Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR2044026 (2005e:57060) 57P10 55U15 57N65
Christensen, René dePont ; Munkholm, Hans J. (DK-SU-DT)
Monoidal controlled Poincaré duality. (English. English
summary)
Forum Math. 16 (2004), no. 4, 519–537.
The authors state and prove a Poincaré duality theorem for (the universal covers of) oriented manifolds equipped with control structures.
This is required for setting up a surgery theory in this context.
The type of control used is monoidal control, specified by a topological space Z together with a submonoid M of the monoid of open
neighbourhoods of the diagonal in Z × Z and some other data. A space
controlled over Z is then a pair (E, p) consisting of a topological space
E and a continuous map p: E → Z, and a monoidally controlled map
(E1 , p1 ) → (E2 , p2 ) is defined to be a continuous map E1 → E2 respecting the projections p1 and p2 up to a distortion bounded by
some fixed relation M ∈ M. It includes previously defined types of
control [see, e.g., D. R. Anderson and H. J. Munkholm, Boundedly
controlled topology, Lecture Notes in Math., 1323, Springer, Berlin,
1988; MR0953961 (89h:57029)] as special cases.
Singular chain, cochain, homology and cohomology functors are
constructed on the category of spaces controlled over Z and take values
in certain abelian functor categories. In contrast, the fundamental
class of an oriented manifold M controlled over Z is represented by a
locally finite top dimensional singular chain in M in the usual sense.
It enters the Poincaré duality map via a cap product pairing. With
these notions at hand, the actual proof of the main theorem follows
from the classical Poincaré-Lefschetz duality theorem.
Bernhard A. Hanke (D-MNCH-MI)
[References]
1. Anderson D. R., Munkholm H. J.: Boundedly Controlled
Topology. Lecture Notes in Math. 1323. Springer-Verlag,
Berlin-Heidelberg-New York 1988 MR0953961 (89h:57029)
2. Anderson D. R., Munkholm H. J.: Continuously Controlled Ktheory with Variable Coefficients. J. Pure Appl. Algebra 145
(2000), 215–266 MR1733430 (2001h:19002)
3. Attie O.: Quasi-isometry Classification of some Manifolds of
Bounded Geometry. Math. Z. 216 (1994), 501–527 MR1288043
(95k:53051)
4. Christensen R. d.: Elements of Monoidally Controlled Algebraic
Topology and Simple Homotopy Theory. Ph.D. Thesis, IMADA,
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
SDU, Odense, Denmark 2001
5. Christensen R. d., Munkholm H. J.: Topology with Monoidal Control. Homology Homotopy Appl. 4 (2002), 213–234 MR1983017
(2004d:55018)
6. Dold A.: Algebraic Topology. Springer-Verlag, Berlin-HeidelbergNew York 1972 MR0415602 (54 #3685)
7. Greenberg M. J., Harper J. R.: Algebraic Topology. A First
Course. Addison-Wesley Publishing Co. 1981 MR0643101
(83b:55001)
8. Higson N., Pedersen E. K., Roe J.: C ∗ -algebras and Controlled
Topology. K-theory 11 (1997), 203–239 MR1451755 (98g:19009)
9. Massey W. S.: Singular Homology Theory. Springer-Verlag, New
York-Berlin-Heidelberg 1980 MR0569059 (81g:55002)
10. Rotman J. J.: An Introduction to Algebraic Topology.
Springer-Verlag, New York-Berlin-Heidelberg 1988 MR0957919
(90e:55001)
11. Schubert H.: Categories. Springer-Verlag, Berlin-Heidelberg 1972
MR0349793 (50 #2286)
12. Vick J. W.: Homology Theory. Springer-Verlag, New York Inc.
1994 MR1254439 (94i:55002)
MR1983017 (2004d:55018) 55U35 18D10 18E10
Christensen, René Depont (DK-SU-DT) ;
Munkholm, Hans Jørgen (DK-SU-DT)
Topology with monoidal control. (English. English
summary)
Homology Homotopy Appl. 4 (2002), no. 1, 213–234 (electronic).
This paper is concerned with foundational aspects of topology with
control. It builds on, and generalizes, [D. R. Anderson and H. J.
Munkholm, Boundedly controlled topology, Lecture Notes in Math.,
1323, Springer, Berlin, 1988; MR0953961 (89h:57029)] by introducing
monoidal control. Connections are made with the coarse structures
and entourages of [N. Higson, E. K. Pedersen and J. Roe, K-Theory 11
(1997), no. 3, 209–239; MR1451755 (98g:19009)]. The new machinery
is illustrated by establishing Hurewicz and Whitehead theorems in
the monoidal context.
C. Bruce Hughes (1-VDB)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR1851260 (2002f:19005) 19D35 19B28 20C05
Munkholm, Hans Jørgen (DK-SU-DT) ; Prassidis, Stratos
On the vanishing of certain K-theory Nil-groups. (English.
English summary)
Cohomological methods in homotopy theory (Bellaterra, 1998),
307–321, Progr. Math., 196, Birkhäuser, Basel, 2001.
Summary: “Let Γi , i = 0, 1, be two groups containing Cp , the cyclic
group of prime order p, as a subgroup of index 2. Let Γ = Γ0 ∗Cp
Γ1 . We show that the Nil-groups appearing in Waldhausen’s splitting
theorem for computing Kj (ZΓ) (j ≤ 1) vanish. Thus, in low degrees,
the K-theory of ZΓ can be computed by a Mayer-Vietoris type exact
sequence involving the K-theory of the integral group rings of the
groups Γ0 , Γ1 and Cp .”
{For the entire collection see MR1851241 (2002c:55002)}
Barry H. Dayton (1-NEIL)
MR1733430 (2001h:19002) 19D10 19J10 57R19
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans Jørgen (1-SRCS)
Continuously controlled K-theory with variable coefficients.
(English. English summary)
J. Pure Appl. Algebra 145 (2000), no. 3, 215–266.
The Whitehead theorem is an algebraic test to determine homotopy
equivalence. When one of the spaces comes equipped with a reference
map to a metric space, one needs an algebraic test to determine
whether the map is a bounded homotopy equivalence. That kind of
algebra has been developed by the reviewer and C. Weibel, but only
under rather restrictive assumptions on fundamental group behaviour.
In the paper under review these kinds of restrictions are removed, and
the kind of algebra needed is developed in very great generality.
Erik K. Pedersen (Binghamton, NY)
[References]
1. D.R. Anderson, F.X. Connolly, Finiteness obstructions and cocompact actions on S m × Rn , Comm. Math. Helv. 68 (1993)
85–110. MR1201203 (93k:57053)
2. D.R. Anderson, F.X. Connolly, S. Ferry, E.K. Pedersen, Algebraic K-theory with continuous control at infinity, J. Pure Appl.
Algebra 94 (1994) 25–47. MR1277522 (95b:19003)
3. D.R. Anderson, F.X. Connolly, H.J. Munkholm, A comparison of
continuously controlled and controlled K-theory, Topology Appl.
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
71 (1996) 9–46. MR1391956 (97h:57054)
4. D.R. Anderson, H.J. Munkholm, Boundedly Controlled Toplogy,
Lecture Notes in Mathematics, vol. 1323, Springer, Berlin, 1988.
MR0953961 (89h:57029)
5. D.R. Anderson, H.J. Munkholm, Geometric modules and algebraic K-Homology theory, K-Theory 3 (1990) 561–602.
MR1071896 (91g:57033)
6. D.R. Anderson, H.J. Munkholm, Geometric modules and quinn
homology theory, K-Theory 7 (1993) 443–475. MR1255061
(95f:55003)
7. G. Carlsson, E.K. Pedersen, Controlled algebra and the novikov
conjectures for K- and L-theory, Topology 34 (1995) 731–758.
MR1341817 (96f:19006)
8. T.A. Chapman, Controlled Simple Homotopy Theory and Applications, Lecture Notes in Mathematics, vol. 1009, Springer,
Berlin, 1983. MR0711363 (85d:57016)
9. S.C. Ferry, E.K. Pedersen, Controlled algebraic Ktheory, preprint, SUNY at Binghamton, Binghamton, NY.
(http://math.binghamton.edu/erik/index.html).
MR1116929
(93a:57021)
10. P. Freyd, Abelian Categories, Harper and Row, New York, 1966.
MR0166240 (29 #3517)
11. D. Grayson, Higher algebraic K-Theory (after D. Quillen), Lecture Notes in Mathematics, vol. 551, Springer, Berlin, 1976, pp.
