Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partiallysummable labels
Shingo Okuyama— joint with Kazuhisa Shimakawa
National Institute of Technology, Kagawa College2018.9.13

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Introduction
Configuration spaces
Main construction and theorem

Outline1

Introduction

2

Configuration spaces
Partial abelian monoid
Configuration space of intervals

3

Main construction and theorem
Configuration space of intervals with partially summablelabelsApproximation map eIM → Ω′ BM
eM → BM
Quasi-fibration sequence eIM → E

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Introduction
Configuration spaces
Main construction and theorem

Approximation to a mapping space
Theorem 1 (Milgram-May-Segal)
There exists a weak homotopy equivalenceC(Rn , X ) → Ωn S n X , if X is path-connected.
An approximation to a mapping space by a configuration
space looks likeC(M, X ) ≃ map(M̂, (τ M ∗ X ))(Milgram-May-Segal, McDuff, ...)
In some cases, a system {C(Mn , X )} can approximate asystem of mapping spaces, that is, a homology theory.
(Segal, Shimakawa, Tamaki,...)
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Introduction
Configuration spaces
Main construction and theorem

Approximation to a mapping space

In most cases, an approximation map
C(M, X )− → map(M̂, (τ M ∗ X )))has geometric or physical iterpretation. So this talk is about
a geometric model of a (mapping) spacewith geometrically constructed approximation map.

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Introduction
Configuration spaces
Main construction and theorem

Group completion

Theorem 2 (Segal-F.Cohen)
C(Rn , X ) → Ωn S n X is a group completion if n ≥ 2.
A group completion of an admissible topological monoid M canbe constructed by a homotopy limit of a (possibly huge)
diagram {· · · → M → M → . . . } given by multiplication byelements taken from each connected component of M.
This talk is also about
A geometric construction of a group completion

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

A partial abelian monoid is ...

A partial abelian monoid isalmost an abelian monoid but with partially defined sum.
suitable for configuration space construction.
the additive part of an F1 -algebra.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Definition 1
A topological partial abelian monoid is a space M with basepoint 0 equipped with a subspace M2 of M × M and a mapµ : M2 → M which satisfies
1
2

3

M ∨ M ⊂ M2 , and µ(m, 0) = µ(0, m) = m, for all m ∈ M,
(m, n) ∈ M2 if and only if (n, m) ∈ M2 ,and µ(m, n) = µ(n, m),(µ(l, m), n) ∈ M2 if and only if (l, µ(m, n)) ∈ M2 , andµ(µ(l, m), n) = µ(l, µ(m, n)).

We denote µ(m, n) = m + n.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

ExamplesExtreme cases:
1

An abelian monoid is a partial abelian monoid.

2

A based space X can be regarded as a trivial partial
abelian monoid by setting X2 = X ∨ X and µ : X ∨ X → X
the folding map. It is called a trivial partial abelianmonoid.

3

Let M be an abelian monoid and N be a subset whichcontains 0. Then N is a partial abelian monoid if we set
N2 = {(n1 , n2 ) | n1 + n2 ∈ N}
and a sum coming from that in M.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

ExamplesN≤1 = {0, 1} and N≤2 = {0, 1, 2} have multiplication tables
0 1
1 ×
and
0 1 2
1 2 ×
2 × ×

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

A product of partial abelian monoids
Y : a topological spacemul(Y ) =

⨿

SPn Y

n≥0

⨿
— the free abelian monoid generated by Y+ = Y {0} with anappropriate topology, or equivalently, as SP∞ Y+ , an infinite
symmetric product introduced by Dold and Thom.
—we think of an element of mul(Y ) as a finite multiset — a
finite “set” with repeated elements.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Summability in a pam
For a finite set S,
σ:S→Y
is a multiset. For a subset T ⊂ S,
σ|T : T ,→ S → Y
is a submultiset.
When Y = M is a partial abelian monoid, we may speak of asummable multiset.
We say that σ is pairwise insummable if, for any subset
T ⊂ S of cardinality two, σ|T is insummable.
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

