We want to begin this section by demonstrating its heuristic idea. We consider as before an exceptional setΣwhich we assume to be subset of a manifold X. Suppose that we have a map Γwhich assigns to each pathγ:I→X having ends inX\Σan element ofZ2. We require that Γ(γ) = 0ifγdoes not meetΣand thatΓis homotopy invariant under homotopies having fixed ends. Now assume that we can find a pathγ inX such that Γ(γ) = 1. By our assumptions we know thatγmeetsΣand that every path homotopic toγwith fixed ends has to meetΣas well.
SinceX is a manifold it cannot be singular at some point and hence we have space in order to deformγ throughX and so to infer thatΣneeds to be large.
Before introducing the rigorous setup for our main theorems of this section, we prove a result about the dimension of subsetsΣof manifolds whose first part is well known, although we could not find any proof in the literature. Here we give a proof which uses Poincaré-Lefschetz duality and, moreover, by this method we get some insight in the topology ofΣfor free.
3.3.1 Lemma. LetX be a connected manifold of dimensionn≥2 andΣ⊂X compact.
i) IfX\Σis not path connected, thendim Σ≥n−1.
ii) Assume that X is moreover orientable and 2 ≤k≤n−1 is a fixed natural number such thatHj(X) = 0 for all1≤j≤k−1. If now
• π1(X\Σ)is abelian and
• πk−1(X\Σ)6= 0, thendim Σ≥n−k.
In both cases Σis not contractible.
Proof. We prove the assertions successively and hence consider at first i):
SinceX\Σis not path connected, we obtain that the reduced homology group H˜0(X\Σ;Z2) is non trivial. Since X is path connected by assumption, the long exact sequence of homology gives
. . .→H1(X, X\Σ;Z2)→H˜0(X\Σ;Z2)→H˜0(X;Z2) = 0
and hence yields a surjective map H1(X, X\Σ;Z2) → H˜0(X \ Σ;Z2) implying the non triviality ofH1(X, X\Σ;Z2). SinceΣis compact, we obtain by Poincaré-Lefschetz duality (cf.
[Do80, Prop. 7.2]) an isomorphism
H1(X, X\Σ;Z2)−∼=→Hˇn−1(Σ;Z2),
showing that Hˇn−1(Σ;Z2) 6= 0. Hence Σ is not contractible and, moreover, the estimate dim Σ≥n−1 follows from lemma 3.1.4.
We now turn to the prove of ii). If π0(X\Σ)6= 0, we obtain from the already proved first assertion thatdim Σ≥n−1> n−k.
Ifπ0(X\Σ) = 0we use the assumption thatπk−1(X\Σ)is non trivial and accordingly we can find a minimal natural number2≤l ≤k such thatπl−1(X\Σ)6= 0but πj(X\Σ) = 0 for all 0 ≤j ≤l−2. Hence in this case X\Σis l−2 connected and we obtain from the Hurewicz theorem, thatHl−1(X\Σ)∼=πl−1(X\Σ)6= 0, where we use thatπ1(X \Σ)is assumed to be abelian ifl= 2. SinceHl−1(X) = 0by assumption, the long exact sequence of homology yields a surjection
Hl(X, X\Σ)→Hl−1(X\Σ)→Hl−1(X) = 0,
showing that Hl(X, X\Σ) 6= 0. Because X is orientable and Σ is compact, we can use Poincaré-Lefschetz duality once again and obtain that
Hˇn−l(Σ)∼=Hl(X, X\Σ)6= 0
is non trivial. HenceΣis not contractible and, by using lemma 3.1.4, we obtain thatdim Σ≥ n−l≥n−k.
In applications we can of course not assume to have any information aboutπ1(X\Σ). Hence we want to reformulate 3.3.1 ii) as the following alternative.
3.3.2 Corollary. LetX be a connected orientable manifold of dimensionnandΣ⊂X compact.
Moreover, let2≤k≤n−1 be a fixed natural number such thatHj(X) = 0 for all1≤j≤k−1 andπk−1(X\Σ)6= 0. Thenπ1(X\Σ)is not abelian ordim Σ≥n−k.
The point in this alternative is that a non abelian group is in particular not trivial which can also be interpreted as a result about the size ofΣif, for example,X is simply connected or even contractible. In the latter case it seems that by geometric intuition one needsΣ “of dimension n−2” in order to makeπ1(X\Σ)non trivial which is in accordance with lemma 3.3.1.
3.3.3 Remark. The second part of lemma 3.3.1 also holds under the weaker assumption that π1(X\Σ)is either trivial or not a perfect group. However, in view of the alternative which we formulated in the corollary above, this does not improve our result much.
Let nowY be a subspace of X such thatY ∩Σ =∅and define fork∈N
Ωk(X, Y) ={f : (Ik, ∂Ik)→(X, Y) :f continuous}.
Throughout, we assume thatY 6=∅such that in particular the caseX = Σis excluded.
If f1, f2 ∈ Ωk(X, Y)such that f1(t1, . . . , tk−1,1) = f2(t1, . . . , tk−1,0), we define their product f1∗f2∈Ωk(X, Y)by
(f1∗f2)(t1, . . . , tk) =
f1(t1, . . . , tk−1,2tk), 0≤tk ≤12 f2(t1, . . . , tk−1,2tk−1), 12 ≤tk≤1 andfˇ1∈Ωk(X, Y)by
fˇ1(t1, . . . , tk) =f1(t1, . . . , tk−1,1−tk).