217–240. MR0574096 (58 #28137)
12. M.J. Greenberg, J.R. Harper, Algebraic Topology; A First Course,
Addison-Wesley, New York, 1981. MR0643101 (83b:55001)
13. C.B. Hughes, L.R. Taylor, S. Weinberger, E.B. Williams, Neighborhoods in stratified spaces, I. Two Strata, Preprint. MR1763954
(2001e:57026)
14. M. Karoubi, Functuers derivé et K-théorie, Lecture Notes in
Mathematics, vol. 136, Springer, Berlin, 1970, pp. 107–186.
MR0265435 (42 #344)
15. E.K. Pedersen, On the K−i functors, J. Algebra 90 (1984) 461–
475. MR0760023 (85k:18019)
16. E.K. Pedersen, C. Weibel, A nonconnective delooping of algebraic
K-theory, Lecture Notes in Mathematics, vol. 1126, Springer,
Berlin, 1985, pp. 166–181. MR0802790 (87b:18012)
17. E.K. Pedersen, C. Weibel, K-Theory homology of spaces, Lecture
Notes in Mathematics, vol. 1370, Springer, Berlin, 1989, pp. 346–
361. MR1000388 (90m:55007)
18. D. Quillen, Higher Algebraic K-theory I, Lecture Notes in Math-
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
19.
20.
21.
22.
23.
24.
ematics, vol. 341, Springer, Berlin, 1973, pp. 77–139. MR0338129
(49 #2895)
F. Quinn, Ends of maps, I, Ann. Math. 110 (1979) 275–331.
MR0549490 (82k:57009)
F. Quinn, Ends of maps, II, Inv. Math. 68 (1982) 353–424.
MR0669423 (84j:57011)
F. Quinn, Geometric algebra and ends of maps, CBMS Lectures
at the University of Notre Dame, 1984 (unpublished).
F. Quinn, Geometric algebra, Lecture Notes in Math., vol. 1126,
Springer, Berlin, 1985, pp. 182–198. MR0802791 (86m:57023)
A.A. Ranicki, M. Yamasaki, Controlled K-theory, Topology Appl.
61 (1995) 1–59. MR1311017 (96b:57027)
L. Siebenmann, D. Sullivan, On complexes that are Lipschitz
manifolds, in: J.C. Cantrell (Ed.), Geometric Topology, Academic
Press, New York, 1979, pp. 503–525. MR0537747 (80h:57027)
MR1721129 (2000k:01045) 01A70 01A60 55-03 57-03
Munkholm, Ellen S. (DK-ODNS-CS) ;
Munkholm, Hans J. (DK-ODNS-CS)
Poul Heegaard.
History of topology, 925–946, North-Holland, Amsterdam, 1999.
The article gives a keen impression of the life of Poul Heegaard,
whose character has remained rather elusive despite the impact of
his ideas. The authors base their description on roughly 130 pages
of handwritten autobiographical notes (now available, in Danish, at
http://www.imada.sdu.dk/∼ hjm/heegaard.html), letters, and interviews with descendents.
Born in Copenhagen in 1871, into an academic family (his father,
Sophus Heegaard, was a Professor of Philosophy), Poul Heegaard
displayed interest in a wide variety of academic subjects even as
a child. His interest in mathematics and in astronomy became the
most notable. He studied at the University of Copenhagen, where he
obtained a masters degree in mathematics with minors in astronomy,
chemistry, and physics, and under the direction of H. G. Zeuthen
wrote a thesis concerning a characterization of algebraic curves in a
surface of second order in terms of the number of intersection points
with the generators in the two generating systems. His years at the
University were somewhat marred by financial difficulties resulting
from the early death of his father.
He then spent a rather unsuccessful semester in Paris. Contrary to
widespread beliefs, he probably had no contact with H. Poincaré during this time. Then he spent a more rewarding semester in Göttingen,
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
where he worked closely with F. Klein. After returning to Copenhagen, he chanced upon a mistake in a paper of Poincaré, involving
Poincaré’s original formulation of the duality theorem, and wrote his
dissertation, “Preliminary studies of a theory of connectivity for algebraic surfaces”, explaining the error. His counterexamples played a
role in the development of algebraic topology, at least in the sense
that they forced Poincaré to rethink his approach.
After finishing his dissertation, Heegaard married, spent 12 years
teaching at a variety of naval and military academies in Copenhagen,
and had 6 children. He described these years as happy years. His writing during this time covered a broad range of scientific subjects, most
notably astronomy, but was of the expository, popularizing sort. He
returned to research when asked to report on H. Poincaré’s Analysis Situs in the Enzyklopädie der Mathematischen Wissenschaften.
This led to his collaboration with M. Dehn and their formulation of
complexus, nexus, and connexus.
In 1910, Heegaard was urged to succeed his mentor, Professor
Zeuthen, who was retiring. Somewhat against his will, he gave in
to this urging and spent 7 years as a professor at the University of
Copenhagen. These years culminated in his resignation, due to the
large workload, the small salary, and the political infighting involved
in the job.
He was then appointed at Kristiania (now Oslo) University where
he worked until he retired in 1941. Mathematically, his most notable
work during this time were annotations to a publication of Sophus
Lie’s collected works, and, perhaps, his failed attempts at the 4-color
theorem.
His personal and political involvement, especially in his later years,
appears to be a story in its own right. He spent many years as a high
ranking Freemason. Then there is some evidence of his involvement
with the Nazis during their occupation of Norway. Specifically, he
presented a show on natural science on the Norwegian Radio in 1944
and 1945, i.e., during a time when only households in which the
majority were members in good standing of the Norwegian Nazi party
were allowed to own radios. His autobiographical notes end just before
this time, abruptly, and in the middle of a sentence. After that, the
only recorded information concerning Poul Heegaard is his obituary
presented by the Norwegian Science Academy in 1948.”
{For the entire collection see MR1674906 (2000g:00032)}
Jennifer Schultens (1-CAD)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR1713206 (2000i:19006) 19D35 20J05 57R99
Munkholm, Hans J. (DK-ODNS) ; Prassidis, Stratos (1-VDB)
Waldhausen’s nil groups and continuously controlled Ktheory. (English. English summary)
Algebraic topology (Kazimierz Dolny, 1997).
Fund. Math. 161 (1999), no. 1-2, 217–224.
Let Γ be a pushout of two groups Γ1 , Γ2 along a common subgroup
G. Then, by a theorem of F. Waldhausen, there is a chain complex
· · · → Kj (ZΓ1 ) ⊕ Kj (ZΓ2 ) → Kj (ZΓ) → Kj−1 (ZG) → · · ·
f j−1 (ZG; Z[Γ1 − G], Z[Γ2 − G]) at the Kj (ZΓ)
which has homology Nil
positions and zero elsewhere.
D. R. Anderson and Munkholm developed [J. Pure Appl. Algebra
145 (2000), no. 3, 215–266 MR 2001h:19002 ] a homology theory K∗cc
defined on the category of diagrams B ← H → X, where H and X
are Hausdorff spaces and B is a compact metric space. In the paper
under review, the theory of the above paper is applied in order to
f group with K cc of the diagram [0, 1] ← H → H,
identify the above Nil
j
where H is the double mapping cylinder of the diagram of classifying
spaces induced by the inclusions G ⊂ Γi . There is a similar theorem
for the case of HNN extensions.
{For the entire collection see MR1713210 (2000e:55001)}
Frans Clauwens (NL-NIJM)
MR1682372 (2000m:01036) 01A70
Munkholm, Ellen S. ; Munkholm, Hans J. (DK-ODNS-DT)
Poul Heegaard (1871–1948), a Danish-Norwegian topologist.
(Danish. English summary)
Normat 46 (1998), no. 4, 145–169, 188.
Summary: “This paper is a description of Poul Heegaard’s life
and mathematics, based in part on a set of his handwritten autobiographical notes in Norwegian from 1945, and located by
the authors in 1996 in the possession of Poul Heegaard’s greatgrandson Poul E. Heegaard, Trondheim, Norway. A transcript
of the notes (in Danish) as well as other material related to the
article and its main character can be found on the Internet at
http://www.imada.ou.dk/∼ hjm/heegaard.html. An English language
article on Poul Heegaard by the same authors (but not a translation
of the present article) has appeared [in History of topology, 925–946,
North-Holland, Amsterdam, 1999; MR1721129 (2000k:01045)].”
