  infinite formal  sum of  singular  simplices with coefficients in  $R$,   and  the $R$-module of  infinite singular chains  in  $Y$  is  denoted  by  $c^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.49994pt(\hskip 1.49994ptY\hskip 1.49994pt)$.

Let  us  turn  to singular chains  in  $X$  and subsets of  $X$.   A singular  simplex $s$  in  $X$  is  called  small   if  $s\hskip 1.00006pt(\hskip 1.49994pt\Delta\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997ptU$  for  some  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$,   and an  infinite singular chain  in  $X$  is  called  small   if  all  its  singular  simplices with non-zero coefficients are small.   Suppose  that  for every  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  a  finite singular chain  $\gamma_{\hskip 0.70004ptU}\hskip 1.99997pt\in\hskip 1.99997ptc\hskip 1.49994pt(\hskip 1.49994ptU\hskip 1.49994pt)$  is  given.   Then,   since  $\mathcal{U}$  is  a star  finite,   the sum

$$\quad\gamma\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998ptU\hskip 1.39998pt\in\hskip 1.39998pt\mathcal{U}}\hskip 1.99997pt\gamma_{\hskip 0.70004ptU}$$

is  a  well  defined  infinite chain.   Clearly,  $\gamma$  is  small.   If  $\mathcal{U}$  is  an  open covering ,   then  $\gamma$  is  locally  finite.   When  a chain  $\gamma$  can  be represented  by  such sum,   we  say  that  $\gamma$  is  $\mathcal{U}$-finite.   Let  $c^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 1.99997pt\mathcal{U}\hskip 1.49994pt)$  be  the $R$-module of  $\mathcal{U}$-finite chains.   Let  us  consider  now  the modules

$$\quad c_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt\prod\nolimits_{\hskip 1.39998pt\sigma\hskip 1.39998pt\in\hskip 1.39998ptN_{\hskip 0.50003ptp}}\hskip 1.99997ptc\hskip 1.00006pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 3.00003pt,$$

where  $p\hskip 1.99997pt\geqslant\hskip 1.99997pt-\hskip 1.99997pt1$.   The maps  $\delta_{\hskip 0.35002ptp}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 0.70004ptp\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)$,  $p\hskip 1.99997pt>\hskip 1.99997pt0$,   are defined as before.   Moreover ,   now  we can define  $\delta_{\hskip 1.04996pt0}$  in  the same way,   except  that  now  the  target  of  $\delta_{\hskip 1.04996pt0}$  is  $c^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   not  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$.   Clearly,  $c^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  is  equal  to  the image of

$$\quad\delta_{\hskip 1.04996pt0}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.00006ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.00006pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.00003pt,$$

and  $\delta_{\hskip 1.04996pt0}$  is  the composition of  the inclusion  $c^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  with a  canonical  map

$$\quad\overline{\delta}_{\hskip 1.04996pt0}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 3.00003pt.$$

Clearly,   $\overline{\delta}_{\hskip 1.04996pt0}$  is  surjective.   As usual,  $\delta_{\hskip 0.35002ptp\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.00006pt\circ\hskip 1.99997pt\delta_{\hskip 0.35002ptp}\hskip 3.99994pt=\hskip 3.99994pt0$  for  every  $p\hskip 1.99997pt\geqslant\hskip 1.99997pt1$.

2.2. Lemma.   If  $\mathcal{U}$  is  star  finite,   then  the  following  sequence  is  exact :

Proof.   It  is  sufficient  to  construct  a contracting  chain  homotopy

$$\quad k_{\hskip 1.39998pt0}\hskip 1.00006pt\colon\hskip 1.00006ptc^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 1.04996pt0}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 1.00006pt,\quad\$$

$$\quad k_{\hskip 1.04996ptp}\hskip 1.00006pt\colon\hskip 1.00006ptc_{\hskip 1.04996ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptc_{\hskip 1.04996ptp\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)\hskip 1.00006pt,$$

where  $p\hskip 1.99997pt\geqslant\hskip 1.99997pt0$.   The construction  is  almost  the same as in  the case of  direct  sums  (instead of  products).   Cf.  [$I_{3}$],   Lemma  3.1.   For every  small  singular simplex $s$ let  us  choose a subset  $U_{\hskip 0.70004pts}\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  such  that  $s\hskip 1.49994pt(\hskip 1.00006pt\Delta\hskip 1.00006pt)\hskip 1.99997pt\subset\hskip 3.00003ptU_{\hskip 0.70004pts}$  and  let  $u_{\hskip 0.35002pts}$  be  the corresponding  vertex of  $N$.   If  $s\hskip 1.00006pt(\hskip 1.00006pt\Delta\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$  for some  $\sigma\hskip 1.99997pt\in\hskip 1.99997ptN_{\hskip 0.70004ptp}$,   then  $s\hskip 1.00006pt*\hskip 1.00006pt\sigma$  denotes $s$ considered as an element  of  $c\hskip 1.49994pt(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt)$.

