Rational homotopy type of free and pointed mapping spaces between spheres
Denote by $map(X, Y )$ (respec. $map ^{\ast}(X, Y )$) the space of free (respec. pointed) maps from $X$ to $Y$ . Whenever $X$ is a finite $CW$-complex and $Y$ is a nilpotent $CW$-complex of finite type over $Q$, then any path component of both map $(X, Y )$ and $map ^{\ast} (X, Y ) $ are nilpotent $...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552817760141312 |
|---|---|
| author | Buijs, U. Murillo, A. Buijs, U. Murillo, A. |
| author_facet | Buijs, U. Murillo, A. Buijs, U. Murillo, A. |
| author_institution_txt_mv | [
{
"author": "U. Buijs",
"institution": "Institut de Mathématique Pure et Appliquée chemin du cyclotron"
},
{
"author": "A. Murillo",
"institution": "Departamento de Álgebra, Geometrı́a y Topologı́a Universidad de Málaga"
}
] |
| author_sort | Buijs, U. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | Denote by $map(X, Y )$ (respec. $map ^{\ast}(X, Y )$) the space of free (respec. pointed) maps from $X$ to $Y$ . Whenever $X$ is a finite $CW$-complex and $Y$ is a nilpotent $CW$-complex of finite type over $Q$, then any path component of both map $(X, Y )$ and $map ^{\ast} (X, Y ) $ are nilpotent $CW$-complexes of finite type over $Q$ and in particular, it can be rationalized in the classical sense. From the Sullivan approach to rational homotopy theory , and based in the fundamental work of Haefliger, there is a standard procedure to obtain Sullivan models of the path components $map _{f}(X, Y )$ and $map ^{\ast}_{f} (X, Y ) $ of $map(X, Y ) $ and $map ^{\ast}(X, Y ) $ respectively, containing the map $f : X → Y $. In this note, we show the advantage of this procedure and use it repeatedly to explicitly describe the rational homotopy type of free and pointed mapping spaces between sphere. |
| first_indexed | 2026-08-04T01:05:28Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 130–139
Urtzi Buijs1 and Aniceto Murillo2
Rational homotopy type of free and
pointed mapping spaces between
spheres
Rational homotopy type of free and pointed
mapping spaces between spheres
Urtzi Buijs∗ and Aniceto Murillo†
March 6, 2013
1 Introduction
Denote by map(X, Y ) (respec. map∗(X, Y )) the space of free (respec. pointed)
maps from X to Y . Whenever X is a finite CW-complex and Y is a nilpo-
tent CW-complex of finite type over Q, then [8] any path component of both
map(X, Y ) and map∗(X, Y ) are nilpotent CW-complexes of finite type over
Q and in particular, it can be rationalized in the classical sense. From the
Sullivan approach to rational homotopy theory [9], and based in the funda-
mental work of Haefliger [7], there is a standard procedure [2, 3] to obtain
Sullivan models of the path components mapf (X, Y ) and map∗f (X, Y ) of
map(X, Y ) and map∗(X, Y ) respectively, containing the map f : X → Y . In
this note, we show the advantage of this procedure and use it repeatedly to
explicitly describe the rational homotopy type of free and pointed mapping
spaces between spheres:
∗Partially supported by the following grants: Ministerio de Educación y Ciencia
MTM2010-15831; Junta de Andalućıa FQM-213; 2009-SGR-119; and U-mobility 246550
of the European Union Seventh Framework Program.
†Partially supported by the University of Torun and the following grants: Ministerio
de Educación y Ciencia MTM2010-18089; FEDER European funds; Junta de Andalućıa
FQM-213 and P07-FQM-2863.
2000 Mathematics Subject Classification: 55P62, 54C35.
Key words and phrases: rational homotopy theory. mapping spaces.
1
1Partially supported by the following grants: Ministerio de Educación y Cien-
cia MTM2010-15831; Junta de Andalucia FQM-213; 2009-SGR-119; and U-mobility
246550 of the European Union Seventh Framework Program.