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR1391956 (97h:57054) 57R67 19D35 19D55
Anderson, Douglas R. (1-SRCS) ;
Connolly, Francis X. (1-NDM) ;
Munkholm, Hans J. (DK-ODNS-DT)
A comparison of continuously controlled and controlled
K-theory. (English. English summary)
Topology Appl. 71 (1996), no. 1, 9–46.
Siebenmann’s work in the 1960s on the obstruction to closing open
manifolds was extended in the 1970s by Chapman and Quinn to open
manifolds with a control map. This led to the modern development of
bounded and controlled algebraic K- and L-theory and their many applications to topology. Unfortunately, the algebraic properties required
for the applications to topology have not always kept pace with these
applications. This paper provides a partial remedy, providing a comparison of the continuously controlled K-theory of Anderson et al. [J.
Pure Appl. Algebra 94 (1994), no. 1, 25–47; MR1277522 (95b:19003)]
and the ε-controlled K-theory of the reviewer and M. Yamasaki
[Topology Appl. 61 (1995), no. 1, 1–59; MR1311017 (96b:57027)], and
also the homology groups of F. Quinn [Invent. Math. 68 (1982), no. 3,
353–424; MR0669423 (84j:57011)]. The treatment is confined to K1 and the lower K-groups, with a conjectural extension to K2 and the
higher K-groups. This paper sins in the other direction, by offering
algebra without applications to topology. But it does have the virtue
of being precise, and it may well be that future applications will make
use of this algebra.
A. A. Ranicki (4-EDIN-MS)
MR1255061 (95f:55003) 55N20 57R67
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans Jørgen (DK-ODNS-DT)
Geometric modules and Quinn homology theory. (English.
English summary)
K-Theory 7 (1993), no. 5, 443–475.
In a paper by F. Quinn [Invent. Math. 68 (1982), no. 3, 353–424;
MR0669423 (84j:57011)] geometric problems centered on the question
“Under what conditions can f : M → X, where M is a manifold and f
a map which is not (necessarily) proper, be completed—by adding a
boundary to M —to a proper map?” Eventually Quinn proceeded to
describe homology theories (Quinn homology theories) which comprise
the obstructions.
In an earlier paper [K-Theory 3 (1990), no. 6, 561–602; MR1071896
(91g:57033)] the authors had constructed spectrum-valued functors
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
for situations p: E → B with E a topological space, p continuous
and B equipped with a Lipschitz equivalence class of metrics. On
objects with a “homotopy colimit structure”—a notion the authors
enjoy to recall in detail in the introduction to the present paper—
they had obtained a homological behaviour. They can now show the
coincidence with an idea of Quinn’s.
The carefully written paper of course uses techniques and results of the paper cited above and a related paper [E. K. Pedersen and C. A. Weibel, in Algebraic topology (Arcata, CA, 1986), 346–
361, Lecture Notes in Math., 1370, Springer, Berlin, 1989; MR1000388
(90m:55007)]. The major new tool is the extensive use of the language
of homotopy colimits [R. W. Thomason, Comm. Algebra 10 (1982),
no. 15, 1589–1668; MR0668580 (83k:18006)]. For instance, the proof of
invariance under subdivision (i.e. homotopy invariance in the setting
of the paper) is just reduced to the cofinality theorem for homotopy
colimits.
Roland Schwänzl (D-OSNB)
MR1111432 (93d:19008) 19J10 55P10
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans Jørgen (DK-ODNS-DT)
The boundedly controlled Whitehead theorem.
Proc. Amer. Math. Soc. 117 (1993), no. 2, 561–568.
The authors [Boundedly controlled topology, Lecture Notes in Math.,
1323, Springer, Berlin, 1988; MR0953961 (89h:57029)] stated and
proved a version of the Whitehead theorem for deciding whether
a map between bc spaces is a bc homotopy equivalence, in terms of a
bc homotopy condition. In this paper a version involving bc homology
for the bc analogue of the universal cover is stated and proved.
{See also the preceding review.}
A. A. Ranicki (4-EDIN-MS)
MR1111431 (93d:19007) 19J10 55P10
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans Jørgen (DK-ODNS-DT)
The bounded and thin Whitehead theorems.
Proc. Amer. Math. Soc. 117 (1993), no. 2, 551–560.
Two further Whitehead theorems are proved in the context of the
authors’ theory of bounded controlled topology, continuing the work in
another paper by them [Proc. Amer. Math. Soc. 117 (1993), no. 2, 561–
568; see the following review]. The bounded Whitehead theorem allows
one to decide whether a map is a bounded homotopy equivalence. The
thin Whitehead theorem allows one to decide whether a map of bound
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
zero admits a homotopy inverse of arbitrarily small bound (also on
the homotopies).
A. A. Ranicki (4-EDIN-MS)
MR1071896 (91g:57033) 57R67 18F25 19M05
Anderson, Douglas R. (DK-ODNS) ;
Munkholm, Hans Jørgen (DK-ODNS)
Geometric modules and algebraic K-homology theory.
K-Theory 3 (1990), no. 6, 561–602.
This paper continues the investigations into the algebraic aspects
of controlled topology of the authors’ earlier work [Boundedly controlled topology, Lecture Notes in Math., 1323, Springer, Berlin, 1988;
MR0953961 (89h:57029)]. There are four new developments. Theorem
A generalizes the result of Pedersen and Weibel characterizing the
generalized homology theory with algebraic K-theory coefficients as
the algebraic K-groups of a category of objects bounded over an open
cone, by working with a wider class of coefficient systems. Theorem
B obtains the analogue of the Atiyah-Hirzebruch spectral sequence
for this generalized homology theory. Theorem C examines geometric modules in the style of Quinn (appropriately generalized to the
wider context) and relates them to the bounded objects over the open
cone, proving that the corresponding additive categories are equivalent. Theorem D relates geometric modules to modules over local
systems of bounded fundamental groups, proving they have isomorphic algebraic K-theories.
A. A. Ranicki (4-EDIN-MS)
MR0953961 (89h:57029) 57R80 57Q10
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans J. (DK-ODNS)
FBoundedly controlled topology.
Foundations of algebraic topology and simple homotopy theory.
Lecture Notes in Mathematics, 1323.
Springer-Verlag, Berlin, 1988. xii+309 pp. $28.60.
ISBN 3-540-19397-9
This book is devoted to a careful development of the bounded version
of controlled topology. In this everything is mapped to a reference
metric space, and appropriate images are required to be bounded.
For example the main result is a version of the h-cobordism theorem
which gives conditions on a manifold which imply that it has a product
structure N × I so that the images of the product arcs {n} × I in the
reference space are bounded.
There is an older controlled theory developed by the reviewer, T.
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
Chapman, S. Ferry, and others. In that, a specific number ε > 0 is
chosen, and images are required to have diameter less than ε. The
two versions agree for the important special cases of open cones on
compact spaces, where radial contraction can be used to convert an
abstract bound into a specific one. But in general they can be very
different.
The bounded theory has the substantial internal advantage over
the ε theory that bounded “morphisms” form a category. This means
that a great deal of classical algebraic and geometric topology can
be extended in a straightforward way (once the proper definitions are
worked out) to the bounded setting. The authors have carried this
out beautifully. They have skillfully and accurately abstracted the key
features and incorporated them into a categorical framework which is
effective without being overwhelming. The exposition is also of a high
order, with examples, discussions of definitions and technical results,
and suggestions on reading order.
The first chapter sets up categorical contexts for the algebraic
material, and the second uses these to develop bounded versions
of homotopy and homology. The third and fourth chapters describe
the bounded Whitehead groups, first geometrically using expansions
and collapses, then algebraically using automorphisms of “modules”.
The fifth chapter demonstrates the isomorphism between the geometric and algebraic descriptions. Chapter 6 contains the main result,
that bounded h-cobordisms are classified by the bounded Whitehead groups defined earlier. There is also an analogue of the duality
theorem for torsions.
Finally, in the seventh chapter, special cases and computations
are described. When the reference space is a Euclidean space the
theory agrees with the one worked out earlier by E. K. Pedersen
[in Transformation groups, Poznań 1985, 306–320, Lecture Notes in
Math., 1217, Springer, Berlin, 1986; MR0874186 (88g:57036)]. Over a
half-open interval it is closely related to L. C. Siebenmann’s infinite
simple homotopy [Indag. Math. 32 (1970), 479–495; MR0287542 (44
#4746)]. In these cases the Whitehead group is independent of the
metric. An example in which it depends strongly on the metric is also
given.