Let  us  define  $k_{\hskip 1.04996ptp}$ on  the chains of  the form  $s\hskip 1.00006pt*\hskip 1.00006pt\sigma$  first.   Suppose  that  $\sigma\hskip 1.99997pt\in\hskip 1.99997ptN_{\hskip 0.70004ptp}$ and  $s$  be a singular $q$-simplex such  that  $s\hskip 1.00006pt(\hskip 1.00006pt\Delta^{\hskip 0.35002ptq}\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$.   Let  $\rho\hskip 3.99994pt=\hskip 3.99994pt\sigma\hskip 1.00006pt\cup\hskip 1.00006pt\{\hskip 1.49994ptu_{\hskip 0.70004pts}\hskip 1.49994pt\}$.   Then  $s\hskip 1.00006pt(\hskip 1.00006pt\Delta\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 1.99997pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt\cap\hskip 1.99997ptU_{\hskip 0.70004pts}\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\rho\hskip 1.99997pt|$  and,   in  particular ,  $\rho$  is  a simplex.   If  $u_{\hskip 0.70004pts}\hskip 1.99997pt\in\hskip 3.00003pt\sigma$,   then  $\rho$  is  a $p$-simplex.   Otherwise,  $\rho$  is  a $(\hskip 1.00006ptp\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.00006pt)$-simplex and  $\sigma\hskip 3.99994pt=\hskip 3.99994pt\partial_{\hskip 0.35002pta}\hskip 1.49994pt\rho$  for some  $a$.   Let

$$\quad k_{\hskip 1.04996ptp}\hskip 1.00006pt(\hskip 1.00006pts\hskip 1.00006pt*\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt0\quad\ \mbox{if}\quad\ u_{\hskip 0.70004pts}\hskip 1.99997pt\in\hskip 3.00003pt\sigma\hskip 3.00003pt,$$

$$\quad k_{\hskip 1.04996ptp}\hskip 1.00006pt(\hskip 1.00006pts\hskip 1.00006pt*\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.99997pt-\hskip 1.99997pt1\hskip 1.49994pt)^{\hskip 0.70004pta}\hskip 1.99997pts\hskip 1.00006pt*\hskip 1.00006pt\rho\hskip 3.99994pt\in\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994pt|\hskip 1.99997pt\rho\hskip 1.99997pt|\hskip 1.49994pt)\quad\ \mbox{if}\quad\ u_{\hskip 0.70004pts}\hskip 1.99997pt\not\in\hskip 3.00003pt\sigma\hskip 3.00003pt.$$

As  in  the case of  $\delta_{\hskip 0.70004ptp}$,   the star  finitness of  $\mathcal{U}$ and $N$ allows  to extend  $k_{\hskip 1.04996ptp}$  to $c_{\hskip 1.04996ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)$  by  linearity.   In  order  to verify  that $k_{\hskip 0.70004pt\bullet}$  is  a contracting  homotopy  it  is  sufficient  to check  that

$$\quad\delta_{\hskip 0.70004ptp\hskip 0.70004pt+\hskip 0.70004pt1}\hskip 1.49994pt\bigl{(}\hskip 1.49994ptk_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994pt\gamma\hskip 1.49994pt)\hskip 1.99997pt\bigr{)}\hskip 3.99994pt+\hskip 3.99994ptk_{\hskip 0.70004ptp\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.49994pt\bigl{(}\hskip 1.99997pt\delta_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994pt\gamma\hskip 1.49994pt)\hskip 1.99997pt\bigr{)}\hskip 3.99994pt=\hskip 3.99994pt\gamma$$

when  $\gamma$  has  the  form  $\gamma\hskip 3.99994pt=\hskip 3.99994pts\hskip 1.00006pt*\hskip 1.00006pt\sigma$.   But  this case  is  exactly  the same as for direct  sums.    $\blacksquare$

