The group of all continuous maps from the circle to the space

The group of all homotopy classes of continuous maps from the circle to the space

The group of all continuous maps from the space to the circle

The group of all homotopy classes of continuous maps from the space to the circle

A group that captures the information about the space's holes

A group that captures the information about the space's homology classes

A group that captures the information about the space's singular homology

A group that captures the information about the space's cohomology

A sequence of abelian groups and homomorphisms

A sequence of vector spaces and linear transformations

A sequence of modules and module homomorphisms

A sequence of rings and ring homomorphisms

The group of all singular homology classes of the space

The group of all homology classes of the space

The group of all singular cohomology classes of the space

The group of all cohomology classes of the space

The group of all singular homology classes of the space

The group of all homology classes of the space

The group of all singular cohomology classes of the space

The group of all cohomology classes of the space

A formula that relates the homology of a product space to the homology of its factors

A formula that relates the cohomology of a product space to the cohomology of its factors

A formula that relates the homology of a space to its cohomology

A formula that relates the cohomology of a space to its homology

A theorem that relates the homology of a manifold to its cohomology

A theorem that relates the cohomology of a manifold to its homology

A theorem that relates the homology of a space to its singular homology

A theorem that relates the cohomology of a space to its singular cohomology

A topological space that is locally Euclidean

A topological space that is locally compact

A topological space that is locally connected

A topological space that is simply connected

A collection of simplices that are glued together along their faces

A collection of vertices, edges, and faces that are glued together

A collection of cells that are glued together along their boundaries

A collection of simplices that are glued together along their boundaries

The homology group of a singular chain complex

The homology group of a simplicial chain complex

The homology group of a CW-complex

The homology group of a manifold

A space that is built up from cells

A space that is built up from simplices

A space that is built up from open sets

A space that is built up from closed sets

A sequence that relates the homology of a space to the homology of its open subsets

A sequence that relates the cohomology of a space to the cohomology of its open subsets

A sequence that relates the homology of a space to its singular homology

A sequence that relates the cohomology of a space to its singular cohomology

$$H_0(S^n) = \mathbb{Z}, H_n(S^n) = \mathbb{Z}, H_i(S^n) = 0 \text{ for } 0 < i < n$$

$$H_0(S^n) = \mathbb{Z}, H_n(S^n) = 0, H_i(S^n) = \mathbb{Z} \text{ for } 0 < i < n$$

$$H_0(S^n) = 0, H_n(S^n) = \mathbb{Z}, H_i(S^n) = \mathbb{Z} \text{ for } 0 < i < n$$

$$H_0(S^n) = 0, H_n(S^n) = 0, H_i(S^n) = \mathbb{Z} \text{ for } 0 < i < n$$

$$H_0(T^2) = \mathbb{Z}, H_1(T^2) = \mathbb{Z}^2, H_2(T^2) = \mathbb{Z}, H_i(T^2) = 0 \text{ for } i > 2$$

$$H_0(T^2) = \mathbb{Z}, H_1(T^2) = \mathbb{Z}, H_2(T^2) = \mathbb{Z}^2, H_i(T^2) = 0 \text{ for } i > 2$$

$$H_0(T^2) = \mathbb{Z}^2, H_1(T^2) = \mathbb{Z}, H_2(T^2) = \mathbb{Z}, H_i(T^2) = 0 \text{ for } i > 2$$

$$H_0(T^2) = \mathbb{Z}, H_1(T^2) = \mathbb{Z}^2, H_2(T^2) = 0, H_i(T^2) = 0 \text{ for } i > 2$$

$$H_0(K) = \mathbb{Z}, H_1(K) = \mathbb{Z}^2, H_2(K) = \mathbb{Z}, H_i(K) = 0 \text{ for } i > 2$$

$$H_0(K) = \mathbb{Z}, H_1(K) = \mathbb{Z}, H_2(K) = \mathbb{Z}^2, H_i(K) = 0 \text{ for } i > 2$$

$$H_0(K) = \mathbb{Z}^2, H_1(K) = \mathbb{Z}, H_2(K) = \mathbb{Z}, H_i(K) = 0 \text{ for } i > 2$$

$$H_0(K) = \mathbb{Z}, H_1(K) = \mathbb{Z}^2, H_2(K) = 0, H_i(K) = 0 \text{ for } i > 2$$