No geometric applications are given. The ε control theory is more
complicated and lacks a decent exposition, but does have powerful
applications to other problems in topology. If the bounded theory
could do the same job then the ε theory would be quickly abandoned
for it. Unfortunately, so far the main applications of the ε theory seem
to be out of reach.
Frank Quinn (1-VAPI)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
19870320
19880401
MR873 280 (88f:57033a) 57Q10 19B99 19M05
Anderson, Douglas R. [Anderson, Douglas Ross] (1-SRCS) ;
Munkholm, Hans Jørgen (DK-ODNS)
A geometric construction of the boundedly controlled
Whitehead group.
Geometry and topology (Athens, Ga., 1985), 13–26, Lecture Notes in
Pure and Appl. Math., 105, Dekker, New York, 1987.
MR0873281 (88f:57033b) 57Q10 19B99 19M05
Anderson, Douglas R. (1-SRCS) ;
Munkholm, Hans Jørgen (DK-ODNS)
An introduction to boundedly controlled simple homotopy
theory.
Geometry and topology (Athens, Ga., 1985), 27–42, Lecture Notes in
Pure and Appl. Math., 105, Dekker, New York, 1987.
The two papers under review are trailers for a comprehensive treatment of simple homotopy theory with bounded control in a metric space, with the aim of securing the foundations for the original work of Chapman, Ferry, Quinn and Pedersen on the thin and
bounded h-cobordism theorems. The idea (going back to Connell and
Hollingsworth) is to enrich the simple homotopy theory of the traditional h-cobordism theorem by involving a topological “control space”
in the algebra. Thus it is necessary to develop the apparatus of CW
complexes, cellular chain complexes, homology, torsion, etc. in the
context of spaces and modules with such control. The geometric construction of the “boundedly controlled Whitehead group” uses the
appropriate notion of a category with elementary expansions, making
use of a paper of B. Eckmann [Symposia mathematica, Vol. V (Rome,
1969/70), 285–299, Academic Press, London, 1971; MR0282367 (43
#8079)].
{For the entire collection see MR0873278 (87j:57001)}
A. A. Ranicki (4-EDIN-MS)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0750685 (85k:18017) 18F25 57Q10
Munkholm, Hans J.
Transfer maps on Ki , i ≤ 1, associated with certain periodic
resolutions.
Algebraic K-theory, number theory, geometry and analysis (Bielefeld,
1982), 245–254, Lecture Notes in Math., 1046, Springer, Berlin, 1984.
ϕ
i
From the introduction: “Let Z → π → ρ → 1 be an exact sequence of
groups with a compatible action ω: ρ → {±1} (i.e. if t = i(1) ∈ π then
gtg −1 = tω(ϕ(g)) for all g ∈ π). If i is monic, then the integral group ring
Zρ is of finite homological dimension when viewed as a Zπ module
via ϕ. Consequently there is defined a ‘transfer map’ ϕ!i : Ki (Zρ) →
Ki (Zπ), i ∈ Z [see, e.g., D. Quillen , Algebraic K-theory I: higher Ktheories (Seattle, Wash., 1972), 85–147, Lecture Notes in Math., 341,
Springer, Berlin, 1973; MR0338129 (49 #2895)]. For i = 0 or 1, ϕ!i
is obtained by mapping the class [P ] of an f.g. projective Zρ-module
[respectively [ᾱ] of an automorphism
P ᾱ: F → F of an f.g. free Zρmodule] to the Euler characteristic i (−1)i [Pi ] of a finite resolution
P
of P by f.g. projectives over Zρ [respectively i (−1)i [αi ] of a lifting
of ᾱ to an automorphism α of a suitable finite resolution of F ].
“When the map i: Z → π is not monic Zρ admits a periodic, but
not a finite, resolution over Zπ, and one may consider the “oneperiod Euler characteristics” [P0 ] − [P1 ], respectively [α0 ] − [α1 ], for
resolutions or liftings as above. It is the purpose of this note to point
out that in this way one gets homomorphisms ϕ!i : Ki (Zρ) → Ki (Zπ),
i ≤ 1.”
For i = 1, the proof proceeds by a direct construction. For i = 0,
the author first interprets K0 (R), R a ring, in terms of projections
α: Rn → Rn and then gives the construction needed.
The note closes by raising several questions.
{For the entire collection see MR0750671 (85g:18001)}.
{For the entire collection see MR0750671 (85g:18001)}
Douglas R. Anderson (1-SRCS)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0712261 (85e:57037) 57R67 18F25
Munkholm, H. J. (1-MD) ; Pedersen, E. K. (DK-ODNS)
The S 1 -transfer in surgery theory.
Trans. Amer. Math. Soc. 280 (1983), no. 1, 277–302.
p
An S 1 -bundle S 1 → E →B induces transfer maps
p∗ : Ln (π1 (B)) → Ln+1 (π1 (E))
in the Wall surgery obstruction groups. The surgery obstruction
σ∗ (f, b) ∈ Ln (π1 (B)) of an n-dimensional normal map (f, b) : M →
X with a π1 -isomorphism reference map X → B is sent to the
surgery obstruction p! σ∗ (f, b) = σ∗ (f ! , b! ) ∈ Ln+1 (π1 (E)) of the (n + 1)dimensional normal map (f ! , b! ) : M ! → X ! between the total spaces
of the induced S 1 -bundles. In this paper the algebraic techniques introduced by the authors in their earlier paper [Comm. Math. Helv.
56 (1981), no. 3, 404–430; MR0639359 (83c:57007)] are extended from
the Whitehead group to the L-groups. A detailed analysis of the
surgery kernels K∗ (f ), K∗ (f ! ) for highly-connected (f, b) : M → X
leads to the algebraic description of p∗ .
{Reviewer’s remark: In Section 7.8 of the reviewer’s book [Exact
sequences in the algebraic theory of surgery, Princeton Univ. Press,
Princeton, N.J., 1981; MR0620795 (82h:57027)] there is outlined an
alternative approach to p∗ , describing σ∗ (f ! , b! ) on the chain level for
all (f, b).}
A. A. Ranicki (4-EDIN-MS)
MR0711057 (85c:57033) 57R67 18F25 57Q10
Munkholm, Hans J. (DK-ODNS)
Rothenberg sequences and the algebraic S 1 -bundle transfer.
Proceedings of the Northwestern Homotopy Theory Conference
(Evanston, Ill., 1982), 255–265, Contemp. Math., 19, Amer. Math.
Soc., Providence, RI, 1983.
Author’s summary: “We show that the algebraic S 1 -bundle transfer
maps on Whitehead groups and Wall groups defined by the author
and E. K. Pedersen [Comment. Math. Helv. 56 (1981), no. 3, 404–
430; MR0639359 (83c:57007); Trans. Amer. Math. Soc. 280 (1983),
no. 1, 277–301] commute with the maps in the exact sequences of
Rothenberg [J. L. Shaneson , Ann. of Math. (2) 90 (1969), 296–334;
MR0246310 (39 #7614)].”
{For the entire collection see MR0711037 (84h:55001)}.
{For the entire collection see MR0711037 (84h:55001)}
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0686160 (84f:18021) 18F25 55R12 57Q12
Munkholm, Hans J. ; Ranicki, Andrew A.
The projective class group transfer induced by an S 1 -bundle.
Current trends in algebraic topology, Part 2 (London, Ont., 1981), pp.
461–484, CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I.,
1982.
Authors’ introduction: “This paper gives an explicit algebraic description of the geometric transfer map induced in the (reduced) projective
p
class groups by an S 1 -bundle S 1 → E −
→B
p∗K0 : K̃0 (Z[ρ]) → K̃0 (Z[π])
with π = π1 (E), ρ = π1 (B). This is the transfer map (1.4) of the
preceding paper [MR0686159 (84f:18020) above], to which we refer
for terminology and background material. In particular, t ∈ π is the
canonical generator of the cyclic group ker(p∗ : π ρ) represented by
the inclusion S 1 → E of a fibre, ϕ: Z[π] Z[π]/(t − 1) = Z[ρ]; r 7→
r is the projection of fundamental group rings induced by p∗ : π ρ,
∼
and Z[π] −
→ Z[π]; r 7→ rt is a ring automorphism determined by the
orientation class w1 (p) ∈ H 1 (B; Z2 ) such that (t − 1)r = rt (t − 1). In
the orientable case w1 (p) = 0, t ∈ π is central and rt = r.
