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These spaces were introduced by K. Borsuk [[#References|[a1]]] as a shape-theoretic analogue of ANR spaces (cf. [[Retract of a topological space|Retract of a topological space]]). FANR is an abbreviation of fundamental absolute neighbourhood retract, where  "fundamental"  refers to the particular technique used by Borsuk in his construction of the shape category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200101.png" /> (cf. [[Shape theory|Shape theory]]). A metric compactum (cf. [[Metric space|Metric space]]; [[Compact space|Compact space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200102.png" /> is an FANR space provided that for every metric compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200103.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200104.png" /> there exist a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200106.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200107.png" /> and a shape retraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200108.png" />, i.e., a shape morphism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f1200109.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001010.png" /> denotes the inclusion mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001011.png" /> is the induced shape morphism. Clearly, every compact ANR is a FANR. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001012.png" /> one obtains FAR spaces (fundamental absolute retracts).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001013.png" /> is shape dominated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001014.png" />, i.e., if there exist shape morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001018.png" /> is an FANR space, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001019.png" />. Consequently, FANR spaces coincide with metric compacta which are shape dominated by compact ANR spaces, or equivalently, by compact polyhedra. FANR spaces are characterized by a form of movability, called strong movability [[#References|[a2]]]. In particular, a FANR is a [[Movable space|movable space]].
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In various constructions and theorems, FANR spaces must be pointed. E.g., if the intersection of two pointed FANR spaces is a pointed FANR, then their union is also a pointed FANR [[#References|[a3]]]. Connected pointed FANR spaces coincide with stable continua, i.e., have the shape of an ANR (equivalently, of a polyhedron) [[#References|[a5]]]. A FANR <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001020.png" /> has the shape of a compact ANR (equivalently, of a compact polyhedron) if and only if its Wall obstruction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001021.png" />. This obstruction is an element of the reduced projective class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001022.png" /> of the first shape group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001023.png" />. There exist FANR spaces for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001024.png" /> [[#References|[a4]]]. A pointed metric continuum of finite shape dimension is a pointed FANR if and only if its homotopy pro-groups are stable, i.e. are isomorphic to groups.
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These spaces were introduced by K. Borsuk [[#References|[a1]]] as a shape-theoretic analogue of ANR spaces (cf. [[Retract of a topological space|Retract of a topological space]]). FANR is an abbreviation of fundamental absolute neighbourhood retract, where  "fundamental" refers to the particular technique used by Borsuk in his construction of the shape category $\operatorname{Sh}$ (cf. [[Shape theory|Shape theory]]). A metric compactum (cf. [[Metric space|Metric space]]; [[Compact space|Compact space]]) $X$ is an FANR space provided that for every metric compactum $Y$ containing $X$ there exist a neighbourhood $U$ of $X$ in $Y$ and a shape retraction $R : U \rightarrow X$, i.e., a shape morphism such that $R S [ i ] = id_X$. Here $i : X \rightarrow U$ denotes the inclusion mapping and $S [ i ]$ is the induced shape morphism. Clearly, every compact ANR is a FANR. For $U = Y$ one obtains FAR spaces (fundamental absolute retracts).
  
