Algebraic Morse theory and homological perturbation theory
We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim.
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Sköldberg, E. 2023-02-26T12:33:18Z 2023-02-26T12:33:18Z 2018 Algebraic Morse theory and homological perturbation theory / E. Sköldberg // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 124–129. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC: Primary 18G35; Secondary 55U15. https://nasplib.isofts.kiev.ua/handle/123456789/188379 We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Algebraic Morse theory and homological perturbation theory Article published earlier |
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Algebraic Morse theory and homological perturbation theory |
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Algebraic Morse theory and homological perturbation theory Sköldberg, E. |
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Algebraic Morse theory and homological perturbation theory |
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Algebraic Morse theory and homological perturbation theory |
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Algebraic Morse theory and homological perturbation theory |
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Algebraic Morse theory and homological perturbation theory |
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algebraic morse theory and homological perturbation theory |
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Sköldberg, E. |
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Sköldberg, E. |
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2018 |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim.
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1726-3255 |
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Algebraic Morse theory and homological perturbation theory / E. Sköldberg // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 124–129. — Бібліогр.: 12 назв. — англ. |
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AT skoldberge algebraicmorsetheoryandhomologicalperturbationtheory |
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2025-11-24T06:53:04Z |
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2025-11-24T06:53:04Z |
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1850843316799668224 |
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“adm-n3” — 2018/10/23 — 15:17 — page 124 — #130
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 26 (2018). Number 1, pp. 124–129
c© Journal “Algebra and Discrete Mathematics”
Algebraic Morse theory and homological
perturbation theory
Emil Sköldberg
Communicated by V. Lyubashenko
Abstract. We show that the main result of algebraic Morse
theory can be obtained as a consequence of the perturbation lemma
of Brown and Gugenheim.
1. Introduction
Robin Forman introduced discrete Morse theory in [For98] as a com-
binatorial adaptation of the classical Morse theory suited for studying
the topology of CW-complexes. Its fundamental idea is also applicable in
purely algebraical situations (see e.g. [Jon03], [Koz05], [JW09], [Skö06]).
Homological perturbation theory on the other hand builds on the
perturbation lemma [Bro65], [Gug72]. In addition to its applications
in algebraic topology, it has also found uses in e.g. the study of group
cohomology [Lam92], [Hue89], resolutions in commutative algebra [JLS02],
as well as in operadic settings, [Ber14].
In this note we show how to derive the main result of algebraic Morse
theory from the perturbation lemma. In related work, Berglund [Ber], has
also treated connections between algebraic Morse theory and homological
perturbation theory.
2010 MSC: Primary 18G35; Secondary 55U15.
Key words and phrases: algebraic Morse theory, homological perturbation
theory, Perturbation Lemma.
“adm-n3” — 2018/10/23 — 15:17 — page 125 — #131
E. Sköldberg 125
2. Definitions
We will briefly review the definitions of the main objects of study.
A contraction is a diagram of chain complexes of (left or right) modules
over a ring R
D C
g
f
h
where f and g are chain maps and h is a degree 1 map satisfying the
identities
fg = 1, gf = 1 + dh+ hd
and
fh = 0, hg = 0, h2 = 0.
A contraction is filtered if there is a bounded below exhaustive filtration on
the complexes which is preserved by the maps f , g and h. A perturbation of
a chain complex C is a map t : C → C of degree −1 such that (d+ t)2 = 0.
Given a perturbation t on C, we let C
t be the complex obtained by
equipping C with the new differential d+ t.
We can now state the perturbation lemma.
Theorem 1 (Brown, Gugenheim).
D C
g
f
h
and a filtration lowering perturbation t of C, the diagram
D
t′
C
t
g′
f ′
h′
where
f ′ = f + fSh, g′ = g + hSg, h′ = h+ hSh, t′ = fSg
and
S =
∞
∑
n=0
t(ht)n
defines a contraction.
“adm-n3” — 2018/10/23 — 15:17 — page 126 — #132
126 Algebraic Morse theory
Let us next review some terminology of algebraic Morse theory. By a
based complex of R-modules we mean a chain complex C of R-modules
together with direct sum decompositions Cn =
⊕
α∈In
Cα where {In} is a
family of mutually disjoint index sets. For f :
⊕
nCn →
⊕
nCn a graded
map, we write fβ,α for the component of f going from Cα to Cβ, and
given a based complex C we construct a digraph G(C) with vertex set
V =
⋃
n In and with a directed edge α → β whenever the component dβ,α
is non-zero.
A subset M of the edges of G(C) such that no vertex is incident to
more than one edge of M is called a Morse matching if, for each edge
α → β in M , the corresponding component dβ,α is an isomorphism, and
furthermore there is a well founded partial order ≺ on each In such that
γ ≺ α whenever there is a path α(n) → β → γ(n) in the graph G(C)M ,
which is the graph obtained from G(C) by reversing the edges from M .
Given the matching M , we define the set M0 to be the vertices that
are not incident to an arrow from M .
For α and β vertices in G(C)M we can now consider all directed paths
from α to β. For each such path γ, we get a map from Cα to Cβ by, for
each edge σ → τ in γ which is not in M take the map dτ,σ, and for each
edge σ → τ in γ which is the reverse of an edge in M take the map −d−1
σ,τ
and composing them. Summing these maps over all paths from α to β
defines the map Γβ,α : Cα → Cβ .
3. The main result
From the based complex C with Cn =
⊕
α∈In
Cα furnished with a
Morse matching M , we define another based complex C̃ by letting it be
isomorphic to C as a graded module, and defining the differential d̃ in C̃
as
d̃(x) =
{
dβ,α(x), if α → β ∈ M,
0, otherwise;
for x ∈ Cα.