Classical  singular  chains.   The above discussion applies,   in  particular ,   to  the case  $\Delta\hskip 3.99994pt=\hskip 3.99994pt\Delta^{q}$,   the standard  geometric $q$-simplex.   In  this case we will  denote  $c_{\hskip 0.70004ptp}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptc\hskip 1.49994pt)$  and  $c^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  by

$$\quad C_{\hskip 0.70004ptp}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC_{\hskip 0.70004ptq}\hskip 1.49994pt)\quad\ \mbox{and}\quad\ C_{\hskip 0.70004ptq}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$$

respectively.   The boundary  maps  $d_{\hskip 0.70004pti}$  turn  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  into a complex.   Let  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  be  the homology  of  this complex.   The morphisms

lead  to homomorphisms of  cohomology  groups,

where  $H_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.00006pt)$  is  the homology  of  $t_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.00006pt)$.   If  $\mathcal{U}$  is  star  finite,   Lemma  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  implies  that  the columns of  (2.2)  are exact.   Together  with  the  already  used  theorem about  double complexes  this implies  that  $\tau_{\hskip 1.04996ptC\hskip 0.70004pt*}$  is  an  isomorphism.   This  leads  to  the canonical  homomorphism

(2.3)

$$\quad\lambda_{\hskip 1.39998ptC\hskip 0.70004pt*}\hskip 1.00006pt\circ\hskip 1.00006pt\tau_{\hskip 1.04996ptC\hskip 0.70004pt*}^{\hskip 0.70004pt-\hskip 0.70004pt1}\hskip 1.99997pt\colon\hskip 1.99997ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 1.49994pt)\hskip 3.00003pt.$$

In  general,   one cannot  replace here  $H_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  by  some homology  independent  of  $\mathcal{U}$.

Comparing  classical  and  generalized chains.   Suppose  that  the functor  $e_{\hskip 0.70004pt\bullet}$  is  equipped  with a natural  transformation  $\varphi_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 0.70004pt\bullet}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pt\bullet}$.   Then  $\varphi$  induces a map  $H_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\widetilde{H}_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$  for every  $Y\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$,   where  $\widetilde{H}_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$  is  the  homology  of  the complex  $e_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   In  particular ,  $\varphi$  induces a map  $H_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\widetilde{H}_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$,   but  this  is  not  what  we are interested  in  now.   Lemma  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  implies  that  the complex  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  is  canonically  isomorphic  to  the cokernel  of

$$\quad\delta_{\hskip 0.70004pt1}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 1.04996pt1}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC_{\hskip 0.70004pt\bullet}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 1.04996pt0}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC_{\hskip 0.70004pt\bullet}\hskip 1.49994pt)\hskip 3.00003pt.$$

In  view of  the definition of  $e_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  this  leads  to a canonical  homomorphism

$$\quad\varphi_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$$

and  hence  to a  comparison  homomorphism

(2.4)

$$\quad\varphi_{\hskip 0.70004pt*}\hskip 1.00006pt\colon\hskip 1.00006ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\widetilde{H}_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.99997ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$$

in  homology  groups.   This  is  the map  we are interested  in.

2.3. Theorem.   If  $\mathcal{U}$  is  a  star  finite  $e_{\bullet}$-acyclic  covering,   then  the  comparison  homomorphism  (2.4)  can  be  factored  through  the  canonical  homomorphism  (2.3).   

Proof.   The natural  transformation  $\varphi_{\hskip 0.70004pt\bullet}$  defines  homomorphisms

$$\quad\varphi_{\hskip 0.70004ptq}\hskip 1.00006pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pte_{\hskip 0.70004ptq}\hskip 0.50003pt\left(\hskip 1.49994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.49994pt\right)\hskip 3.00003pt,$$

which,   in  turn,   lead  to a morphism

$$\quad\varphi_{\hskip 0.70004pt\bullet\hskip 0.70004pt\bullet}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC_{\hskip 0.70004pt\bullet}\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptC_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte_{\hskip 0.70004pt\bullet}\hskip 1.49994pt)\hskip 3.00003pt$$

of  double complexes.   In  turn,  $\varphi_{\hskip 0.70004pt\bullet\hskip 0.70004pt\bullet}$  leads  to a  morphism  $\Phi_{\hskip 0.70004pt\bullet}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)\hskip 3.99994pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 3.99994ptt_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)$  of  total  complexes.   Clearly ,   the diagram

is  commutative and  leads  to  the following  commutative diagram of  homology  groups

where  $H_{\hskip 0.70004pt*}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)$  denotes  the cohomology  of  the  total  complex  $t_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003pte\hskip 1.49994pt)$.