A product of partial abelian monoids
M, N : partial abelian monoids, S : a finite set. Consider thefollowing property for σ : S → M × N :for any subset T , if one of pi ◦ (σ|T ) is pairwise insummablethen the other is summable.
M
p1

T

S

σ

M ×N
p2

N.
We denote by TM,N the subspace of mul(M × N) consisting of
σ with this property.
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

A product of partial abelian monoids
Let ∼ be the least equivalence relation on TM,N which satisfies
the following three conditions:(R1) If m1 or n1 is zero then(m1 , n1 ) ∔ · · · ∔ (mr , nr ) ∼ (m2 , n2 ) ∔ · · · ∔ (mr , mr ),(R2) If m1 = m1′ + m1′′ then(m1 , n1 ) ∔ · · · ∔ (mr , nr )∼ (m1′ , n1 ) ∔ (m1′′ , n1 ) ∔ (m2 , n2 ) ∔ · · · ∔ (mr , nr ),(R3) If n1 = n1′ + n1′′ then(m1 , n1 ) ∔ · · · ∔ (mr , nr )∼ (m1 , n1′ ) ∔ (m1 , n1′′ ) ∔ (m2 , n2 ) ∔ · · · ∔ (mr , nr ).
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

A product of partial abelian monoids

Two elements [α], [β] in M ⊗ N are summable if we can choose
their representatives α, β in TM,N so that their sum α ∔ β taken
in mul(M × N) is contained in TM,N . Thus, M ⊗ N is a partial
abelian monoid in a natural way.
We have a functor⊗ : PAM × PAM → PAM ; (M, N) 7→ M ⊗ N.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Examples1

2

For abelian groups A, B, their product A ⊗ B defined here is
the usual tensor product of modules.
For two based spaces X , X ′ , viewed as trivial partial
abelian monoids, their product X ⊗ X ′ coincides with theirsmash product X ∧ X ′ .

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

ExamplesIntermediate cases:
3

4

X ⊗ N = SP∞ X , the infinite symmetric product on a based
space X of Dold and Thom.
Then X ⊗ M is the configuration space of finite points in X
with labels in M such that only summable labels occur
simultaneously.
m1

m2
mn
X

(m1 , . . . , mn ) ∈ Mn

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Examples5Viewing S 1 as a based space, we get S 1 ⊗ M = BM theclassifying space of a partial abelian monoid. In particular,
if M is a monoid this coincides with the McCord model ofthe classifying space of M.
m1
mn∗ ∈ BM
m2(m1 , . . . , mn ) ∈ Mn
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

ExamplesN≤1 ⊗ M ∼= M for any M. (Indeed, N≤1 = S 0 ).
N≤2 ⊗ N≤2 ∼= N.
If X = {0, 1, . . . , n} is a based set, thenBX ∼= S 1 × · · · × S 1 (n times).

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Examples6

Let X be a compact based space and
M = Gr := ⊔Grn (R∞ ) be the infinite Grassmannian with apartial sum defined only for two vector spaces which areperpendicular to each other. Then X ⊗ Gr = F (X )coincides with the configuration space defined by Segal forconnective K -homology. Tamaki gave a similar
construction, which is enriched by an operad to maketwisting on K -theory, thus larger than X ⊗ Gr .

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Examples8

9

Fin(Y ) : ( finite subsets of a space Y ),— Fin(Y ) is a partial abelian monoid by disjoint union. If
Cn = Fin(Rn ) then Cn ⊗ X = Cn (X ) is the configuration
space of finite points in Rn with labels in X , introduced by
Segal and equivalent to the construction by Milgram andMay.
Fin(R∞ ) ⊗ M = C M (R∞ ) is the configuration space of finite
points in R∞ with labels in M defined by Shimakawa.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Intervals
H = {(u, v ) | u ≤ v } ⊂ R2 , a half-plane in R2 ,P = {±1} : the set of “parities”,
To any point (u, v ; p, q) ∈ H × P 2 with u < v , we assign aninterval
J = {x ∈ R | u <p x <q v } ⊂ R,
where the symbol <p is interpreted as an inequality ≤ or <according as p = +1 or −1.