“Our main results are the following. Proposition 2.1: The projection
of rings ϕ: Z[π] Z[ρ] gives rise to an algebraic transfer map in the
projective class groups
ϕ!0 : K0 (Z[ρ]) → K0 (Z[π]);
im(X) 7→ im(X ! ) − Z[π]n .
Here X ∈ Mn (Z[ρ]) is a projection (i.e. an n × n matrix X with entries
2
in Z[ρ] such that X = X) and X ! ∈ M2n (Z[π]) is the projection
defined by
X
Y
∈ M2n (Z[π])
X! =
t − 1 1 − Xt
for any X, Y ∈ Mn (Z[π]) such that ϕ(X) = X, X(1 − X) = Y (t − 1),
XY = Y X t . Proposition 4.1: The algebraic and geometric transfer
maps in the reduced projective class groups coincide, that is if B, E
are finitely dominated CW complexes
ϕ̃!0 = p∗K0 : K̃0 (Z[ρ]) → K̃0 (Z[π]); [B] 7→ ϕ̃!0 ([B]) = p∗K0 ([B]) = [E]
with [B], [E] the Wall finiteness obstructions.”
{For the entire collection see MR0686135 (83m:55002b)}
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0686159 (84f:18020) 18F25 55R12
Munkholm, Hans J. ; Pedersen, Erik K.
Transfers in algebraic K- and L-theory induced by
S 1 -bundles.
Current trends in algebraic topology, Part 2 (London, Ont., 1981), pp.
451–460, CMS Conf. Proc., 2, Amer. Math. Soc., Providence, R.I.,
1982.
From the text: “Geometric transfers coming from S 1 -bundles: Let
p
S1 → E −
→ B be a fiber bundle with fundamental group sequence
ϕ
i
(1.1) Z −
→π−
→ ρ → {1} and orientation map (1.2) ω: ρ → {±1} =
Aut(Z). Also suppose given orientation maps wB : ρ → {±1}, wE : π →
{±1} such that (1.3) wE (g) = wB (ϕ(g))ω(ϕ(g)), g ∈ π. If B is a finitely
dominated CW complex then so is E and there is a homomorphism
(1.4) p∗K0 : K̃0 (Zρ) → K̃0 (Zπ) mapping the finiteness obstruction σ̃(B)
to σ̃(E) [see K. Ehrlich, J. Pure Appl. Algebra 14 (1979), 131–
136; MR0524182 (80g:55031)]. Let B and B1 be finite complexes. If
f : B1 → B is a homotopy equivalence then so is the pull-back f : E1 →
E and there is a homomorphism [D. R. Anderson, Michigan Math.
J. 21 (1974), 171–180; MR0343283 (49 #8025)] (1.5) p∗Wh : Wh(ρ) →
Wh(π) mapping the Whitehead torsion τ (f ) to τ (f ). Finally, if B is a
(simple) Poincaré duality space while f : N → B is a surgery problem
(we suppress the bundle data from the notation) then the pull-back
f : M → E is again a surgery problem and there is a homomorphism
(1.6) p∗L : Lεl (ρ; wB ) → Lεl+1 (π; wE ) mapping the surgery obstruction for
f to the one for f . Here ε = s or ε = h.
“In Section 2 of this note we announce a complete algebraic description of p∗Wh and p∗L (in terms of matrices). Details have appeared
elsewhere [the authors, Comment. Math. Helv. 56 (1981), 404–430;
MR0639359 (83c:57007); “Wall group transfers for S 1 -bundles. An
algebraic description”, Preprint, Odense Univ., Denmark, 1981; per
bibl.]. The main part of the present paper is Section 3 in which we
prove some vanishing results for p∗Wh .”
{For the entire collection see MR0686135 (83m:55002b)}
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0639359 (83c:57007) 57Q10 18F25
Munkholm, Hans J. ; Pedersen, Erik Kjaer
Whitehead transfers for S 1 -bundles, an algebraic description.
Comment. Math. Helv. 56 (1981), no. 3, 404–430.
f
Given an orientable S 1 -bundle S 1 → E −
→ B with fundamental group
sequence Z → π → ρ → {1} there is defined a geometric transfer map
∗
fWh
: Wh(ρ) → Wh(π) in the Whitehead groups, sending the torsion
τ (h: X1 → X2 ) of a homotopy equivalence h of finite CW complexes
∗
with a π1 -isomorphism reference map X2 → B to the torsion fWh
(x) =
τ (f ∗ h: f ∗ X1 → f ∗ X2 ) of the induced homotopy equivalence f ∗ h which
is equipped with a reference map f ∗ X2 → E. The authors obtain the
∗
following algebraic description of fWh
: Let x ∈ Wh(ρ) be represented
by an n × n invertible matrix A over Z[ρ]; choose lifts of A and B =
−1
A to n × n matrices A and B over Z[π] so that AB = I − (t − 1)C
for some n × n matrix C over Z[π], with t ∈ π the image
of 1 ∈ Z;
A −C
∗
then fWh (x) ∈ Wh(π) is given by the 2n × 2n matrix
t−1 B
over Z[π], which is invertible. The key step in the derivation of this
formula is to show that if h: X1 → X2 has Z[ρ]-module cellular chain
complex kernel
A
→ Z[ρ]n → 0 → · · ·
C(h): · · · → 0 → Z[ρ]n −
then f ∗ h: f ∗ X1 → f ∗ X2 has Z[π]-module cellular chain complex kernel
n (t−1−A)
A
t−1
C(f ∗ h): · · · → 0 → Z[π] −−−−→ Z[π]n ⊕ Z[π]n −−−−→ Z[π]n → 0 → · · · .
A twisted version of the formula, for nonorientable S 1 -bundles is also
obtained.
{Reviewer’s remark: The algebraic methods inaugurated by this
paper have led to analogous algebraic formulae for the S 1 -bundle
transfer maps fK∗ 0 : K̃0 (Z[ρ]) → K0 (Z[π]), fL∗ : Ln (Z[ρ]) → Ln+1 (Z[π]) in
the reduced projective class groups (the value groups of the Wall
finiteness obstruction), and the Wall surgery obstruction groups.
See Section 7.8 of the reviewer’s book [Exact sequences in the algebraic theory of surgery, Princeton Univ. Press, Princeton, N.J.,
1981; MR0620795 (82h:57027)] and the papers “Transfers in algebraic
K- and L-theory induced by S 1 -bundles” by the authors [Symposium
on Algebraic Topology (London, Ont., 1981), Contemporary Mathematics, 12, Amer. Math. Soc., Providence, R.I., 1982] and “The
projective class group transfer induced by an S 1 -bundle” by the first
author and the reviewer [ibid.].}
A. A. Ranicki (Edinburgh)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0631085 (82j:53116) 53C65 53A35
Haagerup, Uffe; Munkholm, Hans J.
Simplices of maximal volume in hyperbolic n-space.
Acta Math. 147 (1981), no. 1-2, 1–11.
The main theorem is that in hyperbolic space of more than one
dimension a simplex is of maximal volume if and only if it is regular
and ideal. This result is new in dimensions n ≥ 4, and there makes
applicable a technique developed by Gromov for proving Mostow’s
rigidity theorem for closed hyperbolic manifolds.
Troels Jørgensen (New York)
MR0601684 (82d:18014) 18F25 55R99 57Q10
Munkholm, Hans J.
Transfer on algebraic K-theory and Whitehead torsion for PL
fibrations.
J. Pure Appl. Algebra 20 (1981), no. 2, 195–225.
If ϕ: π → ρ is a group homomorphism which is the inclusion of π as a
subgroup of finite index in ρ, the group ring Z[ρ] is an f.g. free Z[π]module and there is defined a transfer map ϕ∗ : K∗ (Z[ρ]) → K∗ (Z[π])
in the algebraic K-groups. Milnor’s treatment of Whitehead torsion
in terms of based chain complexes with based homology modules is
here generalized to modules with a based free resolution, allowing the
construction of a transfer map ϕ∗ : K1 (Z[ρ]) → K1 (Z[π]) for a group
homomorphism ϕ: π → ρ such that ν = ker(ϕ) is a group of type (FF),
i.e. such that Z viewed as a Z[ν]-module with trivial ν-action admits
a finite free resolution. Furthermore, Hatcher’s theory of PL fibrations
is used to describe ϕ∗ in terms of the action of the representation ring
G(π), at least in the case when there exists a PL fibration p: E → B
ϕ
r
of compact polyhedra with p∗ : π1 (B) → π → π1 (E) = ρ such that r is
onto and ν = ker(ϕ) is of type (FF).