2Partially supported by the University of Torun and the following grants: Min-
isterio de Educación y Ciencia MTM2010-18089; FEDER European funds; Junta de
Andalucia.
c© U. Buijs, A. Murillo, 2013
Urtzi Buijs and Aniceto Murillo 131
Theorem 1.1. (i) For m odd and any n ≥ 1,
map(Sn, Sm) 'Q
Sm ×K(Z,m− n), if m > n.⋃
N S
m, if m = n.
Sm, if m < n.
map∗(Sn, Sm) 'Q
K(Z,m− n), if m > n.⋃
N ∗, if m = n.
∗, if m < n.
(ii) For m even and any n ≥ 1,
map(Sn, Sm) 'Q
Y, if m > n.
Sm ×K(Z, 2m− n− 1)
⋃
N S
2m−1, if m = n.
Sm ×K(Z, 2m− n− 1), if m < n < 2m− 1.⋃
N S
m, if m = 2n− 1.
Sm, if m = 2n− 1.
map∗(Sn, Sm) 'Q
K(Z,m− n)×K(Z, 2m− n− 1), if m > n.⋃
NK(Z, 2m− n− 1), if m = n.
K(Z, 2m− n− 1), if m < n < 2m− 1.⋃
N ∗, if m = 2n− 1.
∗, if m < n.
Here, 'Q means “rationally homotopy equivalent”;
⋃
denotes the disjoint
union; and Y is a rational space which sits in a fibration of the form
SmQ ×K(Q,m− n)→ Y → K(Q, 2m− n− 1).
We should mention that the above result might be known, or easily deduced
by specialists. However, to our knowledge, it has not been made explicit in
the literature. Thus, this paper reviews in a particular a useful situation, the
general procedure of obtaining the rational homotopy type of both free and
pointed mappping spaces.
Acknowledgement. The second author expresses his gratitude to Prof.
Marek Golasinski from University of Torun, from its support during the
Topology Workshop 2012, where this paper was partially written.
2
132 Rational homotopy type ...
2 Models of mapping spaces between spheres
In this section we prove the theorem above. We highly depend on known
facts and techniques arising from rational homotopy theory. All of them can
be found in the excellent reference [6] which is now standard on the subject.
Here, we simply present a summary of some basic facts.
For any simply connected, or more generally, nilpotent CW-complex of
finite type X, its rationalization XQ is a rational space (i.e., its homotopy
groups are rational vector spaces), together with a map X → XQ inducing
isomorphisms in rational homotopy.
On the other hand, to any space X there corresponds, in a contravariant
way, its minimal Sullivan model which is a particular Sullivan algebra (ΛV, d),
unique up to isomorphism, which algebraically models the rational homotopy
type of the space X, or equivalently, the homotopy type of its rationalization
XQ. By ΛV we mean the free commutative algebra generated by the graded
vector space V , i.e., ΛV = TV/I where TV denotes the tensor algebra over
V and I is the ideal generated by v ⊗ w − (−1)|w||v|w ⊗ v, ∀v, w ∈ V . The
differential d satisfies a certain minimality condition which, in the simply
connectid case it translates to: for any element of v ∈ V , dv is a polynomial
in ΛV with no linear term.
This correspondence yields an equivalence between the homotopy cate-
gories of 1-connected rational spaces of finite type and that of 1-connected
rational commutative differential graded algebras of finite type. Indeed, this
equivalence is the restriction to the appropriate subcategories of the classical
adjoint functors [1]
SimplSets
APL→←
〈 〉
CDGA
between the homotopy categories of commutative differential graded algebras
and simplicial sets.
One can precise, through these functors, the notion of models of non con-
nected spaces. As in [3], a model of a general space X, not necessarily con-
nected, is a Z-graded free CDGA (ΛW,d) such that its simplicial realization
〈(ΛW,d)〉 has the same homotopy tye of the Milnor simplicial approximation
of XQ, S∗(XQ).
We now introduce the Haefliger model [7] of the free and pointed map-
ping spaces map(X, Y ), map∗(X, Y ), via the functorial description of Brown-
Szczarba [2].
3
Urtzi Buijs and Aniceto Murillo 133
Let B be a finite dimensional CDGA (commutative differential graded
algebra) model of the finite CW-complex X and let A = (ΛV, d) be a Sullivan
model of the nilpotent CW-complex of finite type Y .