I = {(u, v ; p, q) ∈ H × P 2 | u < v }.

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Intervals
For J1 , J2 ∈ I,
we denote J1 < J2 if v1 < u2 , where Jk = (uk , vk ; pk , qk ).
Let Lr be the subspace of I r given by{}
J1 < · · · < Jr .
Lr = (J1 , . . . , Jr ) ∈ I r
Then Lr is the configuration space of r bounded intervals in R
with mutually disjoint closures.
Now we define⨿I=
Lrand give it a topology such that cutting-pasting andcreation-annihilation is allowed.
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Intervals

Figure: Cutting-Pasting–two intervals are pasted when meeting endpoints have
opposite parities, that is, one is open and the other is closed,

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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Intervals

Figure: Creation-Annihilation— a half-open interval annihilates when its length approaches
zero.
Then I has a partial abelian monoid structure by thesuperimposition of disjoint configurations.
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Introduction
Configuration spaces
Main construction and theorem

Partial abelian monoid
Configuration space of intervals

Example : I2 = C1 ⊗ I

A point of C1 ⊗ I is
— a finite subset of R1 ,— with labels in I,— in which, points cancollide,— in case labels are
summableC1 ⊗ I is a configuration space of horizontal intervals in R2 .
Let’s denote this space by I2 .
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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MPartially summable labels

Partially summable labels
enrich a configuration space in a certain way.
control a topology of the reproduced configuration space.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MPartially summable labels

Figure: Sum of labels (where red + blue = violet )30 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Elementary configuration
U = (a, b) : an open interval in R.
We consider two special types of elements in mul(I × M).
(E1) e = (J, n) with one of the following :1234

J = (a, b)
J = (a, w) or J = (a, w], a < w < b
J = (w, b) or J = [w, b), a < w < b
J = (w1 , w2 ] or J = [w1 , w2 ), a < w1 < w2 < b(E2) e = (J1 , n) ∔ (J2 , n) with one of the following:
12

J1 = (a, w1 ], J2 = (w2 , b) and a ≤ w1 < w2 < b, or
J1 = (a, w1 ), J2 = [w2 , b) and a < w1 < w2 ≤ b,

where n is a non-zero element in M for both cases.
We call such e an elementary configuration in U. In bothcases, n ∈ M is denoted by n(e).
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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Elementary configurations

E1

E2
a

a

b

b

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MAdmissible multisets
For any ξ = (J1 , m1 ) ∔ · · · ∔ (Jr , mr ) ∈ TI,M ⊂ mul(I × M),
Let ξ|U = (J1 ∩ U, m1 ) ∔ · · · ∔ (Jr ∩ U, mr )).
ξ ∈ TI,M is said to be admissible if for any t ∈ R there exists anopen interval U = (a, b) which contains t such that
ξ|U = e1 +̇ . . . +̇er
for some elementary configurations e1 , . . . , er in U such that(n(e1 ), . . . , n(er )) ∈ Mr .
If, moreover, there exist ε > 0 and an interval U can be taken
as U = (t − ε, t + ε) for all t, then we say that ξ is ε-admissible.
It is clear that ε-admissible elements are ε′ -admissible if ε′ < ε.
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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MAdmissible multisets