A. A. Ranicki (Edinburgh)
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0585656 (81k:53046) 53C35 51N20 57N15
Munkholm, Hans J.
Simplices of maximal volume in hyperbolic space, Gromov’s
norm, and Gromov’s proof of Mostow’s rigidity theorem
(following Thurston).
Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen,
Siegen, 1979), pp. 109–124, Lecture Notes in Math., 788, Springer,
Berlin, 1980.
The author announces his joint result with Haagerup that in a hyperbolic n-space a geodesic n-simplex is of maximal volume if and only if
all its vertices are on the sphere at infinity and all its faces are congruent modulo the isometries of the hyperbolic n-space. This result was
conjectured by Milnor. He then outlines how this result can be used
in a proof of Mostow’s rigidity theorem which was given by W. P.
Thurston [“The geometry and topology of three-manifolds”, Lecture
Notes, Princeton Univ., Princeton, N.J., 1977/78] and was attributed
by Thurston to Gromov.
{For the entire collection see MR0585648 (81i:57001)}
Y.-T. Siu (Stanford, Calif.)
MR0580901 (81j:55016) 55R05
Munkholm, Hans J. ; Pedersen, Erik Kjaer
On the Wall finiteness obstruction for the total space of
certain fibrations.
Trans. Amer. Math. Soc. 261 (1980), no. 2, 529–545.
Let p: E → B be a Serre fibration with fiber F . Assume that B is
finitely dominated and F is a finite complex. The authors study the relationship between the Wall finiteness obstructions w̃(E) ∈ K̃0 (Zπ1 E)
ϕ
s
and w̃(B) ∈ K̃0 (Zπ1 B). If π1 E → π → π1 B denotes a factorization of
p∗ with s surjective, then they compute s∗ w̃(E) under the assumption
that ker ϕ is of type FP and that the covering F corresponding to the
kernel of π1 F → ker ϕ has finitely generated integral homology. The
description involves the transfer ϕ∗ : K0 (Zπ1 B) → K0 (Zπ1 E) (which is
well defined under the assumption made) and the integral representations Hi (F ; Z) of π. In case ϕ is monic one recovers essentially the
main result of the second author and L. R. Taylor [Amer. J. Math.
100 (1978), no. 4, 887–896; MR0509078 (80g:55030)]. If F is aspherical and π1 F → π1 E is injective, the formula reduces to w̃E = ϕ∗ w̃(B).
Several other interesting applications are given and the relationship
with earlier results is extensively discussed (compare in particular articles by D. R. Anderson [ibid. 95 (1973), 281–293; MR0334227 (48
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
#12546)] and K. Ehrlich [J. Pure Appl. Algebra 14 (1979), no. 2,
131–136; MR 80g:550318]).
Guido Mislin (Zurich)
MR0525622 (82g:55020) 55R60 57Q99
Munkholm, Hans J.
A chain level transfer homomorphism for PL fibrations.
Math. Z. 166 (1979), no. 2, 183–186.
Let p: E → B be a simplicial map such that the inverse image of
each simplex is a finite subcomplex. Then, if B 0 is the barycentric
subdivision of B, a canonical homomorphism τ (p)# : C∗ (B 0 ) → C∗ (E 0 )
is defined. Here, C denotes simplicial chains, E 0 the barycentric
subdivision of E, but identified with E as a space so that p is still
simplicial from E 0 to B 0 . τ (p)# is not a chain complex map in general,
but a certain additional condition which holds, e.g., if p is a Serre
fibration, will guarantee this. In this case τ (p)# is a “local formula”
inducing homology and cohomology transfer.
Norman Levitt (New Brunswick, N.J.)
MR0557164 (81e:57024) 57Q10
Munkholm, Hans J.
Whitehead torsion for PL fiber homotopy equivalences.
Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo,
Waterloo, Ont., 1978), pp. 90–101, Lecture Notes in Math., 741,
Springer, Berlin, 1979.
Let E and B be finite connected polyhedra and p: E → B a piecewise
linear Serre fibration. If the inclusion of a subpolyhedron A in B
is a homotopy equivalence, the same will hold for EA ⊂ E, where
EA = p−1 (A). Therefore, Whitehead torsions τ (B, A) ∈ Wh(π1 (B))
and τ (E, EA ) ∈ Wh(π1 (E)) are well defined [cf. J. W. Milnor, Bull.
Amer. Math. Soc. 72 (1966), 358–426; MR0196736 (33 #4922)].
The aim of the paper is to show that the transfer of τ (B, A) under
the homomorphism induced by p on fundamental groups, whenever it
is defined, is τ (E, EA ). For similar results with K0 instead of K1 and
Wh, cf. the article by the author and E. K. Pedersen [Trans. Amer.
Math. Soc. 261 (1980), no. 2, 529–545].
Instead of based chain complexes with based homology as in the
classical treatment of torsion, the author uses chain complexes C∗ such
that each Ci and each Hi (C∗ ) has a preferred based finite resolution.
This makes it possible to define a transfer map ϕ∗ : Wh(ρ) → Wh(π)
for a homomorphism ϕ: π → ρ which is surjective and whose kernel ν,
acting trivially on Z, defines a Zν-module with a finite free resolution.
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
As usual, the transfer ϕ∗ of a Zρ-module M and an endomorphism
f consists essentially in looking at M and f as a Zπ-module and a
Zπ-endomorphism via the ring homomorphism Zπ → Zρ induced by
ϕ.
Under the above assumptions, the main theorem states that
ϕ∗ (τ (B, A)) = τ (E, EA ) where ρ is π1 (B), π is π1 (E) and ϕ is induced by p. In fact the results are more general. In particular, some
relation between τ (B, A) and τ (E, EA ) is given even in the case
where the transfer cannot be defined but where ν is a finite cyclic
group.
The proofs, mostly outlined, are quite tricky, a common feature in
torsion computations, because one has to be rather careful with the
basis elements.
{For the entire collection see MR0557161 (80k:55002)}
Oscar Burlet (Froideville)
MR0509167 (80m:55008) 55P10
Munkholm, H.
Erratum to: “shm maps of differential graded algebras. I. A
characterization up to homotopy” [J. Pure Appl. Algebra 9
(1976), no. 1, 39–46; MR 55 #11247a].
J. Pure Appl. Algebra 13 (1978), no. 3, 335.
In the original paper, the proof that i: DA → DASHh sends homology
isomorphisms into isomorphisms is incomplete; a more subtle complete
proof is given here.
J. Stasheff (Chapel Hill, N.C.)
MR0509162 (80m:55018) 55P60 55P99
Munkholm, Hans J.
DGA algebras as a Quillen model category. Relations to shm
maps.
J. Pure Appl. Algebra 13 (1978), no. 3, 221–232.
From the introduction: “In this paper we deal with categories of differential graded augmented algebras with unit over a field. If the
morphisms are maps preserving all the structure, we call the category
DA; if we allow strongly homotopy-multiplicative maps, we talk of
DASH. On DASH there is a homotopy notion and a resulting homotopy category DASHh . For DA we have the category HoDA obtained
by localizing DA with respect to all homology isomorphisms. We shall
prove that HoDA and DASHh are isomorphic.
“The input in the proof is the universal description of DASHh given
by the author in an earlier paper [same journal 9 (1976/77), no. 1,
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
39–46; MR0438331 (55 #11247a)] and the theory of closed model
categories of Quillen.
“We show that DA is a closed model category in a natural way. The
proof follows closely similar proofs for the category of commutative
differential graded algebras and for cocommutative differential graded
coalgebras. We characterize the cofibrant objects in DA. We prove that
HoDA and DASHh enjoy the same characteristic universal properties,
so they coincide. This makes precise the claim by the reviewer and
Halperin that over a field the category of differential algebras and
shm maps is obtained by localizing with respect to all homology
isomorphisms.”
J. Stasheff (Chapel Hill, N.C.)
19771101
19780501
MR0 438 331 (55 11247a) 55D20
Munkholm, Hans J.
shm maps of differential algebras. I. A characterization up to
homotopy.
J. Pure Appl. Algebra 9 (1976/77), no. 1, 39–46.
MR0438332 (55 11247b) 55H20
Munkholm, Hans J.
shm maps of differential algebras. II. Applications to spaces
with polynomial cohomology.