All FANR spaces are pointed FANR spaces [[#References|[a6]]]. The crucial step in the proof of this important theorem is the following homotopy-theoretic result: On a finite-dimensional [[Polyhedron|polyhedron]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001025.png" /> every homotopy idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001026.png" /> splits, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001027.png" /> implies the existence of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001028.png" /> and of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001030.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120010/f12001032.png" />.
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If $X$ is shape dominated by $X ^ { \prime }$, i.e., if there exist shape morphisms $F : X \rightarrow X ^ { \prime }$ and $G : X ^ { \prime } \rightarrow X$ such that $G F =  \operatorname {id}_X$, and $X ^ { \prime }$ is an FANR space, then so is $X$. Consequently, FANR spaces coincide with metric compacta which are shape dominated by compact ANR spaces, or equivalently, by compact polyhedra. FANR spaces are characterized by a form of movability, called strong movability [[#References|[a2]]]. In particular, a FANR is a [[Movable space|movable space]].
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In various constructions and theorems, FANR spaces must be pointed. E.g., if the intersection of two pointed FANR spaces is a pointed FANR, then their union is also a pointed FANR [[#References|[a3]]]. Connected pointed FANR spaces coincide with stable continua, i.e., have the shape of an ANR (equivalently, of a polyhedron) [[#References|[a5]]]. A FANR $X$ has the shape of a compact ANR (equivalently, of a compact polyhedron) if and only if its Wall obstruction $\sigma( X ) = 0$. This obstruction is an element of the reduced projective class group $\widetilde{ K } ^ { 0 } ( \check{\pi} _ { 1 } ( X , * ) )$ of the first shape group $\check{\pi} _ { 1 } ( X , * )$. There exist FANR spaces for which $\sigma( X ) \neq 0$ [[#References|[a4]]]. A pointed metric continuum of finite shape dimension is a pointed FANR if and only if its homotopy pro-groups are stable, i.e. are isomorphic to groups.
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All FANR spaces are pointed FANR spaces [[#References|[a6]]]. The crucial step in the proof of this important theorem is the following homotopy-theoretic result: On a finite-dimensional [[Polyhedron|polyhedron]] $X$ every homotopy idempotent $f : X \rightarrow X$ splits, i.e., $f ^ { 2 } \simeq f$ implies the existence of a space $Y$ and of mappings $u : Y \rightarrow X$, $v : X \rightarrow Y$, such that $v u \simeq 1_{Y}$, $u v \simeq f$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Borsuk,  "Fundamental retracts and extensions of fundamental sequences"  ''Fund. Math.'' , '''64'''  (1969)  pp. 55–85</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "A note on the theory of shape of compacta"  ''Fund. Math.'' , '''67'''  (1970)  pp. 265–278</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Dydak,  S. Nowak,  S. Strok,  "On the union of two FANR-sets"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys.'' , '''24'''  (1976)  pp. 485–489</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.A. Edwards,  R. Geoghegan,  "Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction"  ''Ann. Math.'' , '''101'''  (1975)  pp. 521–535  (Correction: 104 (1976), 389)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.A. Edwards,  R. Geoghegan,  "Stability theorems in shape and pro-homotopy"  ''Trans. Amer. Math. Soc.'' , '''222'''  (1976)  pp. 389–403</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H.M. Hastings,  A. Heller,  "Homotopy idempotents on finite-dimensional complexes split"  ''Proc. Amer. Math. Soc.'' , '''85'''  (1982)  pp. 619–622</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  K. Borsuk,  "Fundamental retracts and extensions of fundamental sequences"  ''Fund. Math.'' , '''64'''  (1969)  pp. 55–85</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Borsuk,  "A note on the theory of shape of compacta"  ''Fund. Math.'' , '''67'''  (1970)  pp. 265–278</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Dydak,  S. Nowak,  S. Strok,  "On the union of two FANR-sets"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys.'' , '''24'''  (1976)  pp. 485–489</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D.A. Edwards,  R. Geoghegan,  "Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction"  ''Ann. Math.'' , '''101'''  (1975)  pp. 521–535  (Correction: 104 (1976), 389)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D.A. Edwards,  R. Geoghegan,  "Stability theorems in shape and pro-homotopy"  ''Trans. Amer. Math. Soc.'' , '''222'''  (1976)  pp. 389–403</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  H.M. Hastings,  A. Heller,  "Homotopy idempotents on finite-dimensional complexes split"  ''Proc. Amer. Math. Soc.'' , '''85'''  (1982)  pp. 619–622</td></tr></table>