Denote by B] = Hom(B,Q) the differential graded coalgebra, dual of B
and therefore negatively graded, and consider the Z-graded CDGA Λ(A⊗B])
with the natural differential induced by the one on A and by the dual δ of
the differential on B. Now, consider the differential ideal I ⊂ Λ(A ⊗ B])
generated by 1− 1⊗ 1] and by the elements of the form
v1v2 ⊗ β −
∑
j
(−1)|v2||β
′
j |(v1 ⊗ β′j)(v2 ⊗ β′′j ),
with v1, v2 ∈ V , β ∈ B and ∆β =
∑
j β
′
j ⊗ β′′j . Then, the composition
ρ : Λ(V ⊗B]) ↪→ Λ(A⊗B]) � Λ(A⊗B])/I
is an isomorphism of graded algebras [2, Thm.1.2]. Thus, we may consider
on Λ(V ⊗B]) the differential d̃ for which the above becomes an isomorphisms
of CDGA’s. To explicitly determine d̃ on the generator v⊗ β ∈ V ⊗B], first
compute dv ⊗ β + (−1)|v|v ⊗ δβ and then use the relations which generate
the ideal I to express dv ⊗ β as an element of Λ(V ⊗B]).
Then, it turns out [2, Thm.1.3] that
(
Λ(V⊗B]), d̃
)
is a model of map(X, Y ).
Moreover, if B]
+ denotes the subspace of B] of strictly negative elements,(
Λ(V ⊗B]
+), d̃
)
is a model of map∗(X, Y ).
For the model of the components of map(X, Y ) and/or map∗(X, Y ) we
follow the approach and notation of [3, 4]:
For any free CDGA (ΛW,d), in which W is Z-graded, and any algebra
morphism u : ΛW −→ Q consider the differential ideal Ku generated by
A1 ∪ A2 ∪ A3, being
A1 = W<0, A2 = dW 0, A3 = {α− u(α) : α ∈ W 0}.
(ΛW,d)/Ku is again a free CDGA of the form (Λ(W
1 ⊕W≥2), du) in which
W
1
is a complement in W 1 of d(W 0) modulo identifications via A1 and A3,
see [3, §4] for details. Note that, W
1
depends also on u. Moreover, if (ΛW,d)
is a model of a non-connected space X and u corresponds to a 0-simplex of
X, as remarked in [2, 4.3], (Λ(W
1⊕W≥2), du) is a Sullivan model of the path
component of X containing the fixed 0-simplex.
4
134 Rational homotopy type ...
Next, consider
(
Λ(V ⊗B]), d̃
)
the model of map(X, Y ) which we have just
recalled and let ϕ : (ΛV, d)→ B be a model of a given map f : X → Y . The
morphism ϕ clearly induces a natural augmentation which shall be denoted
also by ϕ :
(
Λ(V ⊗B]), d̃
)
→ Q. Applying the process above to this particular
case yields the Sullivan algebra
(
Λ
(
V ⊗B]
1 ⊗ (V ⊗B])≥2
)
, d̃ϕ
)
which constitutes a Sullivan model of mapf (X, Y ). In the same way,
(
Λ
(
V ⊗B]
+
1
⊗ (V ⊗B]
+)≥2
)
, d̃ϕ
)
is a Sullivan model of map∗f (X, Y ).
To prove our Theorem we will apply all of the above to the particular
case of choosing X = Sm and Y = Sn to be spheres, m,n ≥ 1. For it, recall
that, if m is an odd integer, the minimal model of Sm is the exterior algebra
on a generator of degree m with zero differential (Λxm, 0). On the other
hand, if m is even, the minimal model of Sm is (Λxm, y2m−1, d), dxm = 0,
dy2m−1 = x2m. From now on, subscripts will always denote degree.
On the other hand, for any n, a coalgebra model of Sn is B = 〈1, αn〉, in
which αn is a primitive cycle of degree −n, i.e., ∆αn = αn ⊗ 1 + 1⊗ αn.
We will now distinguish different cases:
Case 1: m odd.