We also need a based complex coming from the vertices in M0, so we
define C̃
M by
C̃M
n =
⊕
α∈In∩M0
Cα, d
C̃M = 0,
“adm-n3” — 2018/10/23 — 15:17 — page 127 — #133
E. Sköldberg 127
and maps f̃ : C̃ → C̃
M , g̃ : C̃M → C̃ and h̃ : C̃ → C̃[1] given by
f̃(x) =
{
x, if α ∈ M0,
0, otherwise,
g̃(x) = x, x ∈ Cα.
h̃(x) =
{
−d−1
α,β(x), if β → α ∈ M,
0, otherwise;
With this notation we can now formulate the following lemma.
Lemma 1. The diagram
C̃
M
C̃
g̃
f̃
h̃
is a contraction.
Proof. We first need to verify that f̃ and g̃ are chain maps, which is readily
seen. Next we check the identities
f̃ g̃ = 1, g̃f̃ = 1 + d̃h̃+ h̃d̃.
The first one is obvious, and the second follows from the fact that for a
basis element x ∈ Cα, d̃h̃(x) = −x if there is an edge β → α in M , and 0
otherwise; and similarly h̃d̃(x) = −x if there is an edge α → β in M , and
0 otherwise. The identities
h̃g̃ = 0, f̃ h̃ = 0, h̃2 = 0
follow from that vertices in M0 are not incident to any edge in M (the
first two) and that no vertex is incident to more than one edge in M (the
third).
Let us now define the perturbation t on C̃ as t = d− d̃, where d is the
differential on C, so
t(x) =
∑
α→β 6∈M
dβ,α(x)
for x ∈ Cα. This makes C̃
t and C isomorphic as based complexes.
“adm-n3” — 2018/10/23 — 15:17 — page 128 — #134
128 Algebraic Morse theory
Lemma 2. The diagram
C
M
C
g
f
h
where, for x ∈ Cα with α ∈ In,
dCM (x) =
∑
β∈M0∩In−1
Γβ,α(x), f(x) =
∑
β∈M0∩In
Γβ,α(x),
g(x) =
∑
β∈In
Γβ,α(x), h(x) =
∑
β∈In+1
Γβ,α(x),
is a filtered contraction.
Proof. From Lemma 1 together with the fact that there are no infinite
paths in G(C)M , the Morse graph of C, we can deduce that ht is locally
nilpotent, and we can thus invoke the perturbation lemma. It is not so
hard to see that the perturbed differential on C̃
M is given by
d(x) =
∞
∑
i=0
t(ht)i(x) =
∑
β∈M0∩In−1
Γβ,α(x)
and the maps f , g and h by
f(x) =
∞
∑
i=0
f(ht)i(x) =
∑
β∈M0∩In
Γβ,α(x)
g(x) =
∞
∑
i=0
g(ht)i(x) =
∑
β∈In
Γβ,α(x)
h(x) =
∞
∑
i=0
(ht)ih(x) =
∑
β∈In+1
Γβ,α(x)
where x ∈ Cα.
The above result is also shown (without the use of the perturbation
lemma) in [Ber] using a result from [JW09].
From the preceding lemma, the main result of algebraic Morse theory
now follows.
Theorem 2. Let C be a based complex with a Morse matching M , then
there is a differential on the graded module
⊕
α∈M0 Cα such that the
resulting complex is homotopy equivalent to C.
“adm-n3” — 2018/10/23 — 15:17 — page 129 — #135
E. Sköldberg 129
References
[Ber] Alexander Berglund, Algebraic discrete Morse theory II – extra algebraic
structures, Preprint.
[Ber14] Alexander Berglund, Homological perturbation theory for algebras over operads,
Algebr. Geom. Topol. 14 (2014), no. 5, 2511–2548.
[Bro65] R. Brown, The twisted Eilenberg-Zilber theorem, Simposio di Topologia
(Messina, 1964), Edizioni Oderisi, Gubbio, 1965, pp. 33–37.
[For98] Robin Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), no. 1,
90–145.
[Gug72] V. K. A. M. Gugenheim, On the chain-complex of a fibration, Illinois J. Math.
16 (1972), 398–414.
[Hue89] Johannes Huebschmann, Perturbation theory and free resolutions for nilpotent
groups of class 2, J. Algebra 126 (1989), no. 2, 348–399.
[JLS02] Leif Johansson, Larry Lambe, and Emil Sköldberg, On constructing resolutions
over the polynomial algebra, Homology Homotopy Appl. 4 (2002), no. 2, part
2, 315–336, The Roos Festschrift volume, 2.
[Jon03] Jakob Jonsson, On the topology of simplicial complexes related to 3-connected
and Hamiltonian graphs, J. Combin. Theory Ser. A 104 (2003), no. 1, 169–199.
[JW09] Michael Jöllenbeck and Volkmar Welker, Minimal resolutions via algebraic
discrete Morse theory, Mem. Amer. Math. Soc. 197 (2009), no. 923, vi+74.
[Koz05] Dmitry N. Kozlov, Discrete Morse theory for free chain complexes, C. R.
Math. Acad. Sci. Paris 340 (2005), no. 12, 867–872.
[Lam92] Larry A. Lambe, Homological perturbation theory, Hochschild homology, and
formal groups, Deformation theory and quantum groups with applications to
mathematical physics (Amherst, MA, 1990), Amer. Math. Soc., Providence,
RI, 1992, pp. 183–218.
[Skö06] Emil Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math.
Soc. 358 (2006), no. 1, 115–129 (electronic).
Contact information
E. Sköldberg School of Mathematics,
Statistics and Applied Mathematics,
National University of Ireland,
Galway, Ireland
E-Mail(s): emil.skoldberg@nuigalway.ie
Web-page(s): http://www.maths.nuigalway.ie/
∼emil/
Received by the editors: 23.09.2016.
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