query: Homotopy Groups of $S^l \vee S^k$
positive_passages.docid: 20692
positive_passages.text: Homotopy groups of a wedge sum
negative_passages.docid: 44069
negative_passages.text: Proof that a continuous map to $S^n$ whose image is a proper subset of $S^n$ is null-homotopic
negative_passages.docid: 142698
negative_passages.text: Proving that the fundamental groups of two spaces with same homotopy type are isomorphic
negative_passages.docid: 42412
negative_passages.text: Singular homology $S^{n}$ with rational coefficients
negative_passages.docid: 133413
negative_passages.text: Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)
negative_passages.docid: 37635
negative_passages.text: Diffeomorphism group of the unit circle
negative_passages.docid: 18141
negative_passages.text: Embedding $S_n$ into $A_{n+2}$
negative_passages.docid: 120100
negative_passages.text: Hairy Ball Theorem: homotopy from the identity to the antipodal map
negative_passages.docid: 77175
negative_passages.text: Proof that two spaces that are homotopic have the same de Rham cohomology
negative_passages.docid: 143314
negative_passages.text: Sylow subgroup of $S_{p^2}$
negative_passages.docid: 126930
negative_passages.text: Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
negative_passages.docid: 74383
negative_passages.text: How many group structures make $S^1$ a topological group?
negative_passages.docid: 18687
negative_passages.text: First and second homotopy groups of a connected Lie group
negative_passages.docid: 94018
negative_passages.text: Homology groups of $\mathbb{R}^3\setminus \ S^1$ and $\mathbb{R}^3\setminus E^1_+$
negative_passages.docid: 27670
negative_passages.text: Group homomorphisms $\mathbb{Q}\to\mathbb{R}$
negative_passages.docid: 95686
negative_passages.text: $Lie(G \times H)\cong Lie(G)\oplus Lie(H)$
negative_passages.docid: 83109
negative_passages.text: Formation of the free group on $S$ is a functor