J. Pure Appl. Algebra 9 (1976/77), no. 1, 47–63.
From the author’s introduction to Part I: “In this note we study
the category DASHh of differential graded algebras over a field with
homotopy classes of shm (strongly homotopy multiplicative) maps
as morphisms. Algebraic shm maps were first introduced by Clark.
Stasheff and Halperin indicated how one should be able to use them
to obtain collapse results for the Eilenberg-Moore spectral sequence
for homogeneous spaces. Gugenheim and the present author made a
relatively thorough study of the properties of shm maps. The results
enabled the present author to carry out the program that Stasheff
and Halperin had anticipated.
“Let DA be the category of differential graded algebras with multiplicative differential maps and let i: DA → DASHh be the obvious
functor. In this paper we prove that i is universal among all functors
k: DA → C which satisfy: (i) If f ∈ DA(A, B) is a homology isomorphism then k(f ) is an isomorphism. (ii) If f ∼
= g ∈ DA(A, B) then
k(f ) = k(g).”
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
The reviewer and Halperin claim that over a field DASH is obtained
from DA by “localizing with respect to homology isomorphisms”.
Since the unit R → Λ(y) ⊗ R[x] for a Koszul complex is not an
isomorphism when viewed in DASH this statement is not correct.
The reviewer has tried (not very hard, but so far in vain) to ascertain
whether the claim is true when DASH is replaced by DASHh , i.e.,
whether the above condition (ii) can be left out. (In a recent preprint
(“DGA algebras as a Quillen model category”, Mat. Inst. Odense,
preprint no. 2, 1977) the author extends the present work to prove
that DASHh is in fact a closed model category and is isomorphic
with HoDA, the category obtained by localizing as above.) Part I
has as its main step proving that any homology isomorphism in DA
has an inverse in DASHh . (This is what Halperin and the reviewer
had in mind.) The present paper also establishes related results for
the category of differential graded augmented coalgebras DC and the
corresponding DCSH.
Part II is correctly titled; the author considers “a fibre square
E0
↓
X
→
→
E
↓
B
with H ∗ X, H ∗ B and H ∗ E polynomial and we show that γX and
γE give rise to a small Koszul type complex H ∗ X ⊗t Λ(sV ) ⊗s H ∗ E
whose homology is H ∗ E 0 . Here Λ(sV ) is the exterior coalgebra on
the suspension of the space V of indecomposables of H ∗ B, and t,
s are twisting cochains. Explicit formulas for t and s are obtained
provided Sq1 behaves nicely on H ∗ X, H ∗ E. Examples where Sq1
does behave nicely include generalized Eilenberg-MacLane spaces
but also BSO. Schochet proved that the Eilenberg-Moore spectral
sequence for a fibration of type X → B may be non trivial. We give a
different example of that phenomenon.” The example is very close to
Schochet’s: f : BSO × BSO → K(Z2 , 4) is given by f ∗ (i) = w2 × w2 and
provides independent verification of his result. “Some of the algebra
of the present paper is of interest in itself. In Section 4 we show that
the natural transformation of differential (co-)algebras BA1 × BA2 →
B(A1 × A2 ), Ω(C1 × C2 ) → ΩC1 × ΩC2 have strongly homotopy (co)multiplicative homotopy inverses.”
J. Stasheff
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0350735 (50 3227) 55H20
Munkholm, Hans J.
The Eilenberg-Moore spectral sequence and strongly
homotopy multiplicative maps.
J. Pure Appl. Algebra 5 (1974), 1–50.
The reviewer [Bull. Amer. Math. Soc. 74 (1968), 334–339; MR0239596
(39 #953)] initiated the algebraic analysis of the internal structure of
the Eilenberg-Moore spectral sequence with a view towards proving
collapse theorems and calculating differentials. Since then, there have
been numerous attempts to simplify the proofs and to generalize
the results. Probably the simplest proofs are those by V. K. A.
M. Gugenheim and the reviewer [On the theory and applications of
differential torsion products, Mem. Amer. Math. Soc., No. 142, Amer.
Math. Soc., Providence, R.I., 1974]. The author here attains what
is probably the most general collapse theorem obtainable by such
algebraic techniques. (His methods do not address the problem of
computing differentials when collapse fails.) Take cohomology with
coefficients in a ring R and take fibrations to have trivial local
coefficients. Assume given a fibre square
E −−−−→


y
Y


y
X −−−−→ B
such that H X, H B, and H ∗ Y are polynomial algebras of finite
type and Sq1 vanishes on H ∗ X and H ∗ Y if char R = 2. The author’s
main theorem asserts that H ∗ E is then additively isomorphic to
TorH ∗ B (H ∗ X, H ∗ Y ). (For comparison, the reviewer’s main collapse
theorem had Y contractible and X = BT n , extension to X = BG for
other compact Lie groups with suitably little torsion following by a
naturality argument due to P. Baum.) The basic idea is to prove
that TorC ∗ B (C ∗ X, C ∗ Y ) is isomorphic to TorH ∗ B (H ∗ X, H ∗ Y ) by first
generalizing Tor to a functor defined on a suitable category of diagrams
A ← B → C of differential algebras and homotopy classes of strongly
homotopy multiplicative morphisms thereof and then producing a
homology isomorphism from H ∗ X ← H ∗ B → H ∗ Y to C ∗ X ← C ∗ B →
C ∗ Y in the resulting category. The argument bristles with technical
difficulties and entails a great deal of hard algebraic work, but the
exposition is careful and thorough throughout.
J. P. May
∗
∗
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0347946 (50 445) 18G15 57F30
Gugenheim, V. K. A. M.; Munkholm, H. J.
On the extended functoriality of Tor and Cotor.
J. Pure Appl. Algebra 4 (1974), 9–29.
The authors study a differential algebra A and the “bar construction”,
or “classifying coalgebra”, B(A). They study the category having as
objects differential algebras, and as maps from A to A0 all coalgebra
maps from B(A) to B(A0 ). They also study the dual situation, beginning with a coalgebra and forming the “cobar construction”, or “loop
algebra”. One advantage of this extended category is that the cup
product on C ∗ (X), while not a map of algebras, is a map in the extended category. The authors prove that the differential Tor and Cotor
functors extend to the new categories. They also derive various properties of these functors; in particular, with certain hypotheses, they
describe a multiplicative structure on TorB (M, N ).
J. McNamara
MR0312498 (47 1055) 55H20
Munkholm, Hans J.
A collapse result for the Eilenberg-Moore spectral sequence.
Bull. Amer. Math. Soc. 79 (1973), 115–118.
The computation of H ∗ (G/H) in terms of H ∗ (BH) ← H ∗ (BG) has
received considerable attention over the years. There are currently at
least three attacks which are approaching publication. The author’s is
indicated here with an outline of a proof of the following theorem: Let
F → E →p B be a fibration with E and B 1-connected, H ∗ (E; Z2 ) and
H ∗ (B; Z2 ) polynomial algebras in finitely many variables. Suppose also
that Sqn−1 (y) = 0 for all y ∈ H n E. Then the Eilenberg-Moore spectral
sequence with E2 = TorH ∗ (B;Z2 ) (H ∗ (E; Z2 ), Z2 ) and Er ⇒ H ∗ (F ; Z2 )
collapses.
As is common, the collapse result disguises the fact that the proof
gives directly an isomorphism
TorH ∗ (B) (H ∗ (E); Z2 ) ≈ TorC ∗ (B) (C ∗ (E), Z2 ).
The result itself is distinguished by the appearance of the condition
“S n−1 (y) = 0” rather than by a condition on the spaces (e.g., the
existence of maximal tori).
The basic technique is the use of the category of differential graded
algebras and sh m maps as proposed for this problem by the reviewer
and S. Halperin [Proc. Advanced Study Inst. Algebraic Topology
(Mat. Inst., Aarhus Univ., Aarhus, 1970), Vol. III, pp. 567–577, Mat.
Inst., Aarhus Univ., Aarhus, 1970] and improved by the author and V.
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
K. Gugenheim (“On the extended naturality”, J. Pure Appl. Algebra,
to appear). In verifying the commutativity of the diagram
H ∗B
=⇒ H ∗ E
⇓
C ∗B
=⇒
⇓
C ∗E
in this category, the author is able to reduce the argument to one
with domain Z2 [z1 , z2 ] and range strongly homotopy commutative.
The homotopy classification of such maps is shown to depend on the
cohomology classes of the images f (zi ) and on their ∪1 -products. This
gives an alternate explanation of May’s maxim that ∪1 is enough for
spaces with polynomial cohomology.