The red arrows are isomorphisms.   Indeed,   since  the covering  $\mathcal{U}$  is  $e_{\bullet}$-acyclic,  $\lambda_{\hskip 0.70004pte\hskip 0.70004pt*}$  is  an  isomorphism  by  Lemma  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains,   and  Lemma  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  implies  that  $\tau_{\hskip 1.39998ptC\hskip 0.70004pt*}$  is  always an  isomorphism,   as we already  pointed out.   By  inverting  these  two arrows we get  the commutative diagram

It  follows  that  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\widetilde{H}_{*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  factors  through  the canonical  homomorphism

The  theorem  follows.    $\blacksquare$

3. Compactly  finite  and  $l_{\hskip 1.00806pt1}$-homology

Singular $l_{\hskip 0.70004pt1}$-chains.   Recall  that  for a  topological  space  $Y$  we denote  by  $L_{\hskip 1.04996ptq}\hskip 0.50003pt(\hskip 1.49994ptY\hskip 1.49994pt)$  the real  vector space of  infinite singular $q$-chains in  $Y$  having  finite $l_{\hskip 0.70004pt1}$-norm.   These chains are called  $l_{\hskip 0.70004pt1}$-chains  of  dimension $q$.   There are obvious inclusions  $C_{\hskip 0.70004ptq}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)\hskip 1.99997pt\subset\hskip 3.00003ptL_{\hskip 1.04996ptq}\hskip 0.50003pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   The boundary  maps in  $C_{\hskip 0.70004pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$  extend  by  continuity  to $L_{\hskip 1.04996pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$,  turning  $L_{\hskip 1.04996pt\bullet}\hskip 1.00006pt(\hskip 1.49994ptY\hskip 1.49994pt)$ into a complex.   Its homology  are denoted  by  $H_{\hskip 0.70004pt*}^{{\hskip 0.70004ptl_{\hskip 0.50003pt1}}}\hskip 0.50003pt(\hskip 1.49994ptY\hskip 1.49994pt)$.   In  this section  we will  apply  the  theory  of  Section  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains  to  $e_{\hskip 0.70004pt\bullet}\hskip 3.99994pt=\hskip 3.99994ptL_{\hskip 1.04996pt\bullet}$.

Let  $\mathcal{U}$ be a star  finite covering  of  $X$.   The complex  $L_{\hskip 1.04996pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  admits a description similar  to  the definition of  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$.   Namely,   suppose  that  for every  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  an  $l_{\hskip 0.70004pt1}$-chain  $\gamma_{\hskip 0.70004ptU}\hskip 1.99997pt\in\hskip 1.99997ptL_{\hskip 1.04996ptq}\hskip 0.50003pt(\hskip 1.49994ptU\hskip 1.49994pt)$  is  given.   Then,   since  $\mathcal{U}$  is  a star  finite,   the sum

$$\quad\gamma\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998ptU\hskip 1.39998pt\in\hskip 1.39998pt\mathcal{U}}\hskip 1.99997pt\gamma_{\hskip 0.70004ptU}$$

is  a  well  defined  infinite chain.   Let  $\mathcal{L}_{\hskip 0.70004ptq}^{\hskip 0.70004pt{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  be  the vector space of  such chains.

The inclusions  $L_{\hskip 1.04996ptq}\hskip 0.50003pt(\hskip 1.49994ptU\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004ptq}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$  lead  to a map

$$\quad\delta_{\hskip 1.04996pt0}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 1.04996pt0}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptL_{\hskip 0.70004ptq}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004ptq}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptX\hskip 1.49994pt)$$

having  $\mathcal{L}_{\hskip 0.70004ptq}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$  as  the image.   Let

$$\quad\overline{\delta}_{\hskip 1.04996pt0}\hskip 1.00006pt\colon\hskip 1.00006ptC_{\hskip 1.04996pt0}^{\hskip 0.70004pt\operatorname{inf}}\hskip 1.00006pt(\hskip 1.49994ptN\hskip 0.50003pt,\hskip 3.00003ptL_{\hskip 0.70004ptq}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{L}_{\hskip 0.70004ptq}^{\hskip 0.70004pt{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 3.00003pt$$

be  the map resulting  from changing  the  target  of  $\delta_{\hskip 1.04996pt0}$.