A model of map(Sn, Sm) is therefore,
(Λ(xm ⊗ 1, xm ⊗ αn), 0).
To avoid excessive notation we set xm ⊗ 1 = am and xm ⊗ αn = bm−n and
rewrite the above as:
(Λ(am, bm−n), 0).
On the other hand, taking into account that the evaluation fibration
map∗(Sn, Sm)→ map(Sn, Sm)→ Sm
is modelled by
(Λam, 0)→ (Λ(am, bm−n), 0)→ (Λbm−n, 0),
a model for map∗(Sn, Sm) is simply (Λbm−n, 0).
5
Urtzi Buijs and Aniceto Murillo 135
We now obtain Sullivan models for the path components and identify the
homotopy type of their realizations.
Case 1.1: free maps.
m > n:
In this case (Λ(am, bm−n), 0) is already a Sullivan model as bm−n has
positive degree. Hence the only component of map(Sn, Sn) has the rational
homotopy type of the product Sm ×K(Z,m− n) of Sm with the Eilenberg-
MacLane space of type (Z,m− n).
m = n:
In this case bm−n has degree 0 and there are a countable number of non
homotopic morphisms ϕλ : (Λ(am, bm−n), 0)→ Q, one for each λ ∈ Q, sending
bm−n to 1. Then, the procedure above give rise to a countable number of
components, just like in the integral case, each of which with Sullivan model
(Λam, 0) whose realization is just SmQ .
Observe, as in [5, Ex. 3], that in this case, since map(Sm, Sn) has in-
finitely many components, its rational homology in degree zero is infinite
dimensional. Thus, its rational cohomology, also in degree zero, has un-
countable dimension. This sharply contrasts with the rational cohomology
of its model (Λ(xm ⊗ 1, xm ⊗ αn), 0), which in degree zero has countable di-
mension. This illustrates why, in the non-connected case, a model of a space
does not preserve, in general, rational homotopy invariants.
m < n:
In this case bm−n has negative degree and therefore, it vanishes when
considering models of components. Therefore, there is only one component
with Sullivan model (Λam, 0) whose realization is again SmQ .
Case 1.2: pointed maps.
m > n:
As in this case bm−n is of positive degree there is only one component
with Sullivan model (Λbm−n, 0) whose realization is K(Q,m− n).
m = n:
As in the free case, there are a countable number of non homotopic mor-
phisms ϕλ : (Λbm−n, 0) → Q, one for each λ ∈ Q, sending bm−n to λ. Thus,
6
136 Rational homotopy type ...
when replacing bm−n by λ we obtain Q as a model for the corresponding
component and therefore, each component is rationally trivial.
m < n:
In this case bm−n has negative degree so there is only one component
which is rationally trivial.
Case 2: m even.
In this case, a model of map(Sn, Sm) is again computed via the methods
above:
(Λ(xm ⊗ 1, y2m−1 ⊗ 1, xm ⊗ αn, y2m−1 ⊗ αn), d)
To avoid excessive notation, as before, we set xm ⊗ 1 = am, y2m−1 ⊗ 1 =
c2m−1, xm ⊗ αn = bm−n, y2m−1 ⊗ αn = z2m−n−1 and rewrite this model as:
(Λ(am, c2m−1, bm−n, z2m−n−1), d),
in which the differential is given by
dam = dbm−n = 0, dc2m−1 = a2m, dz2m−n−1 = 2ambm−n.
Concerning pointed maps and taking into account that the evaluation
fibration
map∗(Sn, Sm)→ map(Sn, Sm)→ Sm
is modelled by
(Λ(am, c2m−1), d)→ (Λ(am, c2m−1, bm−n, z2m−n−1), d),→ (Λ(bm−n, z2m−n−1), 0),
a model for map∗(Sn, Sm) is simply (Λ(bm−n, z2m−n−1), 0).
Then, on components:
Case 2.1: free maps.
m > n:
In this case both bm−n, z2m−n−1 have positive degrees so the above is al-
ready a Sullivan model. Hence, there is only one component whose realization
is a space Y which fits in a fibration of the form
SmQ ×K(Q,m− n)→ Y → K(Q, 2m− n− 1).