Let V = (a, b) be and open interval with b − a > ε.
We say that an ε-admissible element ξ is supported by V ifξ|(a+ε/2,b−ε/2) = ξ. If V ⊂ V ′ then ε-admissible elements
supprted by V are supported by V ′ .
Let W , W (ε), and W (ε, V ) be the subspace of TI,M which
consists of admissible elements, ε-admissible elements, andε-admissble elements supported by V , respectively.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MConfiguration space of intervals with partiallysummable labels
Let IM be the image in I ⊗ M of W under the natural mapπ⊗ ◦ πmul . Let also IM (ε) and IM (ε, V ) be the image in I ⊗ M of
W (ε) and W (ε, V ), respectively , under π⊗ ◦ πmul . Then we alter
the topology of IM by the weak topology of the union∪IM =IM (ε, V ).
ε>0,VThus, we have defined a configuration space of intervals
with partially summable labels.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MThickening — Moore type variant
We defineeIM =∪IM (ε, s) × {s} × {ε}ε>0,s≥εand give it the topology as a subspace of IM × R2 .
If s = ε, IM (ε, ε) consists of one point, the element ∅ in IM which
represents the empty configuration. As a base point of eIM , wetake (∅, 1, 1).
Proposition 1
The projection eIM → IM onto the first component is a weakhomotopy equivalence.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MApproximation map eIM → Ω′ BM

Approximation map eIM → Ω′ BM
is defined in 3 steps : disintegration, scanning, andsumming-up.
is shown to be weak equivalence so to constitute a zig-zag
of weak equivalences :IM ← eIM → Ω′ BM.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MDisintegration and scanning1

By definition, for any element ([ξ], s, ε) ∈ eIM and t ∈ R,
ξ|Ut = e1 +̇ . . . +̇er

2

for some elementary configurations e1 , . . . , er in Ut suchthat (n(e1 ), . . . , n(er )) ∈ Mr , where Ut = (t − ε, t + ε).
For any elementary configuration e in Ut , we have awell-definedmap )ω(e) : Vt → S 1 , where(εVt = t − 2 , t + 2ε .

So we have an element in mul(Map(Vt , S 1 ) × M) for eacht ∈ (0, s).
This defines a map
ωt : Ws,ε → mul(Map(Vt , S 1 ) × M).
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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
Mω ′ (J)

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MSumming up
3

The composite of ωt with the sequence of natural maps
mul(Map(Vt , S 1 ) × M) → mul(Map(Vt , S 1 × M))→ Map(Vt , mul(S 1 × M)),maps into Map(Vt , TS 1 ,M ).
Recalling that TS 1 ,M is a subset of mul(S 1 × M) on which
we defined the tensor relations, we have an element in
Map(Vt , S 1 ⊗ M).
This construction is compatible for distinct t’s so that wecan paste local functions to get a global function in
Map((0, s), S 1 ⊗ M).

Thus we get a map α : eIM → Ω′ BM.
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Introduction
Configuration spaces
Main construction and theorem

ωt

Ws,ε

G

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

mul(Map(Vt , S 1 ) × M)µ

G′

mul(Map(Vt , S 1 ) × M)

mul(Map(Vt , S 1 × M))

αt

Map(Vt , TS 1 ,M )Map(Vt , mul(S 1 × M))

Map(Vt , BM).

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Introduction
Configuration spaces
Main construction and theorem

eM
Total space E

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

A space EM is almost IM but
intervals lie in a half line [0, ∞), andThe origin works as a “vanishing point”.
Then
eM =
E

∪

EM (ε, s) × {s} × {ε}ε>0,s≥εis its thickening.
Proposition 2
eM is weakly contractible.
E(proof) Push everything into the origin.
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Introduction
Configuration spaces
Main construction and theorem

eIM ,→ E
eM → BM

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

In IM (ε, s), intervals lie in (0, ∞), so we haveeM .
IM (ε, s) ⊂ EM (ε, s), thus we have an inclusion eIM ,→ E
eM → BM is defined using scanning at the origin.
E2t
1 − 2ttt

