Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$ -manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$ . The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$.

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.

Two locally generic maps $f, g \colon M^n \to \mathbb{R}^{2n - 1}$ are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if $n \neq 3$ and $M^n$ is a closed $n$-manifold then the regular homotopy class of every locally generic map $f \colon M^n \to \mathbb{R}^{2n - 1}$ is completely determined by the number of its singular points provided that $f$ is singular (that is, $f$ is not an immersion).

A proof is given that for each non-negative integer $g$, there is an infinite family of knotted surfaces of genus $g$, none of which is ambient isotopic to itself with the orientation reversed.

This paper gives non-embeddings and non-immersions for the real flag manifolds RF(1, 1, n–2), n > 3 and shows that Lam's immersions for n = 4 and 5 and Stong's result for n = 6 are the best possible.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.