In the following we assume that for anyk∈Nwe have a map
Γk: Ωk(X, Y)→Z2, such that
i) Γk(f) = 0∈Z2 iff(Ik)∩Σ =∅,
ii) Γk(f1∗f2) = Γk(f1) + Γk(f2)wheneverf1∗f2 exists, iii) Γk(f) = Γk(g)iff andg are homotopic inΩk(X, Y).
3.3.4 Lemma. Let πk(X)be trivial and f1, f2∈Ωk(X, Y)be such thatf1|∂Ik=f2|∂Ik. Then
Γk(f1∗fˇ2) = 0.
Hence we obtain in particular
Γk(f1∗fˇ1) = 0.
Proof. At first we want to show thatf1∗fˇ2|∂Ik:∂Ik →Y is homotopic to a constant map by a homotopy inY. Note thatf1∗fˇ2is given by
(f1∗fˇ2)(t1, . . . , tk) =
f1(t1, . . . , tk−1,2tk), 0≤tk≤ 12 f2(t1, . . . , tk−1,2−2tk), 12 ≤tk≤1.
Then, by using the homotopy H:I×∂Ik →Y
H(λ, t1, . . . , tk) =
f1(t1, . . . , tk−1,2tk(1−λ)), 0≤tk≤ 12 f2(t1, . . . , tk−1,(2−2tk)(1−λ)), 12 ≤tk ≤1, we can deformf1∗fˇ2|∂Ik into
(t1, . . . , tk)7→f1(t1, . . . , tk−1,0) =f2(t1, . . . , tk−1,0), (t1, . . . , tk)∈∂Ik. Furthermore we can deform the latter map to a constant one by
H˜ :I×∂Ik →Y, (λ, t1, . . . , tk)7→f1((1−λ)·t1, . . . ,(1−λ)·tk−1,0).
Since(Ik, ∂Ik)has the homotopy extension property, we infer thatf1∗fˇ2is homotopic inside Ωk(X, Y)to a mapg∈Ωk(X, Y)such thatg(∂Ik) =y0for somey0∈Y. Because of the assumed triviality ofπk(X)we finally obtain thatg is homotopic (insideΩk(X, Y)) to the constant map g0 ∈Ωk(X, Y),g0(Ik) =y0. Hence
Γk(f1∗fˇ2) = Γk(g) = Γk(g0) = 0 becausey0∈Y andY ∩Σ =∅.
The following result is the main observation in order to use lemma 3.3.1 to obtain estimates on the dimension ofΣby homotopy.
3.3.5 Lemma. Letf ∈Ωk(X, Y)be such thatΓk(f) = 1∈Z2 andπk(X) = 0. Then πk−1(X\ Σ)6= 0.
Proof. Assume that there existsg∈Ωk(X, Y)such thatg|∂Ik=f |∂Ik andg(Ik)∩Σ =∅. Then by the properties (i),(ii) ofΓk and lemma 3.3.4 we infer
06= Γk(f) = Γk(f) + Γk(ˇg) = Γk(f∗ˇg) = 0;
a contradiction. Hence a mapgwith the required properties cannot exist. Sincef(∂Ik)⊂Y and Y ∩Σ =∅, we conclude that there exists no continuous extension of f |∂Ik: ∂Ik →X\Σ to Ik. Consequently, f |∂Ik is not homotopic to a constant map (cf. [StZi94, 2.3.3]) and so in particular defines a non trivial element inπk−1(X\Σ).
3.3.6 Proposition. LetX be a simply connected manifold of dimensionn≥2. IfΣis compact and there existsf ∈Ω1(X, Y)such that Γ1(f) = 1∈Z2, then
i) dim Σ≥n−1 ii) Σis not contractible.
Proof. By lemma 3.3.5, X \Σ is not path connected and hence the assertion follows from the first part of lemma 3.3.1.
We now turn to the casek≥2 where, according to lemma 3.3.1, we need more assumptions in order to conclude results on the dimension ofΣ.
3.3.7 Proposition. LetX be a connected orientable manifold of dimensionnand2≤k≤n−1 a fixed natural number such thatHj(X) = 0for all 1≤j≤k−1and πk(X) = 0. Assume that the exceptional setΣ⊂X is compact and thatπ1(X\Σ)is abelian. Moreover, assume that there existsf ∈Ωk(X, Y)such thatΓk(f) = 1. Then
i) dim Σ≥n−k.
ii) Σis not contractible.
Proof. By lemma 3.3.5, we haveπk−1(X\Σ)6= 0and hence the assertion follows from the second part of 3.3.1.
3.3.8 Remark. Note that theorem 3.2.1 states that the canonical inclusion ι: Σ ,→ X is not homotopic to a constant map, whereas the propositions 3.3.6 and 3.3.7 assert that the identity map Σ → Σ is not homotopic to a constant map. Hence the conclusion of theorem 3.2.1 is stronger. On the other hand, exceptional sets Σwhich are contractible in X cannot be detected by theorem 3.2.1.
3.3.9 Remark. We want to point out thatX is not assumed to be compact in the propositions 3.3.6 and 3.3.7. On the other hand we need several other restrictive assumptions on the topology of X. This suggests to use these propositions as a kind of local version of 3.2.1. Since X is by assumption a manifold, we can use our results in open coordinate balls B of X. Then B is contractible and hence all assumptions on the triviality of the homology and homotopy groups of B hold. Note that estimates on the dimension of B∩Σ yield estimates on the dimension of all ofΣby remark 3.1.2.