3.1. Lemma.   The  following  sequence  is  exact :

Proof.   The proof  is  completely  similar  to  the proof  of  Lemma  Leray  theorems  for  $l_{\hskip 0.35002pt1}$-norms  of  infinite  chains.   On  the chains of  the form  $s\hskip 1.00006pt*\hskip 1.00006pt\sigma$  the chain  homotopy  $k_{\hskip 1.04996ptp}$  is  defined as before.   The fact  that  $\mathcal{U}$  is  star  finite ensures  that  this definition extends  to  $l_{\hskip 0.70004pt1}$-chains.   The homotopy  identity  holds on  the chains of  the form  $s\hskip 1.00006pt*\hskip 1.00006pt\sigma$  by  the same reason as before and  hence holds on all  $l_{\hskip 0.70004pt1}$-chains.    $\blacksquare$

3.2. Corollary.   The map  $\overline{\delta}_{\hskip 1.04996pt0}$  induces an  isomorphism  $L_{\hskip 1.04996pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathcal{L}_{\hskip 0.70004ptq}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)$.   $\blacksquare$

3.3. Lemma.   If  $\mathcal{U}$ is  a star  finite proper covering,   then a small  chain  is  $\mathcal{U}$-finite  if  and  only  if  it  is  compactly  finite.

Proof.   Let  us  prove  the  “if”  part  first.   Given a small  chain $\gamma$,   let  us  write  it  as a formal  sum

$$\quad\gamma\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 0.70004pti\hskip 0.70004pt\in\hskip 0.70004ptI}\hskip 1.99997pta_{\hskip 0.70004pti}\hskip 1.00006pt\sigma_{\hskip 0.70004pti}$$

of  small  simplices $\sigma_{\hskip 0.70004pti}$  with coefficients  $a_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt0$.   We may  assume  that  $\sigma_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt\sigma_{j}$  if  $i\hskip 3.99994pt\neq\hskip 3.99994ptj$.   For every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  let  us  choose  $U\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  such  that  $\sigma_{\hskip 0.70004pti}$  is  a simplex  in  $U\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)$.   For  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$  let

$$\quad\gamma_{\hskip 0.70004ptU}\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998ptU\hskip 0.70004pt(\hskip 0.70004pti\hskip 1.04996pt)\hskip 1.39998pt=\hskip 2.10002ptU}\hskip 1.99997pta_{\hskip 0.70004pti}\hskip 1.00006pt\sigma_{\hskip 0.70004pti}\hskip 3.00003pt.$$

Clearly,  $\gamma$ is  equal  to  the sum of  chains  $\gamma_{\hskip 0.70004ptU}$.   If  $\gamma$  is  a compactly  finite chain,   then every  $\gamma_{\hskip 0.70004ptU}$  is  a  finite chain,   and  hence  $\gamma$  is  $\mathcal{U}$-finite.

Let  us  prove  the  “only  if”  part.   If  $\gamma$  is  a $\mathcal{U}$-finite chain,   then  $\gamma\hskip 3.99994pt=\hskip 3.99994pt\sum\nolimits_{\hskip 1.39998ptU\hskip 1.39998pt\in\hskip 1.39998pt\mathcal{U}}\hskip 1.99997pt\gamma_{\hskip 0.70004ptU}$,   where each  $\gamma_{\hskip 0.70004ptU}$  is  a  finite chain  in  $U$.   If  $Z\hskip 1.99997pt\subset\hskip 1.99997ptX$  is  compact,   then  $Z$  is  contained  in  the union of  finitely  many  sets  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   Since  $\mathcal{U}$  is  star  finite,   this implies  that  $Z$  intersects only  finitely  many  sets  $U\hskip 1.99997pt\in\hskip 1.99997pt\mathcal{U}$.   Since each  $\gamma_{\hskip 0.70004ptU}$  is  a  finite chain,   it  follows  that  only  finitely  many  simplices entering  $\gamma$  with non-zero coefficients intersect  $Z$.   Hence  $\gamma$  is  compactly  finite.    $\blacksquare$

3.4. Lemma.   If  $\mathcal{U}$ is  a star  finite proper covering,   then  the map  $H_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptH_{\hskip 0.70004pt*}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  induced  by  the inclusion  $C_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{l{\hskip 0.17496pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 0.50003pt,\hskip 3.00003pt\mathcal{U}\hskip 1.49994pt)\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC_{\hskip 0.70004pt\bullet}^{\hskip 0.70004pt\operatorname{c{\hskip 0.35002pt}f}}\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)$  is  an  isomorphism.