# Tagged Questions

Low-dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds and knot theory.

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### Non-uniquness of spheres in the prime decomposition up to isotopy

Peter Scott, in his survey "The Geometries of 3-Manifolds", states that the family of spheres defining the decomposition of a given 3-manifold $M$ into primes is not unique up to isotopy even when $M$ ...
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### Does any technical definition of embedding accept a “non-injective” function as opposed to only “injective”?

Embedding is defined to be a one-to-one structure preserving mapping. My question is if the one-to-one condition is really critical. Like if linear mappings from high-dimensional space to low-...
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### planarity of trivalent graphs with a cyclic ordering on the edges in each vertex

Let $G$ be an (undirected) trivalent graph. For each vertex $v$ of $G$ we choose a cyclic ordering on the edges coming into $v$ (so if vertex $A$ has neighbors $B, C$ and $D$ we decide whether the ...
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### Homology 4-balls with boundary $S^3$

Are there interesting homology 4-balls with boundary $S^3$? Going the other way, must any homology 4-ball with boundary $S^3$ be homotopy equivalent/homeomorphic/diffeomorphic to $B^4$?
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### Let $f$ be a map from the real projective plane to the torus. Show that $f$ must be homotopic to a constant map.

Let $f$ be a map from the real projective plane to the torus. Show that $f$ must be homotopic to a constant map. This is a qual problem. Any help would be appreciated. Thank you.
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### Classification of Bieberbach groups

Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of Four-...
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### Isotopy and homeomorphism

Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$. The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$? I am especially ...
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