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Scanning at the origin
A configuration consisting of two intervals with “red” and “blue”
as a respective label, which maps under p : ẼM → BM to aconfiguration consisting of a point with “violet” as its label.
2u = 1 − 2tt u“red” + “blue” = “violet” so that (“red”, “blue”) ∈ P(“violet”) .
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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MMain theorem
Proposition 3
Let M be a partial abelian monoid whose elements are
eM → BM is aself-insummable. Then the map p : E
quasi-fibration with fiber eIM .
Assuming this, we can state and prove the main theorem :Theorem (O.-Shimakawa)
Let M be a partial abelian monoid whose elements are self
insummable. Then the configuration space IM of intervals in R
with labels in M is weakly homotopy equivalent to ΩBM.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
MProof of the main theorem
In the following commutative diagram, lower horizontal line is
the Serre’s path-loop fibration. The vertical map in the middle isa weak homotopy equivalence, since it is a map between
weakly-contractible spaces, hence so is the vertical map on the
left.
eIM

eM
E

BM

Ω′ BM

P ′ BMBM

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Examples1

Let X be a based set {0, 1, 2} ( trivial partial abelianmonoid ).
0 1 2
1 × ×
2 × ×
Then BX ∼= S 1 ∨ S 1 and we know from the theorem that
IX ≃w Ω(S 1 ∨ S 1 ).
An element of IX can be depicted as follows.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesScanning map eIX → Ω(S 1 ∨ S 1 ) can be graphed as follows.

π0 (IX ) ∼= π1 (S 1 ∨ S 1 ) ∼=Z∗Z48 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Examples2

Let M = N≤1 × N≤1 be the direct product (as a partial
abelian monoid !) of N≤1 with itself.
00 10 01 11
10 × 11 ×
01 11 × ×
11 × × ×Then BM ∼= S 1 × S 1 and we know from the theorem thatIM ≃w Ω(S 1 × S 1 ).
An element of IM looks as follows.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesScanning map eIM → Ω(S 1 × S 1 ) can be graphed as follows.

π0 (IM ) ∼= π1 (S 1 × S 1 ) ∼=Z⊕Z

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Examples3

Let M = C1 is the configuration space of finite subsets of
R1 . Then we know from Milgram-May-Segal’s theorem thatBC1 ∼= C1 ⊗ S 1 ≃w ΩSS 1 = ΩS 2 . Now, our theorem
asserts thatIM ≃w Ω2 S 2 .
So any element of π3 S 2 can be written as a based loop in
IM . A generator of π3 S 2 ∼= Z is given by a Hopf map
η : S3 → S2.

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesCorresponding loop in IM is given by :52 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesCrossing change gives an inverse :53 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesCorresponding loop in IM is given by :54 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesCrossing change gives an inverse :55 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesIndeed, we can paste two surfaces to remove them:(Step 1)56 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesIndeed, we can paste two surfaces to remove them:(Step 2)57 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesIndeed, we can paste two surfaces to remove them:(Step 3)58 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesIndeed, we can paste two surfaces to remove them:(Step 4)59 / 61

Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

ExamplesMoreover, if we use the standard embedding R1 ,→ R2 to get a
map IC1 → IC2 , and this amounts to an embedding ofconfiguration of intervals in R2 into configuration of intervals in
R3 under the standard embedding R2 ,→ R3 . This corresponds
to the suspension map Z ∼= π3 (S 2 ) → π4 (S 3 ) ∼= Z/2Z. So theabove pictures also show the vanishing of 2η in π4 (S 3 ).

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Introduction
Configuration spaces
Main construction and theorem

Configuration space of intervals with partially summable labels
Approximation map eIM → Ω′ BM
e → BM
Quasi-fibration sequence eIM → E
M

Two lemmas for quasi-fibration
For a proof of proposition 3, we may use the Dold-Thom
criterion.
Lemma 1
For any open set V ⊂ Fj BM − Fj−1 BM, there exists a homotopy
equivalence p−1 V ≃ V × eIM , so that V is distinguished.
Lemma 2
There exists an open set O ⊂ Fj BM which contains Fj−1 BM
and homotopies ht : O → O and Ht : p−1 O → p−1 O such that1h0 = idO , ht (Fj−1 BM) ⊂ Fj−1 BM and h1 (O) ⊂ Fj−1 BM,

2

H0 = idp−1 O and p ◦ Ht = ht ◦ p for all t, and3

H1 : p−1 z → p−1 h1 (z) is a weak homotopy equivalence forall z ∈ O.
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