Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:M\rightarrow N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> be a map where M, N are connected, topological manifolds, not necessarily orientable, of dimension <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and n, respectively. Assume that f is either an embedding or an immersion having only double points so that the set of such double points is a submanifold of dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> which is a strong deformation retract of a neighborhood of M. In case f is an embedding, we show that the complement <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N-f(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is either connected or has two connected components and therefore, the number of components is never bigger than two. In the case of an immersion, we set sharp lower bounds for the number of connected components of the complement (one or two, depending on the characteristics of the spaces and of the map f involved) and we show that in many cases the number of connected components can be realized for any integer greater than the corresponding lower bound . We point out that the fact that we are not assuming orientability of the manifolds M and N made possible to lower the least number of connected components of the complement <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N-f(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Jordan theorems for embeddings and immersions in codimension one

期刊论文
发表日期：

2025年10月25日

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Lucília Daruiz Borsari,
Fernanda Soares Pinto Cardona,
Daciberg Lima Gonçalves

摘要

Let $f:M\rightarrow N$ $f : M \to N$ be a map where M, N are connected, topological manifolds, not necessarily orientable, of dimension $n-1$ $n - 1$ and n, respectively. Assume that f is either an embedding or an immersion having only double points so that the set of such double points is a submanifold of dimension $n-2$ $n - 2$ which is a strong deformation retract of a neighborhood of M. In case f is an embedding, we show that the complement $N-f(M)$ $N - f (M)$ is either connected or has two connected components and therefore, the number of components is never bigger than two. In the case of an immersion, we set sharp lower bounds for the number of connected components of the complement (one or two, depending on the characteristics of the spaces and of the map f involved) and we show that in many cases the number of connected components can be realized for any integer greater than the corresponding lower bound . We point out that the fact that we are not assuming orientability of the manifolds M and N made possible to lower the least number of connected components of the complement $N-f(M)$ $N - f (M)$ .