7
Urtzi Buijs and Aniceto Murillo 137
m = n:
Now z2m−n−1 has positive degree but bm−n has degree zero and there are
a countable number of non homotopic morphisms
ϕλ : (Λ(am, c2m−1, bm−n, z2m−n−1), d)→ Q,
one for each λ ∈ Q, sending bm−n to λ. This gives rise to a countable
number of components. If λ 6= 0 then the corresponding component has
Sullivan minimal model (Λc2m−1, 0) whose realization is S2m−1. On the other
hand, if λ = 0, the corresponding component has Sullivan minimal model
(Λ(am, c2m−1, z2m−n−1), d), with dz2m−n−1 = 0 whose realization is of the
rational homotopy type of Sm ×K(Z, 2m− n− 1).
m < n < 2m− 1:
Now bm−n has negative degree but z2m−n−1 has positive degree. Thus,
there is only one component with model (Λ(am, c2m−1, z2m−n−1), d), with
dz2m−n−1 = 0 whose realization is again SmQ ×K(Q, 2m− n− 1).
n = 2m− 1:
Here, bm−n has negative and z2m−n−1 has degree zero. Hence, we have
a countable number of components arising from the CDGA morphisms
ϕλ : (Λ(am, c2m−1, bm−n, z2m−n−1), d) → Q, one for each λ ∈ Q, sending
z2m−n−1 to λ. Each of them produces via the procedure above the same
Sullivan model (Λ(am, c2m−1), d) whose realization is SmQ .
n > 2m− 1:
In this case both bm−n, z2m−n−1 have negative degrees. Hence, there is
only one component with model (Λ(am, c2m−1), d) whose realization is SmQ .
Case 2.2: pointed maps.
m > n:
In this case, both bm−n, z2m−n−1 have positive degrees and the model
(Λ(bm−n, z2m−n−1), 0) is already minimal. Thus, there is one component ra-
tionally equivalent to K(Z,m− n)×K(Z, 2m− n− 1).
m = n:
Now, z2m−n−1 has positive degree but bm−n has degree zero. Thus, as in
precedent cases, it can be replaced by any rational number giving rise to a
8
138 Rational homotopy type ...
countable number of components each of which with model (Λz2m−n−1, 0),
whose realization is K(Q, 2m− n− 1).
m < n < 2m− 1:
Here, z2m−n−1 has positive degree but bm−n is of negative degree. Hence,
there is only one component with model (Λz2m−n−1, 0) whose realization is
again K(Q, 2m− n− 1).
n = 2m− 1:
In this case bm−n has negative degree and z2m−n−1 is of degree zero. Hence,
in the procedure of obtaining components, bm−n vanishes while z2m−n−1 is re-
placed by any rational number giving rise to a countable number of rationally
trivial components.
n > 2m− 1:
Finally, both bm−n, z2m−n−1 have negative degrees and there is only one
component which is rationally trivial.
Summarizing all of the above finishes the proof of our theorem.
References
[1] A. K. Bousfield and V. K. A. M. Gugenheim, On PL De Rahm theory
and rational homotopy type, Mem. Amer. Math. Soc., 179, 1976.
[2] E. H. Brown and R. H. Szczarba, On the rational homotopy type of
function spaces, Trans. Amer. Math. Soc., 349, 1997, 4931–4951.
[3] U. Buijs and A. Murillo, Basic constructions in rational homotopy theory
of function spaces, Annales de l’institut Fourier, 56(3), 2007, 815-838.
[4] U. Buijs and A. Murillo, The rational homotopy Lie algebra of function
spaces, Comment. Math. Helv., 83(4), 2008, 723-739.
[5] U. Buijs, Y. Félix and A. Murillo, Lie models for the components of
sections of a nilpotent fibrations, Trans. Amer. Math. Soc., 361(10),
2009, 5601–5614.
[6] Y. Félix, S. Halperin and J.C. Thomas, Rational Homotopy Theory,
G.T.M. 205, Springer, 2000.
9
Urtzi Buijs and Aniceto Murillo 139
[7] A. Haefliger, Rational homotopy of the space of sections of a nilpotent
bundle, Trans. Amer. Math. Soc., 273, 1982, 609–620.