J. Stasheff
MR0322857 (48 1218) 55C35
Munkholm, Hans J. ; Nakaoka, Minoru
The Borsuk-Ulam theorem and formal group laws.
Osaka J. Math. 9 (1972), 337–349.
Let G be a cyclic group of odd order q and Σ a homotopy (2n +
1)-sphere with a free differentiable G-action. For a continuous map
f : Σ → M , M a differentiable m-manifold, let A(f ) ={x ∈ Σ: f (x) =
f (xg) for all g ∈ G}. The authors have shown previously [the first
author, Math. Scand. 24 (1969), 167–185 (1970); MR0258025 (41
#2672); the second author, Osaka J. Math. 7 (1970), 423–441;
MR0275422 (43 #1179)] that dim A(f ) ≥ 2n + 1 − (p − 1)m if q is
a prime p, and the first author established [ibid. 7 (1970), 451–456;
MR0276960 (43 #2699)] that
(∗)
dim A(f ) ≥ (2n + 1) − (pa − 1)m
− [m(a − 1)pa − (ma + 2)pa−1 + m + 3]
if q is a prime power pa and M is Euclidean space Rm . In the present
paper the authors prove (∗) for q = pa and any differentiable manifold
M . The method of proof is different from the previous ones; the
result follows from a general theorem in connection with the formal
group law for a general cohomology theory specialized to K-theory.
This method also yields a generalized result of J. W. Vick [Bull.
Amer. Math. Soc. 75 (1969), 1017–1019; MR0245023 (39 #6336)] on
equivariant maps from lens spaces to spheres.
A. Aeppli
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0276960 (43 2699) 55.36
Munkholm, Hans J.
On the Borsuk-Ulam theorem for Zpa actions on S 2n−1 and
maps S 2n−1 → Rm .
Osaka J. Math. 7 1970 451–456
The author proves a pa Borsuk-Ulam theorem: For a free Zpa -action
on the (2n − 1)-sphere S 2n−1 , p an odd prime, and a (continuous) map
f : S 2n−1 → Rm , one has dim A(f ) ≥ (2n − 1) − (pa − 1)m − [m(a −
1)pa − (ma + 2)pa−1 + m + 3], where A(f ) ={x ∈ S 2n−1 |f (x) = f (xg)
for all g ∈ G}, G = Zpa . On the other hand, there is a pa BorsukUlam “anti-theorem”: For the standard linear action of Zpa on S 2n−1 ,
let a > 1, pa 6= 9, and let 2n − 1 ≤ (pa − 1)m + (2p − 3)m − 1; then
there exists a map f : S 2n−1 → Rm with A(f ) = ∅. The proofs are
based on the existence of sections in certain bundles, in continuation
of the author’s earlier paper [Math. Scand. 24 (1969) 167–185 (1970);
MR0258025 (41 #2672)].
A. Aeppli
MR0258025 (41 2672) 55.36
Munkholm, Hans Jørgen
Borsuk-Ulam type theorems for proper Zp -actions on (mod p
homology) n-spheres.
Math. Scand. 24 1969 167–185 (1970)
Let Sn be a closed n-manifold which is a mod p homology sphere (p a
prime), and let G = Zp act freely on Sn , µ: Sn × G → Sn . f : Sn → M m
is a “nice” map into the m-manifold M m , i.e., there exists f0 ' f and
there exists y0 ∈ M m such that for all x, f0 (xg) 6= y0 for at most one
g ∈ G. If p = 2, assume that
f∗ = 0: Hn (Sn ; Z2 ) → Hn (M m ; Z2 ).
A(µ; f ) = {x ∈ Sn |f (x) = f (xg) for all g ∈ G} is the coincidence set
of f with respect to µ. cd denotes the cohomological dimension over
the coefficients Zp . The main theorem states that in the described
situation, cd(A(µ; f )) ≥ n − (p − 1)m. For the proof, the bundle
ξµ = (IG → Sn ×G IG → Sn /G)
is considered, fiber IG = augmentation ideal of the group algebra RG.
For ep (ξµ ) = (mod p) Euler class of ξµ , the following proposition holds:
If f : Sn → M m as above, cd(A(µ; f )/G; Zp ) < n − (p − 1)m, f nice,
f∗ = 0: Hn (Sn ; Zp ) → Hn (M m ; Zp ), then (ep (ξµ ))m = 0. On the other
hand, the author shows that (ep (ξµ ))m 6= 0 for G = Zp , n ≥ (p − 1)m,
hence the theorem is established. Application of the theorem to Sn =
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
S n gives the mod p Conner-Floyd theorem (without differentiability
conditions), and M m = Rm gives the mod p Bourgin-Yang theorem.
A. Aeppli
MR0251143 (40 4374) 20.50
Munkholm, Hans Jørgen
Mod 2 cohomology of D2n and its extensions by Z2 . 1969
Conf. on Algebraic Topology (Univ. of Illinois at Chicago Circle,
Chicago, Ill., 1968) pp. 234–252 Univ. of Illinois at Chicago Circle,
Chicago, Ill.
The author studies the problem of computing H ∗ (G; Z2 ) as an algebra
over the Steenrod algebra when G is a group of order 2n . The
method of attack is to write G as an extension 1 → Z2 → G → G → 1,
where H ∗ (G; Z2 ) is known inductively, and to compute the spectral
sequence of the extension by means of cochain operations. Explicit
computations are given for the dihedral group D2n , the generalized
quaternion group Q2n , and all other extensions of D2n by Z2 .
J. P. May
MR0238324 (38 6600) 55.60 57.00
Munkholm, Hans Jørgen
A Borsuk-Ulam theorem for maps from a sphere to a compact
topological manifold.
Illinois J. Math. 13 1969 116–124
From the author’s introduction: “Theorem 1: Let f : S n → M k be a
map from the n-sphere to a compact topological k-manifold M k ; let
A(f ) = {x ∈ S n ; f (x) = f (−x)}. Then (a) if n > k, then dim(A(f )) ≥
n − k, and (b) if n = k and f ∗ : H n (M n ; Z2 ) → H n (S n ; Z2 ) is zero, then
A(f ) 6= ∅.
“If one restricts to manifolds admitting a differentiable structure
the theorem may be found in the work of P. E. Conner and E. E.
Floyd [Differentiable periodic maps, Ergeb. Math. Grenzgeb. (N.F.),
Band 33, Springer, Berlin, 1964; MR0176478 (31 #750)]; the restriction to the case M k = Rk is known as the Bourgin-Yang theorem (see D. G. Bourgin [Comment. Math. Helv. 29 (1955), 199–
214; MR0072469 (17,289d)] and C.-T. Yang [Ann. of Math. (2) 60
(1954), 262–282; MR0065910 (16,502d); ibid. (2) 62 (1955), 271–283;
MR0072470 (17,289e)]); our line of reasoning is close to that of Conner and Floyd [loc. cit.].”
Morton Brown
Results from MathSciNet: Mathematical Reviews on the Web
c Copyright American Mathematical Society 2005
MR0223465 (36 6513) 20.80
Munkholm, Hans Jørgen
Induced monomial representations, Young elements, and
metacyclic groups.
Proc. Amer. Math. Soc. 19 1968 453–458
In the Young theory of the representations of the symmetric groups,
primitive idempotents are constructed by certain combinatorial considerations involving suitable linear characters of appropriate subgroups P, Q. Isolating those features which underlie this construction,
the author gives conditions which allow the same construction to proceed in an arbitrary finite group G. In case Q normalizes P and G =
P Q, the equivalence of such representations is settled as well. In particular, this applies to metacyclic groups.
P. Fong
MR0182903 (32 385) 50.30
Munkholm, Hans Jørgen
On the classification of incidence theorems in plane
projective geometry.
Math. Z. 90 1965 215–230
Es wird der Zusammenhang untersucht zwischen dem von LombardoRadice [Rend. Sem. Mat. Univ. Padova 24 (1955), 312–345;
MR0075609 (17,776e)] eingeführten Begriff “proposizione configurazionale non degenere” und der folgenden Einschränkung des Begriffs
“allgemeiner normaler offener Schließungssatz” (“Schließungssatz”
= “incidence theorem”): Offene Teilebene (“partial plane”) mit
einem Tripel von Punkten, von denen keine zwei in der Teilebene
eine Verbindungsgerade besitzen. Dieser Untersuchung dienen
verschiedene Äquivalenzbegriffe zwischen Schließungssätzen der
angegebenen Art.
G. Pickert