Latest revision as of 15:31, 1 July 2020

These spaces were introduced by K. Borsuk [a1] as a shape-theoretic analogue of ANR spaces (cf. Retract of a topological space). FANR is an abbreviation of fundamental absolute neighbourhood retract, where "fundamental" refers to the particular technique used by Borsuk in his construction of the shape category $\operatorname{Sh}$ (cf. Shape theory). A metric compactum (cf. Metric space; Compact space) $X$ is an FANR space provided that for every metric compactum $Y$ containing $X$ there exist a neighbourhood $U$ of $X$ in $Y$ and a shape retraction $R : U \rightarrow X$, i.e., a shape morphism such that $R S [ i ] = id_X$. Here $i : X \rightarrow U$ denotes the inclusion mapping and $S [ i ]$ is the induced shape morphism. Clearly, every compact ANR is a FANR. For $U = Y$ one obtains FAR spaces (fundamental absolute retracts).

If $X$ is shape dominated by $X ^ { \prime }$, i.e., if there exist shape morphisms $F : X \rightarrow X ^ { \prime }$ and $G : X ^ { \prime } \rightarrow X$ such that $G F = \operatorname {id}_X$, and $X ^ { \prime }$ is an FANR space, then so is $X$. Consequently, FANR spaces coincide with metric compacta which are shape dominated by compact ANR spaces, or equivalently, by compact polyhedra. FANR spaces are characterized by a form of movability, called strong movability [a2]. In particular, a FANR is a movable space.

In various constructions and theorems, FANR spaces must be pointed. E.g., if the intersection of two pointed FANR spaces is a pointed FANR, then their union is also a pointed FANR [a3]. Connected pointed FANR spaces coincide with stable continua, i.e., have the shape of an ANR (equivalently, of a polyhedron) [a5]. A FANR $X$ has the shape of a compact ANR (equivalently, of a compact polyhedron) if and only if its Wall obstruction $\sigma( X ) = 0$. This obstruction is an element of the reduced projective class group $\widetilde{ K } ^ { 0 } ( \check{\pi} _ { 1 } ( X , * ) )$ of the first shape group $\check{\pi} _ { 1 } ( X , * )$. There exist FANR spaces for which $\sigma( X ) \neq 0$ [a4]. A pointed metric continuum of finite shape dimension is a pointed FANR if and only if its homotopy pro-groups are stable, i.e. are isomorphic to groups.

All FANR spaces are pointed FANR spaces [a6]. The crucial step in the proof of this important theorem is the following homotopy-theoretic result: On a finite-dimensional polyhedron $X$ every homotopy idempotent $f : X \rightarrow X$ splits, i.e., $f ^ { 2 } \simeq f$ implies the existence of a space $Y$ and of mappings $u : Y \rightarrow X$, $v : X \rightarrow Y$, such that $v u \simeq 1_{Y}$, $u v \simeq f$.

References

[a1] K. Borsuk, "Fundamental retracts and extensions of fundamental sequences" Fund. Math. , 64 (1969) pp. 55–85
[a2] K. Borsuk, "A note on the theory of shape of compacta" Fund. Math. , 67 (1970) pp. 265–278
[a3] J. Dydak, S. Nowak, S. Strok, "On the union of two FANR-sets" Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 24 (1976) pp. 485–489
[a4] D.A. Edwards, R. Geoghegan, "Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction" Ann. Math. , 101 (1975) pp. 521–535 (Correction: 104 (1976), 389)
[a5] D.A. Edwards, R. Geoghegan, "Stability theorems in shape and pro-homotopy" Trans. Amer. Math. Soc. , 222 (1976) pp. 389–403
[a6] H.M. Hastings, A. Heller, "Homotopy idempotents on finite-dimensional complexes split" Proc. Amer. Math. Soc. , 85 (1982) pp. 619–622
How to Cite This Entry:
FANR space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=FANR_space&oldid=49943
This article was adapted from an original article by S. Mardešić (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article