[8] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and
spaces, North Holland Mathematics Studies 15, North Holland, 1975.
[9] D. Sullivan, Infinitesimal computations in Topology, Publ. Math. de
l’I.H.E.S., 47, 1978, 269–331.
Institut de Mathématique Pure et Appliquée
chemin du cyclotron, 2
Université Catholique de Louvain
B-1348 Louvain-la-Neuve
Belgique
e-mail: urtzibuijs@gmail.com
Departamento de Álgebra, Geometŕıa y Topoloǵıa
Universidad de Málaga
Ap. 59, 29080 Málaga
Spain
e-mail: aniceto@uma.es
10
|
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| last_indexed | 2026-08-04T01:05:28Z |
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| spelling | oai:trim.imath.kiev.ua:article-3122018-02-10T20:56:26Z Rational homotopy type of free and pointed mapping spaces between spheres Раціональний гомотопічний тип просторів відображень між сферами з виділеними точками та без них Buijs, U. Murillo, A. Buijs, U. Murillo, A. Denote by $map(X, Y )$ (respec. $map ^{\ast}(X, Y )$) the space of free (respec. pointed) maps from $X$ to $Y$ . Whenever $X$ is a finite $CW$-complex and $Y$ is a nilpotent $CW$-complex of finite type over $Q$, then any path component of both map $(X, Y )$ and $map ^{\ast} (X, Y ) $ are nilpotent $CW$-complexes of finite type over $Q$ and in particular, it can be rationalized in the classical sense. From the Sullivan approach to rational homotopy theory , and based in the fundamental work of Haefliger, there is a standard procedure to obtain Sullivan models of the path components $map _{f}(X, Y )$ and $map ^{\ast}_{f} (X, Y ) $ of $map(X, Y ) $ and $map ^{\ast}(X, Y ) $ respectively, containing the map $f : X → Y $. In this note, we show the advantage of this procedure and use it repeatedly to explicitly describe the rational homotopy type of free and pointed mapping spaces between sphere. Позначимо через $map(X, Y )$ ( $map ^{\ast}(X, Y )$) простори відображень з $X$ до $Y$ без виділених точок (з виділеними точками) .&nbsp; Кожного разу, коли $X$ - скінченний $CW$-комплекс і&nbsp; $Y$ - нільпотентний $CW$-комплекс&nbsp; скінченного типу над $Q$, будь-яка компонента лінійної зв'язності обох відображень&nbsp;$map (X, Y )$ і $ map ^{\ast} (X, Y ) $&nbsp; є нільпотентним $CW$-комплексом скінченного типу над&nbsp; $Q$ і зокрема, це можна раціоналізувати в класичному сенсі. Згідно підходу Саллівана до раціональної теорії гомотопій, заснованої в фундаментальній роботі Хефлера, існує стандартна процедура отримання моделей Саллівана компоненти лінійної зв'язності $map_{f}(X, Y )$ і $map^{\ast}_{f} (X, Y ) $ просторів $map(X, Y ) $ і $map ^{\ast}(X, Y ) $ відповідно,&nbsp; що містить відображення $f : X → Y $. В роботі ми показуємо перевагу цієї процедури та використовуємо її повторно, щоб явним чином описати раціональний гомотопічний тип просторів відображень між сферою з відміченими точками та без них. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/312 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 130-139 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 130-139 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 130-139 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/312/291 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Buijs, U. Murillo, A. Buijs, U. Murillo, A. Rational homotopy type of free and pointed mapping spaces between spheres |
| title | Rational homotopy type of free and pointed mapping spaces between spheres |
| title_alt | Раціональний гомотопічний тип просторів відображень між сферами з виділеними точками та без них |
| title_full | Rational homotopy type of free and pointed mapping spaces between spheres |
| title_fullStr | Rational homotopy type of free and pointed mapping spaces between spheres |
| title_full_unstemmed | Rational homotopy type of free and pointed mapping spaces between spheres |
| title_short | Rational homotopy type of free and pointed mapping spaces between spheres |
| title_sort | rational homotopy type of free and pointed mapping spaces between spheres |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/312 |
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