Some calculations involving configuration spaces of distinct points
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We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy. | We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy. | ||
− | (b) Since $\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$ to an equivariant $(2n-1)$-subcomplex of $T$. A homotopy between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case | + | (b) Since $\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$ to an equivariant $(2n-1)$-subcomplex of $T$. A homotopy between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on $\widetilde{N}$. |
QED | QED | ||
</wikitex> | </wikitex> |
Revision as of 13:25, 12 April 2020
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. See [Vassiliev1992].
In introducing notation and definitions we follow [Skopenkov2020a].
The deleted product
![\displaystyle \widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.](/images/math/a/8/9/a89a255d83d0596d761097ca80a883da.png)
This is the configuration space of ordered pairs of distinct points of .
Suppose that is an embedding of a subset
.
Then the map
is well-defined by the Gauss formula
![\displaystyle \widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.](/images/math/5/0/5/50535c3b779e89b044011117dc151d48.png)
We have , i.e. this map is equivariant with respect to the `exchanging factors' involution
on
and the antipodal involution on
.
Thus the existence of an equivariant map
is a necessary condition for the embeddability of
in
.
Denote by the set embeddings of
into
up to isotopy.
Let
be the set of equivariant maps
up to equivariant homotopy. Denote by
the equivalence class of the map
.
The Haefliger-Wu invariant
is defined by formula
.
Theorem 1.1.
The Haefliger-Wu invariant is one-to-one either
(a) is a compact connected
-complex and
or
(b) is a compact
-manifold with nonempty boundary,
is
-connected,
,
,
and
.
See [Skopenkov2006, 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1
] for the DIFF case and [Skopenkov2002, Theorem 1.3
] for the PL case.
2 Uniqueness theorems
Lemma 2.1.
Assume that is a compact
-manifold and either
(a) or
(b) is connected and
.
Then each two equivariant maps from to
are equivariantly homotopic.
Hereafter denote by the product
minus tubular neighborhood of the diagonal.
Proof.
Given two equivariant maps take an arbitrary equivariant triangulation
of
.
(a) One can easily construct an equivariant homotopy between restrictions of and
on vertices of
.
By general position a homotopy between
and
on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed
.
We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
(b) Since has non-empty boundary, there exists an equivariant deformational retraction of
to an equivariant
-subcomplex of
. A homotopy between
and
on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on
.
QED
3 References
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
- [Skopenkov2020a] https://www.mccme.ru/circles/oim/eliminat_talk.pdf
- [Vassiliev1992]
V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications., Amer. Math. Soc., Providence, RI, (1992).